Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - modules - algebras.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.1 lcov report (development 28695-49bb1ac00f) Lines: 3471 3497 99.3 %
Date: 2023-09-25 07:47:46 Functions: 304 305 99.7 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2000  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation; either version 2 of the License, or (at your option) any later
       8             : version. It is distributed in the hope that it will be useful, but WITHOUT
       9             : ANY WARRANTY WHATSOEVER.
      10             : 
      11             : Check the License for details. You should have received a copy of it, along
      12             : with the package; see the file 'COPYING'. If not, write to the Free Software
      13             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      14             : #include "pari.h"
      15             : #include "paripriv.h"
      16             : 
      17             : #define DEBUGLEVEL DEBUGLEVEL_alg
      18             : 
      19             : #define dbg_printf(lvl) if (DEBUGLEVEL >= (lvl) + 3) err_printf
      20             : 
      21             : /********************************************************************/
      22             : /**                                                                **/
      23             : /**           ASSOCIATIVE ALGEBRAS, CENTRAL SIMPLE ALGEBRAS        **/
      24             : /**                 contributed by Aurel Page (2014)               **/
      25             : /**                                                                **/
      26             : /********************************************************************/
      27             : static GEN alg_subalg(GEN al, GEN basis);
      28             : static GEN alg_maximal_primes(GEN al, GEN P);
      29             : static GEN algnatmultable(GEN al, long D);
      30             : static GEN _tablemul_ej(GEN mt, GEN x, long j);
      31             : static GEN _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p);
      32             : static GEN _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p);
      33             : static ulong algtracei(GEN mt, ulong p, ulong expo, ulong modu);
      34             : static GEN alg_pmaximal(GEN al, GEN p);
      35             : static GEN alg_maximal(GEN al);
      36             : static GEN algtracematrix(GEN al);
      37             : static GEN algtableinit_i(GEN mt0, GEN p);
      38             : static GEN algbasisrightmultable(GEN al, GEN x);
      39             : static GEN algabstrace(GEN al, GEN x);
      40             : static GEN algbasismul(GEN al, GEN x, GEN y);
      41             : static GEN algbasismultable(GEN al, GEN x);
      42             : static GEN algbasismultable_Flm(GEN mt, GEN x, ulong m);
      43             : 
      44             : static void H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d);
      45             : static GEN H_add(GEN x, GEN y);
      46             : static GEN H_charpoly(GEN x, long v, long abs);
      47             : static GEN H_divl_i(GEN x, GEN y);
      48             : static GEN H_inv(GEN x);
      49             : static GEN H_mul(GEN x, GEN y);
      50             : static GEN H_neg(GEN x);
      51             : static GEN H_norm(GEN x, long abs);
      52             : static GEN H_random(GEN b);
      53             : static GEN H_sqr(GEN x);
      54             : static GEN H_tomatrix(GEN x, long abs);
      55             : static GEN H_trace(GEN x, long abs);
      56             : static GEN mk_C();
      57             : 
      58             : static int
      59      826996 : checkalg_i(GEN al)
      60             : {
      61             :   GEN mt, rnf;
      62             :   long t;
      63      826996 :   if (typ(al) != t_VEC || lg(al) != 12) return 0;
      64      826779 :   mt = alg_get_multable(al);
      65      826779 :   if (typ(mt) != t_VEC || lg(mt) == 1 || typ(gel(mt,1)) != t_MAT) return 0;
      66      826758 :   rnf = alg_get_splittingfield(al);
      67      826758 :   if (isintzero(rnf) || !gequal0(alg_get_char(al)))
      68      362952 :     return 1;
      69      463806 :   if (typ(gel(al,2)) != t_VEC || lg(gel(al,2)) == 1) return 0;
      70             :   /* not checkrnf_i: beware placeholder from alg_csa_table */
      71      463799 :   t = typ(rnf);
      72      463799 :   return t==t_COMPLEX || t==t_REAL || (t==t_VEC && lg(rnf)==13);
      73             : }
      74             : void
      75      956349 : checkalg(GEN al)
      76             : {
      77      956349 :   if (al && !checkalg_i(al))
      78         112 :     pari_err_TYPE("checkalg [please apply alginit()]",al);
      79      956237 : }
      80             : 
      81             : static int
      82      180992 : checklat_i(GEN al, GEN lat)
      83             : {
      84             :   long N,i,j;
      85             :   GEN m,t,c;
      86      180992 :   if (typ(lat)!=t_VEC || lg(lat) != 3) return 0;
      87      180992 :   t = gel(lat,2);
      88      180992 :   if (typ(t) != t_INT && typ(t) != t_FRAC) return 0;
      89      180992 :   if (gsigne(t)<=0) return 0;
      90      180992 :   m = gel(lat,1);
      91      180992 :   if (typ(m) != t_MAT) return 0;
      92      180992 :   N = alg_get_absdim(al);
      93      180992 :   if (lg(m)-1 != N || lg(gel(m,1))-1 != N) return 0;
      94     1628886 :   for (i=1; i<=N; i++)
      95    13031067 :     for (j=1; j<=N; j++) {
      96    11583173 :       c = gcoeff(m,i,j);
      97    11583173 :       if (typ(c) != t_INT) return 0;
      98    11583173 :       if (j<i && signe(gcoeff(m,i,j))) return 0;
      99    11583173 :       if (i==j && !signe(gcoeff(m,i,j))) return 0;
     100             :     }
     101      180985 :   return 1;
     102             : }
     103      180992 : void checklat(GEN al, GEN lat)
     104      180992 : { if (!checklat_i(al,lat)) pari_err_TYPE("checklat [please apply alglathnf()]", lat); }
     105             : 
     106             : /**  ACCESSORS  **/
     107             : long
     108     5630505 : alg_type(GEN al)
     109             : {
     110             :   long t;
     111     5630505 :   if (!al) return al_REAL;
     112     5500354 :   t = typ(alg_get_splittingfield(al));
     113     5500354 :   if (t==t_REAL || t==t_COMPLEX) return al_REAL;
     114     5496938 :   if (isintzero(alg_get_splittingfield(al)) || !gequal0(alg_get_char(al))) return al_TABLE;
     115     3924464 :   switch(typ(gmael(al,2,1))) {
     116      932974 :     case t_MAT: return al_CSA;
     117     2991455 :     case t_INT:
     118             :     case t_FRAC:
     119             :     case t_POL:
     120     2991455 :     case t_POLMOD: return al_CYCLIC;
     121          35 :     default: return al_NULL;
     122             :   }
     123             :   return -1; /*LCOV_EXCL_LINE*/
     124             : }
     125             : long
     126         224 : algtype(GEN al)
     127         224 : { return checkalg_i(al)? alg_type(al): al_NULL; }
     128             : 
     129             : /* absdim == dim for al_TABLE. */
     130             : static long
     131         238 : algreal_dim(GEN al)
     132             : {
     133         238 :   switch(lg(alg_get_multable(al))) {
     134         154 :     case 2: case 3: return 1;
     135          77 :     case 5: return 4;
     136           7 :     default: pari_err_TYPE("algreal_dim", al);
     137             :   }
     138             :   return -1; /*LCOV_EXCL_LINE*/
     139             : }
     140             : long
     141      225155 : alg_get_dim(GEN al)
     142             : {
     143             :   long d;
     144      225155 :   if (!al) return 4;
     145      225155 :   switch(alg_type(al)) {
     146       10787 :     case al_TABLE: return lg(alg_get_multable(al))-1;
     147      214256 :     case al_CSA: return lg(alg_get_relmultable(al))-1;
     148          77 :     case al_CYCLIC: d = alg_get_degree(al); return d*d;
     149          28 :     case al_REAL: return algreal_dim(al);
     150           7 :     default: pari_err_TYPE("alg_get_dim", al);
     151             :   }
     152             :   return -1; /*LCOV_EXCL_LINE*/
     153             : }
     154             : 
     155             : long
     156     1592841 : alg_get_absdim(GEN al)
     157             : {
     158     1592841 :   if (!al) return 4;
     159     1546123 :   switch(alg_type(al)) {
     160      669422 :     case al_TABLE: case al_REAL: return lg(alg_get_multable(al))-1;
     161      113323 :     case al_CSA: return alg_get_dim(al)*nf_get_degree(alg_get_center(al));
     162      763371 :     case al_CYCLIC:
     163      763371 :       return rnf_get_absdegree(alg_get_splittingfield(al))*alg_get_degree(al);
     164           7 :     default: pari_err_TYPE("alg_get_absdim", al);
     165             :   }
     166             :   return -1;/*LCOV_EXCL_LINE*/
     167             : }
     168             : 
     169             : long
     170        1778 : algdim(GEN al, long abs)
     171             : {
     172        1778 :   checkalg(al);
     173        1757 :   if (abs) return alg_get_absdim(al);
     174        1526 :   return alg_get_dim(al);
     175             : }
     176             : 
     177             : /* only cyclic */
     178             : GEN
     179       13818 : alg_get_auts(GEN al)
     180             : {
     181       13818 :   long ta = alg_type(al);
     182       13818 :   if (ta != al_CYCLIC && ta != al_REAL)
     183           0 :     pari_err_TYPE("alg_get_auts [noncyclic algebra]", al);
     184       13818 :   return gel(al,2);
     185             : }
     186             : GEN
     187         112 : alg_get_aut(GEN al)
     188             : {
     189         112 :   long ta = alg_type(al);
     190         112 :   if (ta != al_CYCLIC && ta != al_REAL)
     191           7 :     pari_err_TYPE("alg_get_aut [noncyclic algebra]", al);
     192         105 :   return gel(alg_get_auts(al),1);
     193             : }
     194             : GEN
     195          42 : algaut(GEN al) { checkalg(al); return alg_get_aut(al); }
     196             : GEN
     197       13839 : alg_get_b(GEN al)
     198             : {
     199       13839 :   long ta = alg_type(al);
     200       13839 :   if (ta != al_CYCLIC && ta != al_REAL)
     201           7 :     pari_err_TYPE("alg_get_b [noncyclic algebra]", al);
     202       13832 :   return gel(al,3);
     203             : }
     204             : GEN
     205          56 : algb(GEN al) { checkalg(al); return alg_get_b(al); }
     206             : 
     207             : /* only CSA */
     208             : GEN
     209      216356 : alg_get_relmultable(GEN al)
     210             : {
     211      216356 :   if (alg_type(al) != al_CSA)
     212          14 :     pari_err_TYPE("alg_get_relmultable [algebra not given via mult. table]", al);
     213      216342 :   return gel(al,2);
     214             : }
     215             : GEN
     216          49 : algrelmultable(GEN al) { checkalg(al); return alg_get_relmultable(al); }
     217             : GEN
     218          56 : alg_get_splittingdata(GEN al)
     219             : {
     220          56 :   if (alg_type(al) != al_CSA)
     221          14 :     pari_err_TYPE("alg_get_splittingdata [algebra not given via mult. table]",al);
     222          42 :   return gel(al,3);
     223             : }
     224             : GEN
     225          56 : algsplittingdata(GEN al) { checkalg(al); return alg_get_splittingdata(al); }
     226             : GEN
     227        4102 : alg_get_splittingbasis(GEN al)
     228             : {
     229        4102 :   if (alg_type(al) != al_CSA)
     230           0 :     pari_err_TYPE("alg_get_splittingbasis [algebra not given via mult. table]",al);
     231        4102 :   return gmael(al,3,2);
     232             : }
     233             : GEN
     234        4102 : alg_get_splittingbasisinv(GEN al)
     235             : {
     236        4102 :   if (alg_type(al) != al_CSA)
     237           0 :     pari_err_TYPE("alg_get_splittingbasisinv [algebra not given via mult. table]",al);
     238        4102 :   return gmael(al,3,3);
     239             : }
     240             : 
     241             : /* only cyclic and CSA */
     242             : GEN
     243    14252565 : alg_get_splittingfield(GEN al) { return gel(al,1); }
     244             : GEN
     245         119 : algsplittingfield(GEN al)
     246             : {
     247             :   long ta;
     248         119 :   checkalg(al);
     249         119 :   ta = alg_type(al);
     250         119 :   if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
     251           7 :     pari_err_TYPE("alg_get_splittingfield [use alginit]",al);
     252         112 :   return alg_get_splittingfield(al);
     253             : }
     254             : long
     255     1207605 : alg_get_degree(GEN al)
     256             : {
     257             :   long ta;
     258     1207605 :   ta = alg_type(al);
     259     1207605 :   if (ta == al_REAL) return algreal_dim(al)==1? 1 : 2;
     260     1207521 :   if (ta != al_CYCLIC && ta != al_CSA)
     261          21 :     pari_err_TYPE("alg_get_degree [use alginit]",al);
     262     1207500 :   return rnf_get_degree(alg_get_splittingfield(al));
     263             : }
     264             : long
     265         322 : algdegree(GEN al)
     266             : {
     267         322 :   checkalg(al);
     268         315 :   return alg_get_degree(al);
     269             : }
     270             : 
     271             : GEN
     272      298765 : alg_get_center(GEN al)
     273             : {
     274             :   long ta;
     275      298765 :   ta = alg_type(al);
     276      298765 :   if (ta == al_REAL)
     277             :   {
     278          21 :     if (algreal_dim(al) != 4) return alg_get_splittingfield(al);
     279           7 :     return stor(1,3);
     280             :   }
     281      298744 :   if (ta != al_CSA && ta != al_CYCLIC)
     282           7 :     pari_err_TYPE("alg_get_center [use alginit]",al);
     283      298737 :   return rnf_get_nf(alg_get_splittingfield(al));
     284             : }
     285             : GEN
     286          70 : alg_get_splitpol(GEN al)
     287             : {
     288          70 :   long ta = alg_type(al);
     289          70 :   if (ta != al_CYCLIC && ta != al_CSA)
     290           0 :     pari_err_TYPE("alg_get_splitpol [use alginit]",al);
     291          70 :   return rnf_get_pol(alg_get_splittingfield(al));
     292             : }
     293             : GEN
     294       69916 : alg_get_abssplitting(GEN al)
     295             : {
     296       69916 :   long ta = alg_type(al), prec;
     297       69916 :   if (ta != al_CYCLIC && ta != al_CSA)
     298           0 :     pari_err_TYPE("alg_get_abssplitting [use alginit]",al);
     299       69916 :   prec = nf_get_prec(alg_get_center(al));
     300       69916 :   return rnf_build_nfabs(alg_get_splittingfield(al), prec);
     301             : }
     302             : GEN
     303        1169 : alg_get_hasse_i(GEN al)
     304             : {
     305        1169 :   long ta = alg_type(al);
     306        1169 :   if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
     307           7 :     pari_err_TYPE("alg_get_hasse_i [use alginit]",al);
     308        1162 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     309        1155 :   return gel(al,4);
     310             : }
     311             : GEN
     312         231 : alghassei(GEN al) { checkalg(al); return alg_get_hasse_i(al); }
     313             : GEN
     314        1911 : alg_get_hasse_f(GEN al)
     315             : {
     316        1911 :   long ta = alg_type(al);
     317        1911 :   if (ta != al_CYCLIC && ta != al_CSA)
     318           7 :     pari_err_TYPE("alg_get_hasse_f [use alginit]",al);
     319        1904 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     320        1897 :   return gel(al,5);
     321             : }
     322             : GEN
     323         329 : alghassef(GEN al) { checkalg(al); return alg_get_hasse_f(al); }
     324             : 
     325             : /* all types */
     326             : GEN
     327        2716 : alg_get_basis(GEN al) { return gel(al,7); }
     328             : GEN
     329          63 : algbasis(GEN al) { checkalg(al); return alg_get_basis(al); }
     330             : GEN
     331       60683 : alg_get_invbasis(GEN al) { return gel(al,8); }
     332             : GEN
     333          63 : alginvbasis(GEN al) { checkalg(al); return alg_get_invbasis(al); }
     334             : GEN
     335     2261379 : alg_get_multable(GEN al) { return gel(al,9); }
     336             : GEN
     337         245 : algmultable(GEN al) { checkalg(al); return alg_get_multable(al); }
     338             : GEN
     339     5936143 : alg_get_char(GEN al) { if (!al) return gen_0; return gel(al,10); }
     340             : GEN
     341         112 : algchar(GEN al) { checkalg(al); return alg_get_char(al); }
     342             : GEN
     343      236466 : alg_get_tracebasis(GEN al) { return gel(al,11); }
     344             : 
     345             : /* lattices */
     346             : GEN
     347      244314 : alglat_get_primbasis(GEN lat) { return gel(lat,1); }
     348             : GEN
     349      289905 : alglat_get_scalar(GEN lat) { return gel(lat,2); }
     350             : 
     351             : /** ADDITIONAL **/
     352             : 
     353             : /* is N=smooth*prime? */
     354        1018 : static int Z_easyfactor(GEN N, ulong lim)
     355             : {
     356             :   GEN fa;
     357        1018 :   if (lgefint(N) <= 3) return 1;
     358          91 :   fa = absZ_factor_limit(N, lim);
     359          91 :   return BPSW_psp(veclast(gel(fa,1)));
     360             : }
     361             : 
     362             : /* no garbage collection */
     363             : static GEN
     364         840 : backtrackfacto(GEN y0, long n, GEN red, GEN pl, GEN nf, GEN data, int (*test)(GEN,GEN), GEN* fa, GEN N, GEN I)
     365             : {
     366             :   long b, i;
     367         840 :   ulong lim = 1UL << 17;
     368         840 :   long *v = new_chunk(n+1);
     369         840 :   pari_sp av = avma;
     370         840 :   for (b = 0;; b += (2*b)/(3*n) + 1)
     371          42 :   {
     372             :     GEN ny, y1, y2;
     373         882 :     set_avma(av);
     374        2758 :     for (i = 1; i <= n; i++) v[i] = -b;
     375         882 :     v[n]--;
     376             :     for(;;)
     377             :     {
     378        1127 :       i = n;
     379        1425 :       while (i > 0)
     380        1383 :       { if (v[i] == b) v[i--] = -b; else { v[i]++; break; } }
     381        1127 :       if (i==0) break;
     382             : 
     383        1085 :       y1 = y0;
     384        4576 :       for (i = 1; i <= n; i++) y1 = nfadd(nf, y1, ZC_z_mul(gel(red,i), v[i]));
     385        1085 :       if (!nfchecksigns(nf, y1, pl)) continue;
     386             : 
     387        1018 :       ny = absi_shallow(nfnorm(nf, y1));
     388        1018 :       if (!signe(ny)) continue;
     389        1018 :       ny = diviiexact(ny, gcdii(ny, N));
     390        1018 :       if (!Z_easyfactor(ny, lim)) continue;
     391             : 
     392         962 :       y2 = idealdivexact(nf, y1, idealadd(nf,y1,I));
     393         962 :       *fa = idealfactor(nf, y2);
     394         962 :       if (!data || test(data,*fa)) return y1;
     395             :     }
     396             :   }
     397             : }
     398             : 
     399             : /* if data == NULL, the test is skipped */
     400             : /* in the test, the factorization does not contain the known factors */
     401             : static GEN
     402         840 : factoredextchinesetest(GEN nf, GEN x, GEN y, GEN pl, GEN* fa, GEN data, int (*test)(GEN,GEN))
     403             : {
     404         840 :   pari_sp av = avma;
     405             :   long n,i;
     406         840 :   GEN x1, y0, y1, red, N, I, P = gel(x,1), E = gel(x,2);
     407         840 :   n = nf_get_degree(nf);
     408         840 :   x = idealchineseinit(nf, mkvec2(x,pl));
     409         840 :   x1 = gel(x,1);
     410         840 :   red = lg(x1) == 1? matid(n): gmael(x1,1,1);
     411         840 :   y0 = idealchinese(nf, x, y);
     412             : 
     413         840 :   E = shallowcopy(E);
     414         840 :   if (!gequal0(y0))
     415        2372 :     for (i=1; i<lg(E); i++)
     416             :     {
     417        1532 :       long v = nfval(nf,y0,gel(P,i));
     418        1532 :       if (cmpsi(v, gel(E,i)) < 0) gel(E,i) = stoi(v);
     419             :     }
     420             :   /* N and I : known factors */
     421         840 :   I = factorbackprime(nf, P, E);
     422         840 :   N = idealnorm(nf,I);
     423             : 
     424         840 :   y1 = backtrackfacto(y0, n, red, pl, nf, data, test, fa, N, I);
     425             : 
     426             :   /* restore known factors */
     427        2372 :   for (i=1; i<lg(E); i++) gel(E,i) = stoi(nfval(nf,y1,gel(P,i)));
     428         840 :   *fa = famat_reduce(famat_mul_shallow(*fa, mkmat2(P, E)));
     429         840 :   return gc_all(av, 2, &y1, fa);
     430             : }
     431             : 
     432             : static GEN
     433         581 : factoredextchinese(GEN nf, GEN x, GEN y, GEN pl, GEN* fa)
     434         581 : { return factoredextchinesetest(nf,x,y,pl,fa,NULL,NULL); }
     435             : 
     436             : /** OPERATIONS ON ASSOCIATIVE ALGEBRAS algebras.c **/
     437             : 
     438             : /*
     439             : Convention:
     440             : (K/F,sigma,b) = sum_{i=0..n-1} u^i*K
     441             : t*u = u*sigma(t)
     442             : 
     443             : Natural basis:
     444             : 1<=i<=d*n^2
     445             : b_i = u^((i-1)/(dn))*ZKabs.((i-1)%(dn)+1)
     446             : 
     447             : Integral basis:
     448             : Basis of some order.
     449             : 
     450             : al:
     451             : 1- rnf of the cyclic splitting field of degree n over the center nf of degree d
     452             : 2- VEC of aut^i 1<=i<=n if n>1, or i=0 if n=1
     453             : 3- b in nf
     454             : 4- infinite hasse invariants (mod n) : VECSMALL of size r1, values only 0 or n/2 (if integral)
     455             : 5- finite hasse invariants (mod n) : VEC[VEC of primes, VECSMALL of hasse inv mod n]
     456             : 6- nf of the splitting field (absolute)
     457             : 7* dn^2*dn^2 matrix expressing the integral basis in terms of the natural basis
     458             : 8* dn^2*dn^2 matrix expressing the natural basis in terms of the integral basis
     459             : 9* VEC of dn^2 matrices giving the dn^2*dn^2 left multiplication tables of the integral basis
     460             : 10* characteristic of the base field (used only for algebras given by a multiplication table)
     461             : 11* trace of basis elements
     462             : 
     463             : If al is given by a multiplication table (al_TABLE), only the * fields are present.
     464             : */
     465             : 
     466             : /* assumes same center and same variable */
     467             : /* currently only works for coprime degrees */
     468             : GEN
     469          84 : algtensor(GEN al1, GEN al2, long maxord) {
     470          84 :   pari_sp av = avma;
     471             :   long v, k, d1, d2;
     472             :   GEN nf, P1, P2, aut1, aut2, b1, b2, C, rnf, aut, b, x1, x2, al;
     473             : 
     474          84 :   checkalg(al1);
     475          70 :   checkalg(al2);
     476          63 :   if (alg_type(al1) != al_CYCLIC  || alg_type(al2) != al_CYCLIC)
     477          21 :     pari_err_IMPL("tensor of noncyclic algebras"); /* TODO: do it. */
     478             : 
     479          42 :   nf = alg_get_center(al1);
     480          42 :   if (!gequal(alg_get_center(al2),nf))
     481           7 :     pari_err_OP("tensor product [not the same center]", al1, al2);
     482             : 
     483          35 :   P1=alg_get_splitpol(al1); aut1=alg_get_aut(al1); b1=alg_get_b(al1);
     484          35 :   P2=alg_get_splitpol(al2); aut2=alg_get_aut(al2); b2=alg_get_b(al2);
     485          35 :   v=varn(P1);
     486             : 
     487          35 :   d1=alg_get_degree(al1);
     488          35 :   d2=alg_get_degree(al2);
     489          35 :   if (ugcd(d1,d2) != 1)
     490           7 :     pari_err_IMPL("tensor of cyclic algebras of noncoprime degrees"); /* TODO */
     491             : 
     492          28 :   if (d1==1) return gcopy(al2);
     493          21 :   if (d2==1) return gcopy(al1);
     494             : 
     495          14 :   C = nfcompositum(nf, P1, P2, 3);
     496          14 :   rnf = rnfinit(nf,gel(C,1));
     497          14 :   x1 = gel(C,2);
     498          14 :   x2 = gel(C,3);
     499          14 :   k = itos(gel(C,4));
     500          14 :   aut = gadd(gsubst(aut2,v,x2),gmulsg(k,gsubst(aut1,v,x1)));
     501          14 :   b = nfmul(nf,nfpow_u(nf,b1,d2),nfpow_u(nf,b2,d1));
     502          14 :   al = alg_cyclic(rnf,aut,b,maxord);
     503          14 :   return gerepilecopy(av,al);
     504             : }
     505             : 
     506             : /* M an n x d Flm of rank d, n >= d. Initialize Mx = y solver */
     507             : static GEN
     508        4383 : Flm_invimage_init(GEN M, ulong p)
     509             : {
     510        4383 :   GEN v = Flm_indexrank(M, p), perm = gel(v,1);
     511        4383 :   GEN MM = rowpermute(M, perm); /* square invertible */
     512        4383 :   return mkvec2(Flm_inv(MM,p), perm);
     513             : }
     514             : /* assume Mx = y has a solution, v = Flm_invimage_init(M,p); return x */
     515             : static GEN
     516      244362 : Flm_invimage_pre(GEN v, GEN y, ulong p)
     517             : {
     518      244362 :   GEN inv = gel(v,1), perm = gel(v,2);
     519      244362 :   return Flm_Flc_mul(inv, vecsmallpermute(y, perm), p);
     520             : }
     521             : 
     522             : GEN
     523        5747 : algradical(GEN al)
     524             : {
     525        5747 :   pari_sp av = avma;
     526             :   GEN I, x, traces, K, MT, P, mt;
     527             :   long l,i,ni, n;
     528             :   ulong modu, expo, p;
     529        5747 :   checkalg(al);
     530        5747 :   if (alg_type(al) != al_TABLE) return gen_0;
     531        5656 :   P = alg_get_char(al);
     532        5656 :   mt = alg_get_multable(al);
     533        5656 :   n = alg_get_absdim(al);
     534        5656 :   dbg_printf(1)("algradical: char=%Ps, dim=%d\n", P, n);
     535        5656 :   traces = algtracematrix(al);
     536        5656 :   if (!signe(P))
     537             :   {
     538         518 :     dbg_printf(2)(" char 0, computing kernel...\n");
     539         518 :     K = ker(traces);
     540         518 :     dbg_printf(2)(" ...done.\n");
     541         518 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     542          70 :     return gerepileupto(av, K);
     543             :   }
     544        5138 :   dbg_printf(2)(" char>0, computing kernel...\n");
     545        5138 :   K = FpM_ker(traces, P);
     546        5138 :   dbg_printf(2)(" ...done.\n");
     547        5138 :   ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     548        2999 :   if (abscmpiu(P,n)>0) return gerepileupto(av, K);
     549             : 
     550             :   /* tough case, p <= n. Ronyai's algorithm */
     551        2382 :   p = P[2]; l = 1;
     552        2382 :   expo = p; modu = p*p;
     553        2382 :   dbg_printf(2)(" char>0, hard case.\n");
     554        4810 :   while (modu<=(ulong)n) { l++; modu *= p; }
     555        2382 :   MT = ZMV_to_FlmV(mt, modu);
     556        2382 :   I = ZM_to_Flm(K,p); /* I_0 */
     557        6436 :   for (i=1; i<=l; i++) {/*compute I_i, expo = p^i, modu = p^(l+1) > n*/
     558             :     long j, lig,col;
     559        4383 :     GEN v = cgetg(ni+1, t_VECSMALL);
     560        4383 :     GEN invI = Flm_invimage_init(I, p);
     561        4383 :     dbg_printf(2)(" computing I_%d:\n", i);
     562        4383 :     traces = cgetg(ni+1,t_MAT);
     563       29022 :     for (j = 1; j <= ni; j++)
     564             :     {
     565       24639 :       GEN M = algbasismultable_Flm(MT, gel(I,j), modu);
     566       24639 :       uel(v,j) = algtracei(M, p,expo,modu);
     567             :     }
     568       29022 :     for (col=1; col<=ni; col++)
     569             :     {
     570       24639 :       GEN t = cgetg(n+1,t_VECSMALL); gel(traces,col) = t;
     571       24639 :       x = gel(I, col); /*col-th basis vector of I_{i-1}*/
     572      269001 :       for (lig=1; lig<=n; lig++)
     573             :       {
     574      244362 :         GEN y = _tablemul_ej_Fl(MT,x,lig,p);
     575      244362 :         GEN z = Flm_invimage_pre(invI, y, p);
     576      244362 :         uel(t,lig) = Flv_dotproduct(v, z, p);
     577             :       }
     578             :     }
     579        4383 :     dbg_printf(2)(" computing kernel...\n");
     580        4383 :     K = Flm_ker(traces, p);
     581        4383 :     dbg_printf(2)(" ...done.\n");
     582        4383 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     583        4054 :     I = Flm_mul(I,K,p);
     584        4054 :     expo *= p;
     585             :   }
     586        2053 :   return Flm_to_ZM(I);
     587             : }
     588             : 
     589             : /* compute the multiplication table of the element x, where mt is a
     590             :  * multiplication table in an arbitrary ring */
     591             : static GEN
     592         427 : Rgmultable(GEN mt, GEN x)
     593             : {
     594         427 :   long i, l = lg(x);
     595         427 :   GEN z = NULL;
     596        5796 :   for (i = 1; i < l; i++)
     597             :   {
     598        5369 :     GEN c = gel(x,i);
     599        5369 :     if (!gequal0(c))
     600             :     {
     601         644 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
     602         644 :       z = z? RgM_add(z, M): M;
     603             :     }
     604             :   }
     605         427 :   return z;
     606             : }
     607             : 
     608             : static GEN
     609          49 : change_Rgmultable(GEN mt, GEN P, GEN Pi)
     610             : {
     611             :   GEN mt2;
     612          49 :   long lmt = lg(mt), i;
     613          49 :   mt2 = cgetg(lmt,t_VEC);
     614         476 :   for (i=1;i<lmt;i++) {
     615         427 :     GEN mti = Rgmultable(mt,gel(P,i));
     616         427 :     gel(mt2,i) = RgM_mul(Pi, RgM_mul(mti,P));
     617             :   }
     618          49 :   return mt2;
     619             : }
     620             : 
     621             : static GEN
     622       21091 : alg_quotient0(GEN al, GEN S, GEN Si, long nq, GEN p, long maps)
     623             : {
     624       21091 :   GEN mt = cgetg(nq+1,t_VEC), P, Pi, d;
     625             :   long i;
     626       21091 :   dbg_printf(3)("  alg_quotient0: char=%Ps, dim=%d, dim I=%d\n", p, alg_get_absdim(al), lg(S)-1);
     627       85062 :   for (i=1; i<=nq; i++) {
     628       63971 :     GEN mti = algbasismultable(al,gel(S,i));
     629       63971 :     if (signe(p)) gel(mt,i) = FpM_mul(Si, FpM_mul(mti,S,p), p);
     630        5390 :     else          gel(mt,i) = RgM_mul(Si, RgM_mul(mti,S));
     631             :   }
     632       21091 :   if (!signe(p) && !isint1(Q_denom(mt))) {
     633          35 :     dbg_printf(3)("  bad case: denominator=%Ps\n", Q_denom(mt));
     634          35 :     P = Q_remove_denom(Si,&d);
     635          35 :     P = ZM_hnf(P);
     636          35 :     P = RgM_Rg_div(P,d);
     637          35 :     Pi = RgM_inv(P);
     638          35 :     mt = change_Rgmultable(mt,P,Pi);
     639          35 :     Si = RgM_mul(P,Si);
     640          35 :     S = RgM_mul(S,Pi);
     641             :   }
     642       21091 :   al = algtableinit_i(mt,p);
     643       21091 :   if (maps) al = mkvec3(al,Si,S); /* algebra, proj, lift */
     644       21091 :   return al;
     645             : }
     646             : 
     647             : /* quotient of an algebra by a nontrivial two-sided ideal */
     648             : GEN
     649        2799 : alg_quotient(GEN al, GEN I, long maps)
     650             : {
     651        2799 :   pari_sp av = avma;
     652             :   GEN p, IS, ISi, S, Si;
     653             :   long n, ni;
     654             : 
     655        2799 :   checkalg(al);
     656        2799 :   if (alg_type(al) != al_TABLE) pari_err_TYPE("alg_quotient [not a table algebra]", al);
     657        2792 :   p = alg_get_char(al);
     658        2792 :   n = alg_get_absdim(al);
     659        2792 :   ni = lg(I)-1;
     660             : 
     661             :   /* force first vector of complement to be the identity */
     662        2792 :   IS = shallowconcat(I, gcoeff(alg_get_multable(al),1,1));
     663        2792 :   if (signe(p)) {
     664        2764 :     IS = FpM_suppl(IS,p);
     665        2764 :     ISi = FpM_inv(IS,p);
     666             :   }
     667             :   else {
     668          28 :     IS = suppl(IS);
     669          28 :     ISi = RgM_inv(IS);
     670             :   }
     671        2792 :   S = vecslice(IS, ni+1, n);
     672        2792 :   Si = rowslice(ISi, ni+1, n);
     673        2792 :   return gerepilecopy(av, alg_quotient0(al, S, Si, n-ni, p, maps));
     674             : }
     675             : 
     676             : static GEN
     677       28491 : image_keep_first(GEN m, GEN p) /* assume first column is nonzero or m==0, no GC */
     678             : {
     679             :   GEN ir, icol, irow, M, c, x;
     680             :   long i;
     681       28491 :   if (gequal0(gel(m,1))) return zeromat(nbrows(m),0);
     682             : 
     683       28477 :   if (signe(p)) ir = FpM_indexrank(m,p);
     684        1498 :   else          ir = indexrank(m);
     685             : 
     686       28477 :   icol = gel(ir,2);
     687       28477 :   if (icol[1]==1) return extract0(m,icol,NULL);
     688             : 
     689          10 :   irow = gel(ir,1);
     690          10 :   M = extract0(m, irow, icol);
     691          10 :   c = extract0(gel(m,1), irow, NULL);
     692          10 :   if (signe(p)) x = FpM_FpC_invimage(M,c,p);
     693           0 :   else          x = inverseimage(M,c); /* TODO modulo a small prime */
     694             : 
     695          10 :   for (i=1; i<lg(x); i++)
     696             :   {
     697          10 :     if (!gequal0(gel(x,i)))
     698             :     {
     699          10 :       icol[i] = 1;
     700          10 :       vecsmall_sort(icol);
     701          10 :       return extract0(m,icol,NULL);
     702             :     }
     703             :   }
     704             : 
     705             :   return NULL; /* LCOV_EXCL_LINE */
     706             : }
     707             : 
     708             : /* z[1],...z[nz] central elements such that z[1]A + z[2]A + ... + z[nz]A = A
     709             :  * is a direct sum. idempotents ==> first basis element is identity */
     710             : GEN
     711        8653 : alg_centralproj(GEN al, GEN z, long maps)
     712             : {
     713        8653 :   pari_sp av = avma;
     714             :   GEN S, U, Ui, alq, p;
     715        8653 :   long i, iu, lz = lg(z), ta;
     716             : 
     717        8653 :   checkalg(al);
     718        8653 :   ta = alg_type(al);
     719        8653 :   if (ta != al_TABLE) pari_err_TYPE("algcentralproj [not a table algebra]", al);
     720        8646 :   if (typ(z) != t_VEC) pari_err_TYPE("alcentralproj",z);
     721        8639 :   p = alg_get_char(al);
     722        8639 :   dbg_printf(3)("  alg_centralproj: char=%Ps, dim=%d, #z=%d\n", p, alg_get_absdim(al), lz-1);
     723        8639 :   S = cgetg(lz,t_VEC); /* S[i] = Im(z_i) */
     724       26952 :   for (i=1; i<lz; i++)
     725             :   {
     726       18313 :     GEN mti = algbasismultable(al, gel(z,i));
     727       18313 :     gel(S,i) = image_keep_first(mti,p);
     728             :   }
     729        8639 :   U = shallowconcat1(S); /* U = [Im(z_1)|Im(z_2)|...|Im(z_nz)], n x n */
     730        8639 :   if (lg(U)-1 < alg_get_absdim(al)) pari_err_TYPE("alcentralproj [z[i]'s not surjective]",z);
     731        8632 :   if (signe(p)) Ui = FpM_inv(U,p);
     732         749 :   else          Ui = RgM_inv(U);
     733             :   if (!Ui) pari_err_BUG("alcentralproj"); /*LCOV_EXCL_LINE*/
     734             : 
     735        8632 :   alq = cgetg(lz,t_VEC);
     736       26931 :   for (iu=0,i=1; i<lz; i++)
     737             :   {
     738       18299 :     long nq = lg(gel(S,i))-1, ju = iu + nq;
     739       18299 :     GEN Si = rowslice(Ui, iu+1, ju);
     740       18299 :     gel(alq, i) = alg_quotient0(al,gel(S,i),Si,nq,p,maps);
     741       18299 :     iu = ju;
     742             :   }
     743        8632 :   return gerepilecopy(av, alq);
     744             : }
     745             : 
     746             : /* al is an al_TABLE */
     747             : static GEN
     748       18988 : algtablecenter(GEN al)
     749             : {
     750       18988 :   pari_sp av = avma;
     751             :   long n, i, j, k, ic;
     752             :   GEN C, cij, mt, p;
     753             : 
     754       18988 :   n = alg_get_absdim(al);
     755       18988 :   mt = alg_get_multable(al);
     756       18988 :   p = alg_get_char(al);
     757       18988 :   C = cgetg(n+1,t_MAT);
     758       92137 :   for (j=1; j<=n; j++)
     759             :   {
     760       73149 :     gel(C,j) = cgetg(n*n-n+1,t_COL);
     761       73149 :     ic = 1;
     762      594117 :     for (i=2; i<=n; i++) {
     763      520968 :       if (signe(p)) cij = FpC_sub(gmael(mt,i,j),gmael(mt,j,i),p);
     764       52318 :       else          cij = RgC_sub(gmael(mt,i,j),gmael(mt,j,i));
     765     7284828 :       for (k=1; k<=n; k++, ic++) gcoeff(C,ic,j) = gel(cij, k);
     766             :     }
     767             :   }
     768       18988 :   if (signe(p)) return gerepileupto(av, FpM_ker(C,p));
     769        1645 :   else          return gerepileupto(av, ker(C));
     770             : }
     771             : 
     772             : GEN
     773        4886 : algcenter(GEN al)
     774             : {
     775        4886 :   checkalg(al);
     776        4886 :   if (alg_type(al)==al_TABLE) return algtablecenter(al);
     777          49 :   return alg_get_center(al);
     778             : }
     779             : 
     780             : /* Only in positive characteristic. Assumes that al is semisimple. */
     781             : GEN
     782        4414 : algprimesubalg(GEN al)
     783             : {
     784        4414 :   pari_sp av = avma;
     785             :   GEN p, Z, F, K;
     786             :   long nz, i;
     787        4414 :   checkalg(al);
     788        4414 :   p = alg_get_char(al);
     789        4414 :   if (!signe(p)) pari_err_DOMAIN("algprimesubalg","characteristic","=",gen_0,p);
     790             : 
     791        4400 :   Z = algtablecenter(al);
     792        4400 :   nz = lg(Z)-1;
     793        4400 :   if (nz==1) return Z;
     794             : 
     795        2839 :   F = cgetg(nz+1, t_MAT);
     796       14734 :   for (i=1; i<=nz; i++) {
     797       11895 :     GEN zi = gel(Z,i);
     798       11895 :     gel(F,i) = FpC_sub(algpow(al,zi,p),zi,p);
     799             :   }
     800        2839 :   K = FpM_ker(F,p);
     801        2839 :   return gerepileupto(av, FpM_mul(Z,K,p));
     802             : }
     803             : 
     804             : static GEN
     805       14777 : out_decompose(GEN t, GEN Z, GEN P, GEN p)
     806             : {
     807       14777 :   GEN ali = gel(t,1), projm = gel(t,2), liftm = gel(t,3), pZ;
     808       14777 :   if (signe(p)) pZ = FpM_image(FpM_mul(projm,Z,p),p);
     809        1407 :   else          pZ = image(RgM_mul(projm,Z));
     810       14777 :   return mkvec5(ali, projm, liftm, pZ, P);
     811             : }
     812             : /* fa factorization of charpol(x) */
     813             : static GEN
     814        7427 : alg_decompose_from_facto(GEN al, GEN x, GEN fa, GEN Z, long mini)
     815             : {
     816        7427 :   long k = lgcols(fa)-1, k2 = mini? 1: k/2;
     817        7427 :   GEN v1 = rowslice(fa,1,k2);
     818        7427 :   GEN v2 = rowslice(fa,k2+1,k);
     819        7427 :   GEN alq, P, Q, p = alg_get_char(al);
     820        7427 :   dbg_printf(3)("  alg_decompose_from_facto\n");
     821        7427 :   if (signe(p)) {
     822        6706 :     P = FpXV_factorback(gel(v1,1), gel(v1,2), p, 0);
     823        6706 :     Q = FpXV_factorback(gel(v2,1), gel(v2,2), p, 0);
     824        6706 :     P = FpX_mul(P, FpXQ_inv(P,Q,p), p);
     825             :   }
     826             :   else {
     827         721 :     P = factorback(v1);
     828         721 :     Q = factorback(v2);
     829         721 :     P = RgX_mul(P, RgXQ_inv(P,Q));
     830             :   }
     831        7427 :   P = algpoleval(al, P, x);
     832        7427 :   if (signe(p)) Q = FpC_sub(col_ei(lg(P)-1,1), P, p);
     833         721 :   else          Q = gsub(gen_1, P);
     834        7427 :   if (gequal0(P) || gequal0(Q)) return NULL;
     835        7427 :   alq = alg_centralproj(al, mkvec2(P,Q), 1);
     836             : 
     837        7427 :   P = out_decompose(gel(alq,1), Z, P, p); if (mini) return P;
     838        7350 :   Q = out_decompose(gel(alq,2), Z, Q, p);
     839        7350 :   return mkvec2(P,Q);
     840             : }
     841             : 
     842             : static GEN
     843       11886 : random_pm1(long n)
     844             : {
     845       11886 :   GEN z = cgetg(n+1,t_VECSMALL);
     846             :   long i;
     847       52135 :   for (i = 1; i <= n; i++) z[i] = random_bits(5)%3 - 1;
     848       11886 :   return z;
     849             : }
     850             : 
     851             : static GEN alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt);
     852             : /* Try to split al using x's charpoly. Return gen_0 if simple, NULL if failure.
     853             :  * And a splitting otherwise
     854             :  * If pt_primelt!=NULL, compute a primitive element of the center when simple */
     855             : static GEN
     856       13853 : try_fact(GEN al, GEN x, GEN zx, GEN Z, GEN Zal, long mini, GEN* pt_primelt)
     857             : {
     858       13853 :   GEN z, dec0, dec1, cp = algcharpoly(Zal,zx,0,1), fa, p = alg_get_char(al);
     859             :   long nfa, e;
     860       13853 :   dbg_printf(3)("  try_fact: zx=%Ps\n", zx);
     861       13853 :   if (signe(p)) fa = FpX_factor(cp,p);
     862        1281 :   else          fa = factor(cp);
     863       13853 :   dbg_printf(3)("  charpoly=%Ps\n", fa);
     864       13853 :   nfa = nbrows(fa);
     865       13853 :   if (nfa == 1) {
     866        6426 :     if (signe(p)) e = gel(fa,2)[1];
     867         560 :     else          e = itos(gcoeff(fa,1,2));
     868        6426 :     if (e == 1) {
     869        3689 :       if (pt_primelt != NULL) *pt_primelt = mkvec2(x, cp);
     870        3689 :       return gen_0;
     871             :     }
     872        2737 :     else return NULL;
     873             :   }
     874        7427 :   dec0 = alg_decompose_from_facto(al, x, fa, Z, mini);
     875        7427 :   if (!dec0) return NULL;
     876        7427 :   if (!mini) return dec0;
     877          77 :   dec1 = alg_decompose(gel(dec0,1), gel(dec0,4), 1, pt_primelt);
     878          77 :   z = gel(dec0,5);
     879          77 :   if (!isintzero(dec1)) {
     880           7 :     if (signe(p)) z = FpM_FpC_mul(gel(dec0,3),dec1,p);
     881           7 :     else          z = RgM_RgC_mul(gel(dec0,3),dec1);
     882             :   }
     883          77 :   return z;
     884             : }
     885             : static GEN
     886           7 : randcol(long n, GEN b)
     887             : {
     888           7 :   GEN N = addiu(shifti(b,1), 1);
     889             :   long i;
     890           7 :   GEN res =  cgetg(n+1,t_COL);
     891          63 :   for (i=1; i<=n; i++)
     892             :   {
     893          56 :     pari_sp av = avma;
     894          56 :     gel(res,i) = gerepileuptoint(av, subii(randomi(N),b));
     895             :   }
     896           7 :   return res;
     897             : }
     898             : /* Return gen_0 if already simple. mini: only returns a central idempotent
     899             :  * corresponding to one simple factor
     900             :  * if pt_primelt!=NULL, sets it to a primitive element of the center when simple */
     901             : static GEN
     902       20363 : alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt)
     903             : {
     904             :   pari_sp av;
     905             :   GEN Zal, x, zx, rand, dec0, B, p;
     906       20363 :   long i, nz = lg(Z)-1;
     907             : 
     908       20363 :   if (nz == 1) {
     909        9247 :     if (pt_primelt != 0) *pt_primelt = mkvec2(zerocol(alg_get_dim(al)), pol_x(0));
     910        9247 :     return gen_0;
     911             :   }
     912       11116 :   p = alg_get_char(al);
     913       11116 :   dbg_printf(2)(" alg_decompose: char=%Ps, dim=%d, dim Z=%d\n", p, alg_get_absdim(al), nz);
     914       11116 :   Zal = alg_subalg(al,Z);
     915       11116 :   Z = gel(Zal,2);
     916       11116 :   Zal = gel(Zal,1);
     917       11116 :   av = avma;
     918             : 
     919       11116 :   rand = random_pm1(nz);
     920       11116 :   zx = zc_to_ZC(rand);
     921       11116 :   if (signe(p)) {
     922       10143 :     zx = FpC_red(zx,p);
     923       10143 :     x = ZM_zc_mul(Z,rand);
     924       10143 :     x = FpC_red(x,p);
     925             :   }
     926         973 :   else x = RgM_zc_mul(Z,rand);
     927       11116 :   dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     928       11116 :   if (dec0) return dec0;
     929        2681 :   set_avma(av);
     930             : 
     931        2737 :   for (i=2; i<=nz; i++)
     932             :   {
     933        2730 :     dec0 = try_fact(al,gel(Z,i),col_ei(nz,i),Z,Zal,mini,pt_primelt);
     934        2730 :     if (dec0) return dec0;
     935          56 :     set_avma(av);
     936             :   }
     937           7 :   B = int2n(10);
     938             :   for (;;)
     939           0 :   {
     940           7 :     GEN x = randcol(nz,B), zx = ZM_ZC_mul(Z,x);
     941           7 :     dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     942           7 :     if (dec0) return dec0;
     943           0 :     set_avma(av);
     944             :   }
     945             : }
     946             : 
     947             : static GEN
     948       16765 : alg_decompose_total(GEN al, GEN Z, long maps)
     949             : {
     950             :   GEN dec, sc, p;
     951             :   long i;
     952             : 
     953       16765 :   dec = alg_decompose(al, Z, 0, NULL);
     954       16765 :   if (isintzero(dec))
     955             :   {
     956        9415 :     if (maps) {
     957        6783 :       long n = alg_get_absdim(al);
     958        6783 :       al = mkvec3(al, matid(n), matid(n));
     959             :     }
     960        9415 :     return mkvec(al);
     961             :   }
     962        7350 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
     963        7350 :   sc = cgetg(lg(dec), t_VEC);
     964       22050 :   for (i=1; i<lg(sc); i++) {
     965       14700 :     GEN D = gel(dec,i), a = gel(D,1), Za = gel(D,4);
     966       14700 :     GEN S = alg_decompose_total(a, Za, maps);
     967       14700 :     gel(sc,i) = S;
     968       14700 :     if (maps)
     969             :     {
     970       10444 :       GEN projm = gel(D,2), liftm = gel(D,3);
     971       10444 :       long j, lS = lg(S);
     972       28335 :       for (j=1; j<lS; j++)
     973             :       {
     974       17891 :         GEN Sj = gel(S,j), p2 = gel(Sj,2), l2 = gel(Sj,3);
     975       17891 :         if (p) p2 = FpM_mul(p2, projm, p);
     976          49 :         else   p2 = RgM_mul(p2, projm);
     977       17891 :         if (p) l2 = FpM_mul(liftm, l2, p);
     978          49 :         else   l2 = RgM_mul(liftm, l2);
     979       17891 :         gel(Sj,2) = p2;
     980       17891 :         gel(Sj,3) = l2;
     981             :       }
     982             :     }
     983             :   }
     984        7350 :   return shallowconcat1(sc);
     985             : }
     986             : 
     987             : static GEN
     988       11172 : alg_subalg(GEN al, GEN basis)
     989             : {
     990       11172 :   GEN invbasis, mt, p = alg_get_char(al);
     991       11172 :   long i, j, n = lg(basis)-1;
     992             : 
     993       11172 :   if (!signe(p)) p = NULL;
     994       11172 :   basis = shallowmatconcat(mkvec2(col_ei(n,1), basis));
     995       11172 :   if (p)
     996             :   {
     997       10178 :     basis = image_keep_first(basis,p);
     998       10178 :     invbasis = FpM_inv(basis,p);
     999             :   }
    1000             :   else
    1001             :   { /* FIXME use an integral variant of image_keep_first */
    1002         994 :     basis = QM_ImQ_hnf(basis);
    1003         994 :     invbasis = RgM_inv(basis);
    1004             :   }
    1005       11172 :   mt = cgetg(n+1,t_VEC);
    1006       11172 :   gel(mt,1) = matid(n);
    1007       37568 :   for (i = 2; i <= n; i++)
    1008             :   {
    1009       26396 :     GEN mtx = cgetg(n+1,t_MAT), x = gel(basis,i);
    1010       26396 :     gel(mtx,1) = col_ei(n,i);
    1011      167652 :     for (j = 2; j <= n; j++)
    1012             :     {
    1013      141256 :       GEN xy = algmul(al, x, gel(basis,j));
    1014      141256 :       if (p) gel(mtx,j) = FpM_FpC_mul(invbasis, xy, p);
    1015       29701 :       else   gel(mtx,j) = RgM_RgC_mul(invbasis, xy);
    1016             :     }
    1017       26396 :     gel(mt,i) = mtx;
    1018             :   }
    1019       11172 :   return mkvec2(algtableinit_i(mt,p), basis);
    1020             : }
    1021             : 
    1022             : GEN
    1023          70 : algsubalg(GEN al, GEN basis)
    1024             : {
    1025          70 :   pari_sp av = avma;
    1026             :   GEN p;
    1027          70 :   checkalg(al);
    1028          70 :   if (alg_type(al) == al_REAL) pari_err_TYPE("algsubalg [real algebra]", al);
    1029          63 :   if (typ(basis) != t_MAT) pari_err_TYPE("algsubalg",basis);
    1030          56 :   p = alg_get_char(al);
    1031          56 :   if (signe(p)) basis = RgM_to_FpM(basis,p);
    1032          56 :   return gerepilecopy(av, alg_subalg(al,basis));
    1033             : }
    1034             : 
    1035             : static int
    1036       11921 : cmp_algebra(GEN x, GEN y)
    1037             : {
    1038             :   long d;
    1039       11921 :   d = gel(x,1)[1] - gel(y,1)[1]; if (d) return d < 0? -1: 1;
    1040       10724 :   d = gel(x,1)[2] - gel(y,1)[2]; if (d) return d < 0? -1: 1;
    1041       10724 :   return cmp_universal(gel(x,2), gel(y,2));
    1042             : }
    1043             : 
    1044             : GEN
    1045        4512 : algsimpledec_ss(GEN al, long maps)
    1046             : {
    1047        4512 :   pari_sp av = avma;
    1048             :   GEN Z, p, r, res, perm;
    1049             :   long i, l, n;
    1050        4512 :   checkalg(al);
    1051        4512 :   p = alg_get_char(al);
    1052        4512 :   dbg_printf(1)("algsimpledec_ss: char=%Ps, dim=%d\n", p, alg_get_absdim(al));
    1053        4512 :   if (signe(p))                     Z = algprimesubalg(al);
    1054         266 :   else if (alg_type(al)!=al_TABLE)  Z = gen_0;
    1055         245 :   else                              Z = algtablecenter(al);
    1056             : 
    1057        4512 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1058        2447 :     n = alg_get_absdim(al);
    1059        2447 :     set_avma(av);
    1060        2447 :     if (!maps) return mkveccopy(al);
    1061        2300 :     retmkvec(mkvec3(gcopy(al), matid(n), matid(n)));
    1062             :   }
    1063        2065 :   res = alg_decompose_total(al, Z, maps);
    1064        2065 :   l = lg(res); r = cgetg(l, t_VEC);
    1065       11480 :   for (i = 1; i < l; i++)
    1066             :   {
    1067        9415 :     GEN A = maps? gmael(res,i,1): gel(res,i);
    1068        9415 :     gel(r,i) = mkvec2(mkvecsmall2(alg_get_dim(A), lg(algtablecenter(A))),
    1069             :                       alg_get_multable(A));
    1070             :   }
    1071        2065 :   perm = gen_indexsort(r, (void*)cmp_algebra, &cmp_nodata);
    1072        2065 :   return gerepilecopy(av, vecpermute(res, perm));
    1073             : }
    1074             : 
    1075             : GEN
    1076         777 : algsimpledec(GEN al, long maps)
    1077             : {
    1078         777 :   pari_sp av = avma;
    1079             :   int ss;
    1080         777 :   GEN rad, dec, res, proj=NULL, lift=NULL;
    1081         777 :   rad = algradical(al);
    1082         777 :   ss = gequal0(rad);
    1083         777 :   if (!ss)
    1084             :   {
    1085          42 :     al = alg_quotient(al, rad, maps);
    1086          42 :     if (maps) {
    1087          14 :       proj = gel(al,2);
    1088          14 :       lift = gel(al,3);
    1089          14 :       al = gel(al,1);
    1090             :     }
    1091             :   }
    1092         777 :   dec = algsimpledec_ss(al, maps);
    1093         777 :   if (!ss && maps) /* update maps */
    1094             :   {
    1095          14 :     GEN p = alg_get_char(al);
    1096             :     long i;
    1097          42 :     for (i=1; i<lg(dec); i++)
    1098             :     {
    1099          28 :       if (signe(p))
    1100             :       {
    1101          14 :         gmael(dec,i,2) = FpM_mul(gmael(dec,i,2), proj, p);
    1102          14 :         gmael(dec,i,3) = FpM_mul(lift, gmael(dec,i,3), p);
    1103             :       }
    1104             :       else
    1105             :       {
    1106          14 :         gmael(dec,i,2) = RgM_mul(gmael(dec,i,2), proj);
    1107          14 :         gmael(dec,i,3) = RgM_mul(lift, gmael(dec,i,3));
    1108             :       }
    1109             :     }
    1110             :   }
    1111         777 :   res = mkvec2(rad, dec);
    1112         777 :   return gerepilecopy(av,res);
    1113             : }
    1114             : 
    1115             : static GEN alg_idempotent(GEN al, long n, long d);
    1116             : static GEN
    1117        6482 : try_split(GEN al, GEN x, long n, long d)
    1118             : {
    1119        6482 :   GEN cp, p = alg_get_char(al), fa, e, pol, exp, P, Q, U, u, mx, mte, ire;
    1120        6482 :   long nfa, i, smalldim = alg_get_absdim(al)+1, dim, smalli = 0;
    1121        6482 :   cp = algcharpoly(al,x,0,1);
    1122        6482 :   fa = FpX_factor(cp,p);
    1123        6482 :   nfa = nbrows(fa);
    1124        6482 :   if (nfa == 1) return NULL;
    1125        3052 :   pol = gel(fa,1);
    1126        3052 :   exp = gel(fa,2);
    1127             : 
    1128             :   /* charpoly is always a d-th power */
    1129        9254 :   for (i=1; i<lg(exp); i++) {
    1130        6209 :     if (exp[i]%d) pari_err(e_MISC, "the algebra must be simple (try_split 1)");
    1131        6202 :     exp[i] /= d;
    1132             :   }
    1133        3045 :   cp = FpXV_factorback(gel(fa,1), gel(fa,2), p, 0);
    1134             : 
    1135             :   /* find smallest Fp-dimension of a characteristic space */
    1136        9247 :   for (i=1; i<lg(pol); i++) {
    1137        6202 :     dim = degree(gel(pol,i))*exp[i];
    1138        6202 :     if (dim < smalldim) {
    1139        3115 :       smalldim = dim;
    1140        3115 :       smalli = i;
    1141             :     }
    1142             :   }
    1143        3045 :   i = smalli;
    1144        3045 :   if (smalldim != n) return NULL;
    1145             :   /* We could also compute e*al*e and try again with this smaller algebra */
    1146             :   /* Fq-rank 1 = Fp-rank n idempotent: success */
    1147             : 
    1148             :   /* construct idempotent */
    1149        3031 :   mx = algbasismultable(al,x);
    1150        3031 :   P = gel(pol,i);
    1151        3031 :   P = FpX_powu(P, exp[i], p);
    1152        3031 :   Q = FpX_div(cp, P, p);
    1153        3031 :   e = algpoleval(al, Q, mkvec2(x,mx));
    1154        3031 :   U = FpXQ_inv(Q, P, p);
    1155        3031 :   u = algpoleval(al, U, mkvec2(x,mx));
    1156        3031 :   e = algbasismul(al, e, u);
    1157        3031 :   mte = algbasisrightmultable(al,e);
    1158        3031 :   ire = FpM_indexrank(mte,p);
    1159        3031 :   if (lg(gel(ire,1))-1 != smalldim*d) pari_err(e_MISC, "the algebra must be simple (try_split 2)");
    1160             : 
    1161        3024 :   return mkvec3(e,mte,ire);
    1162             : }
    1163             : 
    1164             : /*
    1165             :  * Given a simple algebra al of dimension d^2 over its center of degree n,
    1166             :  * find an idempotent e in al with rank n (which is minimal).
    1167             : */
    1168             : static GEN
    1169        3038 : alg_idempotent(GEN al, long n, long d)
    1170             : {
    1171        3038 :   pari_sp av = avma;
    1172        3038 :   long i, N = alg_get_absdim(al);
    1173        3038 :   GEN e, p = alg_get_char(al), x;
    1174        6377 :   for(i=2; i<=N; i++) {
    1175        6321 :     x = col_ei(N,i);
    1176        6321 :     e = try_split(al, x, n, d);
    1177        6307 :     if (e) return e;
    1178        3339 :     set_avma(av);
    1179             :   }
    1180             :   for(;;) {
    1181         161 :     x = random_FpC(N,p);
    1182         161 :     e = try_split(al, x, n, d);
    1183         161 :     if (e) return e;
    1184         105 :     set_avma(av);
    1185             :   }
    1186             : }
    1187             : 
    1188             : static GEN
    1189        3857 : try_descend(GEN M, GEN B, GEN p, long m, long n, long d)
    1190             : {
    1191        3857 :   GEN B2 = cgetg(m+1,t_MAT), b;
    1192        3857 :   long i, j, k=0;
    1193       11011 :   for (i=1; i<=d; i++)
    1194             :   {
    1195        7154 :     k++;
    1196        7154 :     b = gel(B,i);
    1197        7154 :     gel(B2,k) = b;
    1198       17248 :     for (j=1; j<n; j++)
    1199             :     {
    1200       10094 :       k++;
    1201       10094 :       b = FpM_FpC_mul(M,b,p);
    1202       10094 :       gel(B2,k) = b;
    1203             :     }
    1204             :   }
    1205        3857 :   if (!signe(FpM_det(B2,p))) return NULL;
    1206        3437 :   return FpM_inv(B2,p);
    1207             : }
    1208             : 
    1209             : /* Given an m*m matrix M with irreducible charpoly over F of degree n,
    1210             :  * let K = F(M), which is a field, and write m=d*n.
    1211             :  * Compute the d-dimensional K-vector space structure on V=F^m induced by M.
    1212             :  * Return [B,C] where:
    1213             :  *  - B is m*d matrix over F giving a K-basis b_1,...,b_d of V
    1214             :  *  - C is d*m matrix over F[x] expressing the canonical F-basis of V on the b_i
    1215             :  * Currently F = Fp TODO extend this. */
    1216             : static GEN
    1217        3437 : descend_i(GEN M, long n, GEN p)
    1218             : {
    1219             :   GEN B, C;
    1220             :   long m,d,i;
    1221             :   pari_sp av;
    1222        3437 :   m = lg(M)-1;
    1223        3437 :   d = m/n;
    1224        3437 :   B = cgetg(d+1,t_MAT);
    1225        3437 :   av = avma;
    1226             : 
    1227             :   /* try a subset of the canonical basis */
    1228        9751 :   for (i=1; i<=d; i++)
    1229        6314 :     gel(B,i) = col_ei(m,n*(i-1)+1);
    1230        3437 :   C = try_descend(M,B,p,m,n,d);
    1231        3437 :   if (C) return mkvec2(B,C);
    1232         385 :   set_avma(av);
    1233             : 
    1234             :   /* try smallish elements */
    1235        1155 :   for (i=1; i<=d; i++)
    1236         770 :     gel(B,i) = FpC_red(zc_to_ZC(random_pm1(m)),p);
    1237         385 :   C = try_descend(M,B,p,m,n,d);
    1238         385 :   if (C) return mkvec2(B,C);
    1239          35 :   set_avma(av);
    1240             : 
    1241             :   /* try random elements */
    1242             :   for (;;)
    1243             :   {
    1244         105 :     for (i=1; i<=d; i++)
    1245          70 :       gel(B,i) = random_FpC(m,p);
    1246          35 :     C = try_descend(M,B,p,m,n,d);
    1247          35 :     if (C) return mkvec2(B,C);
    1248           0 :     set_avma(av);
    1249             :   }
    1250             : }
    1251             : static GEN
    1252       15568 : RgC_contract(GEN C, long n, long v) /* n>1 */
    1253             : {
    1254             :   GEN C2, P;
    1255             :   long m, d, i, j;
    1256       15568 :   m = lg(C)-1;
    1257       15568 :   d = m/n;
    1258       15568 :   C2 = cgetg(d+1,t_COL);
    1259       43344 :   for (i=1; i<=d; i++)
    1260             :   {
    1261       27776 :     P = pol_xn(n-1,v);
    1262      105728 :     for (j=1; j<=n; j++)
    1263       77952 :       gel(P,j+1) = gel(C,n*(i-1)+j);
    1264       27776 :     P = normalizepol(P);
    1265       27776 :     gel(C2,i) = P;
    1266             :   }
    1267       15568 :   return C2;
    1268             : }
    1269             : static GEN
    1270        3437 : RgM_contract(GEN A, long n, long v) /* n>1 */
    1271             : {
    1272        3437 :   GEN A2 = cgetg(lg(A),t_MAT);
    1273             :   long i;
    1274       19005 :   for (i=1; i<lg(A2); i++)
    1275       15568 :     gel(A2,i) = RgC_contract(gel(A,i),n,v);
    1276        3437 :   return A2;
    1277             : }
    1278             : static GEN
    1279        3437 : descend(GEN M, long n, GEN p, long v)
    1280             : {
    1281        3437 :   GEN res = descend_i(M,n,p);
    1282        3437 :   gel(res,2) = RgM_contract(gel(res,2),n,v);
    1283        3437 :   return res;
    1284             : }
    1285             : 
    1286             : /* isomorphism of Fp-vector spaces M_d(F_p^n) -> (F_p)^(d^2*n) */
    1287             : static GEN
    1288       29939 : Fq_mat2col(GEN M, long d, long n)
    1289             : {
    1290       29939 :   long N = d*d*n, i, j, k;
    1291       29939 :   GEN C = cgetg(N+1, t_COL);
    1292       90160 :   for (i=1; i<=d; i++)
    1293      191632 :     for (j=1; j<=d; j++)
    1294      400526 :       for (k=0; k<n; k++)
    1295      269115 :         gel(C,n*(d*(i-1)+j-1)+k+1) = polcoef_i(gcoeff(M,i,j),k,-1);
    1296       29939 :   return C;
    1297             : }
    1298             : 
    1299             : static GEN
    1300        3752 : alg_finite_csa_split(GEN al, long v)
    1301             : {
    1302             :   GEN Z, e, mte, ire, primelt, b, T, M, proje, lifte, extre, p, B, C, mt, mx, map, mapi, T2, ro;
    1303        3752 :   long n, d, N = alg_get_absdim(al), i;
    1304        3752 :   p = alg_get_char(al);
    1305             :   /* compute the center */
    1306        3752 :   Z = algcenter(al);
    1307             :   /* TODO option to give the center as input instead of computing it */
    1308        3752 :   n = lg(Z)-1;
    1309             : 
    1310             :   /* compute a minimal rank idempotent e */
    1311        3752 :   if (n==N) {
    1312         707 :     d = 1;
    1313         707 :     e = col_ei(N,1);
    1314         707 :     mte = matid(N);
    1315         707 :     ire = mkvec2(identity_perm(n),identity_perm(n));
    1316             :   }
    1317             :   else {
    1318        3045 :     d = usqrt(N/n);
    1319        3045 :     if (d*d*n != N) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 1)");
    1320        3038 :     e = alg_idempotent(al,n,d);
    1321        3024 :     mte = gel(e,2);
    1322        3024 :     ire = gel(e,3);
    1323        3024 :     e = gel(e,1);
    1324             :   }
    1325             : 
    1326             :   /* identify the center */
    1327        3731 :   if (n==1)
    1328             :   {
    1329         287 :     T = pol_x(v);
    1330         287 :     primelt = gen_0;
    1331             :   }
    1332             :   else
    1333             :   {
    1334        3444 :     b = alg_decompose(al, Z, 1, &primelt);
    1335        3444 :     if (!gequal0(b)) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 2)");
    1336        3437 :     T = gel(primelt,2);
    1337        3437 :     primelt = gel(primelt,1);
    1338        3437 :     setvarn(T,v);
    1339             :   }
    1340             : 
    1341             :   /* use the ffinit polynomial */
    1342        3724 :   if (n>1)
    1343             :   {
    1344        3437 :     T2 = init_Fq(p,n,v);
    1345        3437 :     setvarn(T,fetch_var_higher());
    1346        3437 :     ro = FpXQX_roots(T2,T,p);
    1347        3437 :     ro = gel(ro,1);
    1348        3437 :     primelt = algpoleval(al,ro,primelt);
    1349        3437 :     T = T2;
    1350             :   }
    1351             : 
    1352             :   /* descend al*e to a vector space over the center */
    1353             :   /* lifte: al*e -> al ; proje: al*e -> al */
    1354        3724 :   lifte = shallowextract(mte,gel(ire,2));
    1355        3724 :   extre = shallowmatextract(mte,gel(ire,1),gel(ire,2));
    1356        3724 :   extre = FpM_inv(extre,p);
    1357        3724 :   proje = rowpermute(mte,gel(ire,1));
    1358        3724 :   proje = FpM_mul(extre,proje,p);
    1359        3724 :   if (n==1)
    1360             :   {
    1361         287 :     B = lifte;
    1362         287 :     C = proje;
    1363             :   }
    1364             :   else
    1365             :   {
    1366        3437 :     M = algbasismultable(al,primelt);
    1367        3437 :     M = FpM_mul(M,lifte,p);
    1368        3437 :     M = FpM_mul(proje,M,p);
    1369        3437 :     B = descend(M,n,p,v);
    1370        3437 :     C = gel(B,2);
    1371        3437 :     B = gel(B,1);
    1372        3437 :     B = FpM_mul(lifte,B,p);
    1373        3437 :     C = FqM_mul(C,proje,T,p);
    1374             :   }
    1375             : 
    1376             :   /* compute the isomorphism */
    1377        3724 :   mt = alg_get_multable(al);
    1378        3724 :   map = cgetg(N+1,t_VEC);
    1379        3724 :   M = cgetg(N+1,t_MAT);
    1380       33663 :   for (i=1; i<=N; i++)
    1381             :   {
    1382       29939 :     mx = gel(mt,i);
    1383       29939 :     mx = FpM_mul(mx,B,p);
    1384       29939 :     mx = FqM_mul(C,mx,T,p);
    1385       29939 :     gel(map,i) = mx;
    1386       29939 :     gel(M,i) = Fq_mat2col(mx,d,n);
    1387             :   }
    1388        3724 :   mapi = FpM_inv(M,p);
    1389        3724 :   if (!mapi) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 3)");
    1390        3717 :   return mkvec3(T,map,mapi);
    1391             : }
    1392             : 
    1393             : GEN
    1394        3766 : algsplit(GEN al, long v)
    1395             : {
    1396        3766 :   pari_sp av = avma;
    1397             :   GEN res, T, map, mapi, ff, p;
    1398             :   long i,j,k,li,lj;
    1399        3766 :   checkalg(al);
    1400        3759 :   p = alg_get_char(al);
    1401        3759 :   if (gequal0(p))
    1402           7 :     pari_err_IMPL("splitting a characteristic 0 algebra over its center");
    1403        3752 :   res = alg_finite_csa_split(al, v);
    1404        3717 :   T = gel(res,1);
    1405        3717 :   map = gel(res,2);
    1406        3717 :   mapi = gel(res,3);
    1407        3717 :   ff = Tp_to_FF(T,p);
    1408       33593 :   for (i=1; i<lg(map); i++)
    1409             :   {
    1410       29876 :     li = lg(gel(map,i));
    1411       89908 :     for (j=1; j<li; j++)
    1412             :     {
    1413       60032 :       lj = lg(gmael(map,i,j));
    1414      190876 :       for (k=1; k<lj; k++)
    1415      130844 :         gmael3(map,i,j,k) = Fq_to_FF(gmael3(map,i,j,k),ff);
    1416             :     }
    1417             :   }
    1418             : 
    1419        3717 :   return gerepilecopy(av, mkvec2(map,mapi));
    1420             : }
    1421             : 
    1422             : /* multiplication table sanity checks */
    1423             : static GEN
    1424       38129 : check_mt_noid(GEN mt, GEN p)
    1425             : {
    1426             :   long i, l;
    1427       38129 :   GEN MT = cgetg_copy(mt, &l);
    1428       38129 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1429      185525 :   for (i = 1; i < l; i++)
    1430             :   {
    1431      147445 :     GEN M = gel(mt,i);
    1432      147445 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1433      147417 :     if (p) M = RgM_to_FpM(M,p);
    1434      147417 :     gel(MT,i) = M;
    1435             :   }
    1436       38080 :   return MT;
    1437             : }
    1438             : static GEN
    1439       37625 : check_mt(GEN mt, GEN p)
    1440             : {
    1441             :   long i;
    1442             :   GEN MT;
    1443       37625 :   MT = check_mt_noid(mt, p);
    1444       37625 :   if (!MT || !ZM_isidentity(gel(MT,1))) return NULL;
    1445      144274 :   for (i=2; i<lg(MT); i++)
    1446      106677 :     if (ZC_is_ei(gmael(MT,i,1)) != i) return NULL;
    1447       37597 :   return MT;
    1448             : }
    1449             : 
    1450             : static GEN
    1451         168 : check_relmt(GEN nf, GEN mt)
    1452             : {
    1453         168 :   long i, l = lg(mt), j, k;
    1454         168 :   GEN MT = gcopy(mt), a, b, d;
    1455         168 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1456         658 :   for (i = 1; i < l; i++)
    1457             :   {
    1458         511 :     GEN M = gel(MT,i);
    1459         511 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1460        2618 :     for (k = 1; k < l; k++)
    1461       13083 :       for (j = 1; j < l; j++)
    1462             :       {
    1463       10976 :         a = gcoeff(M,j,k);
    1464       10976 :         if (typ(a)==t_INT) continue;
    1465        1771 :         b = algtobasis(nf,a);
    1466        1771 :         d = Q_denom(b);
    1467        1771 :         if (!isint1(d))
    1468          14 :           pari_err_DOMAIN("alg_csa_table", "denominator(mt)", "!=", gen_1, mt);
    1469        1757 :         gcoeff(M,j,k) = lift(basistoalg(nf,b));
    1470             :       }
    1471         497 :     if (i > 1 && RgC_is_ei(gel(M,1)) != i) return NULL; /* i = 1 checked at end */
    1472         490 :     gel(MT,i) = M;
    1473             :   }
    1474         147 :   if (!RgM_isidentity(gel(MT,1))) return NULL;
    1475         147 :   return MT;
    1476             : }
    1477             : 
    1478             : int
    1479         511 : algisassociative(GEN mt0, GEN p)
    1480             : {
    1481         511 :   pari_sp av = avma;
    1482             :   long i, j, k, n;
    1483             :   GEN M, mt;
    1484             : 
    1485         511 :   if (checkalg_i(mt0)) { p = alg_get_char(mt0); mt0 = alg_get_multable(mt0); }
    1486         511 :   if (typ(p) != t_INT) pari_err_TYPE("algisassociative",p);
    1487         504 :   mt = check_mt_noid(mt0, isintzero(p)? NULL: p);
    1488         504 :   if (!mt) pari_err_TYPE("algisassociative (mult. table)", mt0);
    1489         469 :   if (!ZM_isidentity(gel(mt,1))) return gc_bool(av,0);
    1490         455 :   n = lg(mt)-1;
    1491         455 :   M = cgetg(n+1,t_MAT);
    1492        3542 :   for (j=1; j<=n; j++) gel(M,j) = cgetg(n+1,t_COL);
    1493        3542 :   for (i=1; i<=n; i++)
    1494             :   {
    1495        3087 :     GEN mi = gel(mt,i);
    1496       35182 :     for (j=1; j<=n; j++) gcoeff(M,i,j) = gel(mi,j); /* ei.ej */
    1497             :   }
    1498        3073 :   for (i=2; i<=n; i++) {
    1499        2625 :     GEN mi = gel(mt,i);
    1500       28973 :     for (j=2; j<=n; j++) {
    1501      368291 :       for (k=2; k<=n; k++) {
    1502             :         GEN x, y;
    1503      341943 :         if (signe(p)) {
    1504      242039 :           x = _tablemul_ej_Fp(mt,gcoeff(M,i,j),k,p);
    1505      242039 :           y = FpM_FpC_mul(mi,gcoeff(M,j,k),p);
    1506             :         }
    1507             :         else {
    1508       99904 :           x = _tablemul_ej(mt,gcoeff(M,i,j),k);
    1509       99904 :           y = RgM_RgC_mul(mi,gcoeff(M,j,k));
    1510             :         }
    1511             :         /* not cmp_universal: must not fail on 0 == Mod(0,2) for instance */
    1512      341943 :         if (!gequal(x,y)) return gc_bool(av,0);
    1513             :       }
    1514             :     }
    1515             :   }
    1516         448 :   return gc_bool(av,1);
    1517             : }
    1518             : 
    1519             : int
    1520         371 : algiscommutative(GEN al) /* assumes e_1 = 1 */
    1521             : {
    1522             :   long i,j,k,N,sp;
    1523             :   GEN mt,a,b,p;
    1524         371 :   checkalg(al);
    1525         371 :   if (alg_type(al) != al_TABLE) return alg_get_degree(al)==1;
    1526         308 :   N = alg_get_absdim(al);
    1527         308 :   mt = alg_get_multable(al);
    1528         308 :   p = alg_get_char(al);
    1529         308 :   sp = signe(p);
    1530        1449 :   for (i=2; i<=N; i++)
    1531        9464 :     for (j=2; j<=N; j++)
    1532       85820 :       for (k=1; k<=N; k++) {
    1533       77553 :         a = gcoeff(gel(mt,i),k,j);
    1534       77553 :         b = gcoeff(gel(mt,j),k,i);
    1535       77553 :         if (sp) {
    1536       73423 :           if (cmpii(Fp_red(a,p), Fp_red(b,p))) return 0;
    1537             :         }
    1538        4130 :         else if (gcmp(a,b)) return 0;
    1539             :       }
    1540         252 :   return 1;
    1541             : }
    1542             : 
    1543             : int
    1544         371 : algissemisimple(GEN al)
    1545             : {
    1546         371 :   pari_sp av = avma;
    1547             :   GEN rad;
    1548         371 :   checkalg(al);
    1549         371 :   if (alg_type(al) != al_TABLE) return 1;
    1550         308 :   rad = algradical(al);
    1551         308 :   set_avma(av);
    1552         308 :   return gequal0(rad);
    1553             : }
    1554             : 
    1555             : /* ss : known to be semisimple */
    1556             : int
    1557         280 : algissimple(GEN al, long ss)
    1558             : {
    1559         280 :   pari_sp av = avma;
    1560             :   GEN Z, dec, p;
    1561         280 :   checkalg(al);
    1562         280 :   if (alg_type(al) != al_TABLE) return 1;
    1563         224 :   if (!ss && !algissemisimple(al)) return 0;
    1564             : 
    1565         182 :   p = alg_get_char(al);
    1566         182 :   if (signe(p)) Z = algprimesubalg(al);
    1567          91 :   else          Z = algtablecenter(al);
    1568             : 
    1569         182 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1570         105 :     set_avma(av);
    1571         105 :     return 1;
    1572             :   }
    1573          77 :   dec = alg_decompose(al, Z, 1, NULL);
    1574          77 :   set_avma(av);
    1575          77 :   return gequal0(dec);
    1576             : }
    1577             : 
    1578             : static long
    1579         329 : is_place_emb(GEN nf, GEN pl)
    1580             : {
    1581             :   long r, r1, r2;
    1582         329 :   if (typ(pl) != t_INT) pari_err_TYPE("is_place_emb", pl);
    1583         315 :   if (signe(pl)<=0) pari_err_DOMAIN("is_place_emb", "pl", "<=", gen_0, pl);
    1584         308 :   nf_get_sign(nf,&r1,&r2); r = r1+r2;
    1585         308 :   if (cmpiu(pl,r)>0) pari_err_DOMAIN("is_place_emb", "pl", ">", utoi(r), pl);
    1586         294 :   return itou(pl);
    1587             : }
    1588             : 
    1589             : static long
    1590         294 : alghasse_emb(GEN al, long emb)
    1591             : {
    1592         294 :   GEN nf = alg_get_center(al);
    1593         294 :   long r1 = nf_get_r1(nf);
    1594         294 :   return (emb <= r1)? alg_get_hasse_i(al)[emb]: 0;
    1595             : }
    1596             : 
    1597             : static long
    1598         399 : alghasse_pr(GEN al, GEN pr)
    1599             : {
    1600         399 :   GEN hf = alg_get_hasse_f(al);
    1601         399 :   long i = tablesearch(gel(hf,1), pr, &cmp_prime_ideal);
    1602         399 :   return i? gel(hf,2)[i]: 0;
    1603             : }
    1604             : 
    1605             : static long
    1606         763 : alghasse_0(GEN al, GEN pl)
    1607             : {
    1608             :   long ta;
    1609             :   GEN pr, nf;
    1610         763 :   ta = alg_type(al);
    1611         763 :   if (ta == al_REAL) return algreal_dim(al)!=1;
    1612         742 :   if (!pl)
    1613           7 :     pari_err(e_MISC, "must provide a place pl");
    1614         735 :   if (ta == al_CSA)
    1615           7 :     pari_err_IMPL("computation of Hasse invariants over table CSA");
    1616         728 :   if ((pr = get_prid(pl))) return alghasse_pr(al, pr);
    1617         329 :   nf = alg_get_center(al);
    1618         329 :   return alghasse_emb(al, is_place_emb(nf, pl));
    1619             : }
    1620             : GEN
    1621         238 : alghasse(GEN al, GEN pl)
    1622             : {
    1623             :   long h;
    1624         238 :   checkalg(al);
    1625         238 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("alghasse [use alginit]",al);
    1626         231 :   h = alghasse_0(al,pl);
    1627         182 :   return sstoQ(h, alg_get_degree(al));
    1628             : }
    1629             : 
    1630             : /* h >= 0, d >= 0 */
    1631             : static long
    1632         812 : indexfromhasse(long h, long d) { return d/ugcd(h,d); }
    1633             : 
    1634             : long
    1635         812 : algindex(GEN al, GEN pl)
    1636             : {
    1637             :   long d, res, i, l, ta;
    1638             :   GEN hi, hf;
    1639             : 
    1640         812 :   checkalg(al);
    1641         805 :   ta = alg_type(al);
    1642         805 :   if (ta == al_TABLE) pari_err_TYPE("algindex [use alginit]",al);
    1643         798 :   if (ta == al_REAL) return algreal_dim(al)==1 ? 1 : 2;
    1644         714 :   d = alg_get_degree(al);
    1645         714 :   if (pl) return indexfromhasse(alghasse_0(al,pl), d);
    1646             : 
    1647             :   /* else : global index */
    1648         182 :   res = 1;
    1649         182 :   hi = alg_get_hasse_i(al); l = lg(hi);
    1650         308 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hi[i],d));
    1651         182 :   hf = gel(alg_get_hasse_f(al), 2); l = lg(hf);
    1652         336 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hf[i],d));
    1653         182 :   return res;
    1654             : }
    1655             : 
    1656             : int
    1657         224 : algisdivision(GEN al, GEN pl)
    1658             : {
    1659         224 :   checkalg(al);
    1660         224 :   if (alg_type(al) == al_TABLE) {
    1661          21 :     if (!algissimple(al,0)) return 0;
    1662          14 :     if (algiscommutative(al)) return 1;
    1663           7 :     pari_err_IMPL("algisdivision for table algebras");
    1664             :   }
    1665         203 :   return algindex(al,pl) == alg_get_degree(al);
    1666             : }
    1667             : 
    1668             : int
    1669         406 : algissplit(GEN al, GEN pl)
    1670             : {
    1671         406 :   checkalg(al);
    1672         406 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algissplit [use alginit]", al);
    1673         392 :   return algindex(al,pl) == 1;
    1674             : }
    1675             : 
    1676             : int
    1677         203 : algisramified(GEN al, GEN pl) { return !algissplit(al,pl); }
    1678             : 
    1679             : GEN
    1680          98 : algramifiedplaces(GEN al)
    1681             : {
    1682          98 :   pari_sp av = avma;
    1683             :   GEN ram, hf, hi, Lpr;
    1684             :   long r1, count, i, ta;
    1685          98 :   checkalg(al);
    1686          98 :   ta = alg_type(al);
    1687          98 :   if (ta != al_CSA && ta != al_CYCLIC)
    1688          14 :     pari_err_TYPE("algramifiedplaces [not a central simple algebra"
    1689             :         " over a number field]", al);
    1690          84 :   r1 = nf_get_r1(alg_get_center(al));
    1691          84 :   hi = alg_get_hasse_i(al);
    1692          84 :   hf = alg_get_hasse_f(al);
    1693          84 :   Lpr = gel(hf,1);
    1694          84 :   hf = gel(hf,2);
    1695          84 :   ram = cgetg(r1+lg(Lpr), t_VEC);
    1696          84 :   count = 0;
    1697         280 :   for (i=1; i<=r1; i++)
    1698         196 :     if (hi[i]) {
    1699          91 :       count++;
    1700          91 :       gel(ram,count) = stoi(i);
    1701             :     }
    1702         286 :   for (i=1; i<lg(Lpr); i++)
    1703         202 :     if (hf[i]) {
    1704          77 :       count++;
    1705          77 :       gel(ram,count) = gel(Lpr,i);
    1706             :     }
    1707          84 :   setlg(ram, count+1);
    1708          84 :   return gerepilecopy(av, ram);
    1709             : }
    1710             : 
    1711             : /** OPERATIONS ON ELEMENTS operations.c **/
    1712             : 
    1713             : static long
    1714     1037796 : alg_model0(GEN al, GEN x)
    1715             : {
    1716     1037796 :   long t, N = alg_get_absdim(al), lx = lg(x), d, n, D, i;
    1717     1037796 :   if (typ(x) == t_MAT) return al_MATRIX;
    1718      991673 :   if (typ(x) != t_COL) return al_INVALID;
    1719      991610 :   if (N == 1) {
    1720        2842 :     if (lx != 2) return al_INVALID;
    1721        2821 :     switch(typ(gel(x,1)))
    1722             :     {
    1723        1799 :       case t_INT: case t_FRAC: return al_TRIVIAL; /* cannot distinguish basis and alg from size */
    1724        1015 :       case t_POL: case t_POLMOD: return al_ALGEBRAIC;
    1725           7 :       default: return al_INVALID;
    1726             :     }
    1727             :   }
    1728             : 
    1729      988768 :   switch(alg_type(al)) {
    1730      557358 :     case al_TABLE:
    1731      557358 :       if (lx != N+1) return al_INVALID;
    1732      557337 :       return al_BASIS;
    1733      345338 :     case al_CYCLIC:
    1734      345338 :       d = alg_get_degree(al);
    1735      345338 :       if (lx == N+1) return al_BASIS;
    1736       93674 :       if (lx == d+1) return al_ALGEBRAIC;
    1737          14 :       return al_INVALID;
    1738       86072 :     case al_CSA:
    1739       86072 :       D = alg_get_dim(al);
    1740       86072 :       n = nf_get_degree(alg_get_center(al));
    1741       86072 :       if (n == 1) {
    1742        1316 :         if (lx != D+1) return al_INVALID;
    1743        4109 :         for (i=1; i<=D; i++) {
    1744        3409 :           t = typ(gel(x,i));
    1745        3409 :           if (t == t_POL || t == t_POLMOD)  return al_ALGEBRAIC;
    1746             :             /* TODO t_COL for coefficients in basis form ? */
    1747             :         }
    1748         700 :         return al_BASIS;
    1749             :       }
    1750             :       else {
    1751       84756 :         if (lx == N+1) return al_BASIS;
    1752       22582 :         if (lx == D+1) return al_ALGEBRAIC;
    1753           7 :         return al_INVALID;
    1754             :       }
    1755             :   }
    1756             :   return al_INVALID; /* LCOV_EXCL_LINE */
    1757             : }
    1758             : 
    1759             : static void
    1760     1037656 : checkalgx(GEN x, long model)
    1761             : {
    1762             :   long t, i;
    1763     1037656 :   switch(model) {
    1764      871875 :     case al_BASIS:
    1765     9085260 :       for (i=1; i<lg(x); i++) {
    1766     8213392 :         t = typ(gel(x,i));
    1767     8213392 :         if (t != t_INT && t != t_FRAC)
    1768           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1769             :       }
    1770      871868 :       return;
    1771      119658 :     case al_TRIVIAL:
    1772             :     case al_ALGEBRAIC:
    1773      405601 :       for (i=1; i<lg(x); i++) {
    1774      285950 :         t = typ(gel(x,i));
    1775      285950 :         if (t != t_INT && t != t_FRAC && t != t_POL && t != t_POLMOD)
    1776             :           /* TODO t_COL ? */
    1777           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1778             :       }
    1779      119651 :       return;
    1780             :   }
    1781             : }
    1782             : 
    1783             : long
    1784     1037796 : alg_model(GEN al, GEN x)
    1785             : {
    1786     1037796 :   long res = alg_model0(al, x);
    1787     1037796 :   if (res == al_INVALID) pari_err_TYPE("alg_model", x);
    1788     1037656 :   checkalgx(x, res); return res;
    1789             : }
    1790             : 
    1791             : static long
    1792      462630 : H_model0(GEN x)
    1793             : {
    1794             :   long i;
    1795      462630 :   switch(typ(x))
    1796             :   {
    1797       15218 :     case t_INT:
    1798             :     case t_FRAC:
    1799             :     case t_REAL:
    1800             :     case t_COMPLEX:
    1801       15218 :       return H_SCALAR;
    1802       10157 :     case t_MAT:
    1803       10157 :       return H_MATRIX;
    1804      437143 :     case t_COL:
    1805      437143 :       if (lg(x)!=5) return H_INVALID;
    1806     2185603 :       for (i=1; i<=4; i++) if (!is_real_t(typ(gel(x,i)))) return H_INVALID;
    1807      437115 :       return H_QUATERNION;
    1808         112 :     default:
    1809         112 :       return al_INVALID;
    1810             :   }
    1811             : }
    1812             : 
    1813             : static long
    1814      462630 : H_model(GEN x)
    1815             : {
    1816      462630 :   long res = H_model0(x);
    1817      462630 :   if (res == H_INVALID) pari_err_TYPE("H_model", x);
    1818      462490 :   return res;
    1819             : }
    1820             : 
    1821             : static GEN
    1822         756 : alC_add_i(GEN al, GEN x, GEN y, long lx)
    1823             : {
    1824         756 :   GEN A = cgetg(lx, t_COL);
    1825             :   long i;
    1826        2296 :   for (i=1; i<lx; i++) gel(A,i) = algadd(al, gel(x,i), gel(y,i));
    1827         749 :   return A;
    1828             : }
    1829             : static GEN
    1830         406 : alM_add(GEN al, GEN x, GEN y)
    1831             : {
    1832         406 :   long lx = lg(x), l, j;
    1833             :   GEN z;
    1834         406 :   if (lg(y) != lx) pari_err_DIM("alM_add (rows)");
    1835         392 :   if (lx == 1) return cgetg(1, t_MAT);
    1836         385 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1837         385 :   if (lgcols(y) != l) pari_err_DIM("alM_add (columns)");
    1838        1127 :   for (j = 1; j < lx; j++) gel(z,j) = alC_add_i(al, gel(x,j), gel(y,j), l);
    1839         371 :   return z;
    1840             : }
    1841             : static GEN
    1842       17745 : H_add(GEN x, GEN y)
    1843             : {
    1844       17745 :   long tx = H_model(x), ty = H_model(y);
    1845       17724 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_add", x, y);
    1846       17710 :   if (tx>ty) { swap(x,y); lswap(tx,ty); }
    1847       17710 :   switch (tx)
    1848             :   {
    1849         105 :     case H_MATRIX: /* both H_MATRIX */ return alM_add(NULL, x, y);
    1850       16681 :     case H_QUATERNION: /* both H_QUATERNION */ return gadd(x,y);
    1851         924 :     case H_SCALAR:
    1852         924 :       if (ty == H_SCALAR) return gadd(x,y);
    1853             :       else /* ty == H_QUATERNION */
    1854             :       {
    1855         217 :         pari_sp av = avma;
    1856         217 :         GEN res = gcopy(y), im;
    1857         217 :         gel(res,1) = gadd(gel(res,1), real_i(x));
    1858         217 :         im = imag_i(x);
    1859         217 :         if (im != gen_0) gel(res,2) = gadd(gel(res,2), im);
    1860         217 :         return gerepileupto(av, res);
    1861             :       }
    1862             :   }
    1863             :   return NULL; /*LCOV_EXCL_LINE*/
    1864             : }
    1865             : GEN
    1866       54845 : algadd(GEN al, GEN x, GEN y)
    1867             : {
    1868       54845 :   pari_sp av = avma;
    1869             :   long tx, ty;
    1870             :   GEN p;
    1871       54845 :   checkalg(al);
    1872       54845 :   if (alg_type(al)==al_REAL) return H_add(x,y);
    1873       37100 :   tx = alg_model(al,x);
    1874       37093 :   ty = alg_model(al,y);
    1875       37093 :   p = alg_get_char(al);
    1876       37093 :   if (signe(p)) return FpC_add(x,y,p);
    1877       36960 :   if (tx==ty) {
    1878       36078 :     if (tx!=al_MATRIX) return gadd(x,y);
    1879         301 :     return gerepilecopy(av, alM_add(al,x,y));
    1880             :   }
    1881         882 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1882         882 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1883         882 :   return gerepileupto(av, gadd(x,y));
    1884             : }
    1885             : 
    1886             : static GEN
    1887          98 : H_neg(GEN x)
    1888             : {
    1889          98 :   (void)H_model(x);
    1890          70 :   return gneg(x);
    1891             : }
    1892             : 
    1893             : GEN
    1894         245 : algneg(GEN al, GEN x)
    1895             : {
    1896         245 :   checkalg(al);
    1897         245 :   if (alg_type(al)==al_REAL) return H_neg(x);
    1898         147 :   (void)alg_model(al,x);
    1899         140 :   return gneg(x);
    1900             : }
    1901             : 
    1902             : static GEN
    1903         210 : alC_sub_i(GEN al, GEN x, GEN y, long lx)
    1904             : {
    1905             :   long i;
    1906         210 :   GEN A = cgetg(lx, t_COL);
    1907         630 :   for (i=1; i<lx; i++) gel(A,i) = algsub(al, gel(x,i), gel(y,i));
    1908         210 :   return A;
    1909             : }
    1910             : static GEN
    1911         126 : alM_sub(GEN al, GEN x, GEN y)
    1912             : {
    1913         126 :   long lx = lg(x), l, j;
    1914             :   GEN z;
    1915         126 :   if (lg(y) != lx) pari_err_DIM("alM_sub (rows)");
    1916         119 :   if (lx == 1) return cgetg(1, t_MAT);
    1917         112 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1918         112 :   if (lgcols(y) != l) pari_err_DIM("alM_sub (columns)");
    1919         315 :   for (j = 1; j < lx; j++) gel(z,j) = alC_sub_i(al, gel(x,j), gel(y,j), l);
    1920         105 :   return z;
    1921             : }
    1922             : GEN
    1923        1120 : algsub(GEN al, GEN x, GEN y)
    1924             : {
    1925             :   long tx, ty;
    1926        1120 :   pari_sp av = avma;
    1927             :   GEN p;
    1928        1120 :   checkalg(al);
    1929        1120 :   if (alg_type(al)==al_REAL) return gerepileupto(av, algadd(NULL,x,gneg(y)));
    1930         966 :   tx = alg_model(al,x);
    1931         959 :   ty = alg_model(al,y);
    1932         959 :   p = alg_get_char(al);
    1933         959 :   if (signe(p)) return FpC_sub(x,y,p);
    1934         868 :   if (tx==ty) {
    1935         546 :     if (tx != al_MATRIX) return gsub(x,y);
    1936         126 :     return gerepilecopy(av, alM_sub(al,x,y));
    1937             :   }
    1938         322 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1939         322 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1940         322 :   return gerepileupto(av, gsub(x,y));
    1941             : }
    1942             : 
    1943             : static GEN
    1944        1659 : algalgmul_cyc(GEN al, GEN x, GEN y)
    1945             : {
    1946        1659 :   pari_sp av = avma;
    1947        1659 :   long n = alg_get_degree(al), i, k;
    1948             :   GEN xalg, yalg, res, rnf, auts, sum, b, prod, autx;
    1949        1659 :   rnf = alg_get_splittingfield(al);
    1950        1659 :   auts = alg_get_auts(al);
    1951        1659 :   b = alg_get_b(al);
    1952             : 
    1953        1659 :   xalg = cgetg(n+1, t_COL);
    1954        4935 :   for (i=0; i<n; i++)
    1955        3276 :     gel(xalg,i+1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    1956             : 
    1957        1659 :   yalg = cgetg(n+1, t_COL);
    1958        4935 :   for (i=0; i<n; i++) gel(yalg,i+1) = rnfbasistoalg(rnf,gel(y,i+1));
    1959             : 
    1960        1659 :   res = cgetg(n+1,t_COL);
    1961        4935 :   for (k=0; k<n; k++) {
    1962        3276 :     gel(res,k+1) = gmul(gel(xalg,k+1),gel(yalg,1));
    1963        5166 :     for (i=1; i<=k; i++) {
    1964        1890 :       autx = poleval(gel(xalg,k-i+1),gel(auts,i));
    1965        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1966        1890 :       gel(res,k+1) = gadd(gel(res,k+1), prod);
    1967             :     }
    1968             : 
    1969        3276 :     sum = gen_0;
    1970        5166 :     for (; i<n; i++) {
    1971        1890 :       autx = poleval(gel(xalg,k+n-i+1),gel(auts,i));
    1972        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1973        1890 :       sum = gadd(sum,prod);
    1974             :     }
    1975        3276 :     sum = gmul(b,sum);
    1976             : 
    1977        3276 :     gel(res,k+1) = gadd(gel(res,k+1),sum);
    1978             :   }
    1979             : 
    1980        1659 :   return gerepilecopy(av, res);
    1981             : }
    1982             : 
    1983             : static GEN
    1984      205394 : _tablemul(GEN mt, GEN x, GEN y)
    1985             : {
    1986      205394 :   pari_sp av = avma;
    1987      205394 :   long D = lg(mt)-1, i;
    1988      205394 :   GEN res = NULL;
    1989     1930922 :   for (i=1; i<=D; i++) {
    1990     1725528 :     GEN c = gel(x,i);
    1991     1725528 :     if (!gequal0(c)) {
    1992      990444 :       GEN My = RgM_RgC_mul(gel(mt,i),y);
    1993      990444 :       GEN t = RgC_Rg_mul(My,c);
    1994      990444 :       res = res? RgC_add(res,t): t;
    1995             :     }
    1996             :   }
    1997      205394 :   if (!res) { set_avma(av); return zerocol(D); }
    1998      204491 :   return gerepileupto(av, res);
    1999             : }
    2000             : 
    2001             : static GEN
    2002      192758 : _tablemul_Fp(GEN mt, GEN x, GEN y, GEN p)
    2003             : {
    2004      192758 :   pari_sp av = avma;
    2005      192758 :   long D = lg(mt)-1, i;
    2006      192758 :   GEN res = NULL;
    2007     2254758 :   for (i=1; i<=D; i++) {
    2008     2062000 :     GEN c = gel(x,i);
    2009     2062000 :     if (signe(c)) {
    2010      329401 :       GEN My = FpM_FpC_mul(gel(mt,i),y,p);
    2011      329401 :       GEN t = FpC_Fp_mul(My,c,p);
    2012      329401 :       res = res? FpC_add(res,t,p): t;
    2013             :     }
    2014             :   }
    2015      192758 :   if (!res) { set_avma(av); return zerocol(D); }
    2016      192219 :   return gerepileupto(av, res);
    2017             : }
    2018             : 
    2019             : /* x*ej */
    2020             : static GEN
    2021       99904 : _tablemul_ej(GEN mt, GEN x, long j)
    2022             : {
    2023       99904 :   pari_sp av = avma;
    2024       99904 :   long D = lg(mt)-1, i;
    2025       99904 :   GEN res = NULL;
    2026     1563793 :   for (i=1; i<=D; i++) {
    2027     1463889 :     GEN c = gel(x,i);
    2028     1463889 :     if (!gequal0(c)) {
    2029      114415 :       GEN My = gel(gel(mt,i),j);
    2030      114415 :       GEN t = RgC_Rg_mul(My,c);
    2031      114415 :       res = res? RgC_add(res,t): t;
    2032             :     }
    2033             :   }
    2034       99904 :   if (!res) { set_avma(av); return zerocol(D); }
    2035       99764 :   return gerepileupto(av, res);
    2036             : }
    2037             : static GEN
    2038      242039 : _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p)
    2039             : {
    2040      242039 :   pari_sp av = avma;
    2041      242039 :   long D = lg(mt)-1, i;
    2042      242039 :   GEN res = NULL;
    2043     4364787 :   for (i=1; i<=D; i++) {
    2044     4122748 :     GEN c = gel(x,i);
    2045     4122748 :     if (!gequal0(c)) {
    2046      289954 :       GEN My = gel(gel(mt,i),j);
    2047      289954 :       GEN t = FpC_Fp_mul(My,c,p);
    2048      289954 :       res = res? FpC_add(res,t,p): t;
    2049             :     }
    2050             :   }
    2051      242039 :   if (!res) { set_avma(av); return zerocol(D); }
    2052      241927 :   return gerepileupto(av, res);
    2053             : }
    2054             : 
    2055             : static GEN
    2056      244362 : _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p)
    2057             : {
    2058      244362 :   pari_sp av = avma;
    2059      244362 :   long D = lg(mt)-1, i;
    2060      244362 :   GEN res = NULL;
    2061     3944134 :   for (i=1; i<=D; i++) {
    2062     3699772 :     ulong c = x[i];
    2063     3699772 :     if (c) {
    2064      384594 :       GEN My = gel(gel(mt,i),j);
    2065      384594 :       GEN t = Flv_Fl_mul(My,c, p);
    2066      384594 :       res = res? Flv_add(res,t, p): t;
    2067             :     }
    2068             :   }
    2069      244362 :   if (!res) { set_avma(av); return zero_Flv(D); }
    2070      244362 :   return gerepileupto(av, res);
    2071             : }
    2072             : 
    2073             : static GEN
    2074         686 : algalgmul_csa(GEN al, GEN x, GEN y)
    2075             : {
    2076         686 :   GEN z, nf = alg_get_center(al);
    2077             :   long i;
    2078         686 :   z = _tablemul(alg_get_relmultable(al), x, y);
    2079        2485 :   for (i=1; i<lg(z); i++)
    2080        1799 :     gel(z,i) = basistoalg(nf,gel(z,i));
    2081         686 :   return z;
    2082             : }
    2083             : 
    2084             : /* assumes x and y in algebraic form */
    2085             : static GEN
    2086        2345 : algalgmul(GEN al, GEN x, GEN y)
    2087             : {
    2088        2345 :   switch(alg_type(al))
    2089             :   {
    2090        1659 :     case al_CYCLIC: return algalgmul_cyc(al, x, y);
    2091         686 :     case al_CSA: return algalgmul_csa(al, x, y);
    2092             :   }
    2093             :   return NULL; /*LCOV_EXCL_LINE*/
    2094             : }
    2095             : 
    2096             : static GEN
    2097      397466 : algbasismul(GEN al, GEN x, GEN y)
    2098             : {
    2099      397466 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    2100      397466 :   if (signe(p)) return _tablemul_Fp(mt, x, y, p);
    2101      204708 :   return _tablemul(mt, x, y);
    2102             : }
    2103             : 
    2104             : /* x[i,]*y. Assume lg(x) > 1 and 0 < i < lgcols(x) */
    2105             : static GEN
    2106      119651 : alMrow_alC_mul_i(GEN al, GEN x, GEN y, long i, long lx)
    2107             : {
    2108      119651 :   pari_sp av = avma;
    2109      119651 :   GEN c = algmul(al,gcoeff(x,i,1),gel(y,1)), ZERO;
    2110             :   long k;
    2111      119651 :   ZERO = zerocol(alg_get_absdim(al));
    2112      273308 :   for (k = 2; k < lx; k++)
    2113             :   {
    2114      153657 :     GEN t = algmul(al, gcoeff(x,i,k), gel(y,k));
    2115      153657 :     if (!gequal(t,ZERO)) c = algadd(al, c, t);
    2116             :   }
    2117      119651 :   return gerepilecopy(av, c);
    2118             : }
    2119             : /* return x * y, 1 < lx = lg(x), l = lgcols(x) */
    2120             : static GEN
    2121       54502 : alM_alC_mul_i(GEN al, GEN x, GEN y, long lx, long l)
    2122             : {
    2123       54502 :   GEN z = cgetg(l,t_COL);
    2124             :   long i;
    2125      174153 :   for (i=1; i<l; i++) gel(z,i) = alMrow_alC_mul_i(al,x,y,i,lx);
    2126       54502 :   return z;
    2127             : }
    2128             : static GEN
    2129       25627 : alM_mul(GEN al, GEN x, GEN y)
    2130             : {
    2131       25627 :   long j, l, lx=lg(x), ly=lg(y);
    2132             :   GEN z;
    2133       25627 :   if (ly==1) return cgetg(1,t_MAT);
    2134       25529 :   if (lx != lgcols(y)) pari_err_DIM("alM_mul");
    2135       25508 :   if (lx==1) return zeromat(0, ly-1);
    2136       25501 :   l = lgcols(x); z = cgetg(ly,t_MAT);
    2137       80003 :   for (j=1; j<ly; j++) gel(z,j) = alM_alC_mul_i(al,x,gel(y,j),lx,l);
    2138       25501 :   return z;
    2139             : }
    2140             : 
    2141             : static void
    2142      205583 : H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d)
    2143             : {
    2144      205583 :   switch(H_model(x))
    2145             :   {
    2146        5173 :     case H_SCALAR:
    2147        5173 :       *a = real_i(x);
    2148        5173 :       *b = imag_i(x);
    2149        5173 :       *c = gen_0;
    2150        5173 :       *d = gen_0;
    2151        5173 :       return;
    2152      200410 :     case H_QUATERNION:
    2153      200410 :       *a = gel(x,1);
    2154      200410 :       *b = gel(x,2);
    2155      200410 :       *c = gel(x,3);
    2156      200410 :       *d = gel(x,4);
    2157      200410 :       return;
    2158             :     default: *a = *b = *c = *d = NULL; return; /*LCOV_EXCL_LINE*/
    2159             :   }
    2160             : }
    2161             : static GEN
    2162      108101 : H_mul(GEN x, GEN y)
    2163             : {
    2164      108101 :   pari_sp av = avma;
    2165             :   GEN a,b,c,d,u,v,w,z;
    2166      108101 :   long tx = H_model(x), ty = H_model(y);
    2167      108087 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_mul", x, y);
    2168      108080 :   if (tx == H_MATRIX) /* both H_MATRIX */ return alM_mul(NULL, x, y);
    2169      103789 :   if (tx == H_SCALAR && ty == H_SCALAR) return gmul(x,y);
    2170      102592 :   H_compo(x,&a,&b,&c,&d);
    2171      102592 :   H_compo(y,&u,&v,&w,&z);
    2172      102592 :   return gerepilecopy(av,mkcol4(
    2173             :         gsub(gmul(a,u), gadd(gadd(gmul(b,v),gmul(c,w)),gmul(d,z))),
    2174             :         gsub(gadd(gmul(a,v),gadd(gmul(b,u),gmul(c,z))), gmul(d,w)),
    2175             :         gsub(gadd(gmul(a,w),gadd(gmul(c,u),gmul(d,v))), gmul(b,z)),
    2176             :         gsub(gadd(gmul(a,z),gadd(gmul(b,w),gmul(d,u))), gmul(c,v))
    2177             :         ));
    2178             : }
    2179             : 
    2180             : GEN
    2181      475676 : algmul(GEN al, GEN x, GEN y)
    2182             : {
    2183      475676 :   pari_sp av = avma;
    2184             :   long tx, ty;
    2185      475676 :   checkalg(al);
    2186      475676 :   if (alg_type(al)==al_REAL) return H_mul(x,y);
    2187      367827 :   tx = alg_model(al,x);
    2188      367813 :   ty = alg_model(al,y);
    2189      367813 :   if (tx==al_MATRIX) {
    2190       20832 :     if (ty==al_MATRIX) return alM_mul(al,x,y);
    2191           7 :     pari_err_TYPE("algmul", y);
    2192             :   }
    2193      346981 :   if (signe(alg_get_char(al))) return algbasismul(al,x,y);
    2194      205135 :   if (tx==al_TRIVIAL) retmkcol(gmul(gel(x,1),gel(y,1)));
    2195      205030 :   if (tx==al_ALGEBRAIC && ty==al_ALGEBRAIC) return algalgmul(al,x,y);
    2196      203504 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2197      203504 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    2198      203504 :   return gerepileupto(av,algbasismul(al,x,y));
    2199             : }
    2200             : 
    2201             : static GEN
    2202         329 : H_sqr(GEN x)
    2203             : {
    2204         329 :   pari_sp av = avma;
    2205         329 :   long tx = H_model(x);
    2206             :   GEN a,b,c,d;
    2207         308 :   if (tx == H_SCALAR) return gsqr(x);
    2208         224 :   if (tx == H_MATRIX) return H_mul(x,x);
    2209         119 :   H_compo(x,&a,&b,&c,&d);
    2210         119 :   return gerepilecopy(av, mkcol4(
    2211             :         gsub(gsqr(a), gadd(gsqr(b),gadd(gsqr(c),gsqr(d)))),
    2212             :         gshift(gmul(a,b),1),
    2213             :         gshift(gmul(a,c),1),
    2214             :         gshift(gmul(a,d),1)
    2215             :         ));
    2216             : }
    2217             : 
    2218             : GEN
    2219       51031 : algsqr(GEN al, GEN x)
    2220             : {
    2221       51031 :   pari_sp av = avma;
    2222             :   long tx;
    2223       51031 :   checkalg(al);
    2224       50996 :   if (alg_type(al)==al_REAL) return H_sqr(x);
    2225       50667 :   tx = alg_model(al,x);
    2226       50597 :   if (tx==al_MATRIX) return gerepilecopy(av,alM_mul(al,x,x));
    2227       50086 :   if (signe(alg_get_char(al))) return algbasismul(al,x,x);
    2228        2205 :   if (tx==al_TRIVIAL) retmkcol(gsqr(gel(x,1)));
    2229        2023 :   if (tx==al_ALGEBRAIC) return algalgmul(al,x,x);
    2230        1204 :   return gerepileupto(av,algbasismul(al,x,x));
    2231             : }
    2232             : 
    2233             : static GEN
    2234        8946 : algmtK2Z_cyc(GEN al, GEN m)
    2235             : {
    2236        8946 :   pari_sp av = avma;
    2237        8946 :   GEN nf = alg_get_abssplitting(al), res, mt, rnf = alg_get_splittingfield(al), c, dc;
    2238        8946 :   long n = alg_get_degree(al), N = nf_get_degree(nf), Nn, i, j, i1, j1;
    2239        8946 :   Nn = N*n;
    2240        8946 :   res = zeromatcopy(Nn,Nn);
    2241       40936 :   for (i=0; i<n; i++)
    2242      192808 :   for (j=0; j<n; j++) {
    2243      160818 :     c = gcoeff(m,i+1,j+1);
    2244      160818 :     if (!gequal0(c)) {
    2245       31990 :       c = rnfeltreltoabs(rnf,c);
    2246       31990 :       c = algtobasis(nf,c);
    2247       31990 :       c = Q_remove_denom(c,&dc);
    2248       31990 :       mt = zk_multable(nf,c);
    2249       31990 :       if (dc) mt = ZM_Z_div(mt,dc);
    2250      297220 :       for (i1=1; i1<=N; i1++)
    2251     2923228 :       for (j1=1; j1<=N; j1++)
    2252     2657998 :         gcoeff(res,i*N+i1,j*N+j1) = gcoeff(mt,i1,j1);
    2253             :     }
    2254             :   }
    2255        8946 :   return gerepilecopy(av,res);
    2256             : }
    2257             : 
    2258             : static GEN
    2259         917 : algmtK2Z_csa(GEN al, GEN m)
    2260             : {
    2261         917 :   pari_sp av = avma;
    2262         917 :   GEN nf = alg_get_center(al), res, mt, c, dc;
    2263         917 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), D, i, j, i1, j1;
    2264         917 :   D = d2*n;
    2265         917 :   res = zeromatcopy(D,D);
    2266        5362 :   for (i=0; i<d2; i++)
    2267       30562 :   for (j=0; j<d2; j++) {
    2268       26117 :     c = gcoeff(m,i+1,j+1);
    2269       26117 :     if (!gequal0(c)) {
    2270        3794 :       c = algtobasis(nf,c);
    2271        3794 :       c = Q_remove_denom(c,&dc);
    2272        3794 :       mt = zk_multable(nf,c);
    2273        3794 :       if (dc) mt = ZM_Z_div(mt,dc);
    2274       12852 :       for (i1=1; i1<=n; i1++)
    2275       32340 :       for (j1=1; j1<=n; j1++)
    2276       23282 :         gcoeff(res,i*n+i1,j*n+j1) = gcoeff(mt,i1,j1);
    2277             :     }
    2278             :   }
    2279         917 :   return gerepilecopy(av,res);
    2280             : }
    2281             : 
    2282             : /* assumes al is a CSA or CYCLIC */
    2283             : static GEN
    2284        9863 : algmtK2Z(GEN al, GEN m)
    2285             : {
    2286        9863 :   switch(alg_type(al))
    2287             :   {
    2288        8946 :     case al_CYCLIC: return algmtK2Z_cyc(al, m);
    2289         917 :     case al_CSA: return algmtK2Z_csa(al, m);
    2290             :   }
    2291             :   return NULL; /*LCOV_EXCL_LINE*/
    2292             : }
    2293             : 
    2294             : /* left multiplication table, as a vector space of dimension n over the splitting field (by right multiplication) */
    2295             : static GEN
    2296       11564 : algalgmultable_cyc(GEN al, GEN x)
    2297             : {
    2298       11564 :   pari_sp av = avma;
    2299       11564 :   long n = alg_get_degree(al), i, j;
    2300             :   GEN res, rnf, auts, b, pol;
    2301       11564 :   rnf = alg_get_splittingfield(al);
    2302       11564 :   auts = alg_get_auts(al);
    2303       11564 :   b = alg_get_b(al);
    2304       11564 :   pol = rnf_get_pol(rnf);
    2305             : 
    2306       11564 :   res = zeromatcopy(n,n);
    2307       48860 :   for (i=0; i<n; i++)
    2308       37296 :     gcoeff(res,i+1,1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    2309             : 
    2310       48860 :   for (i=0; i<n; i++) {
    2311      104706 :     for (j=1; j<=i; j++)
    2312       67410 :       gcoeff(res,i+1,j+1) = gmodulo(poleval(gcoeff(res,i-j+1,1),gel(auts,j)),pol);
    2313      104706 :     for (; j<n; j++)
    2314       67410 :       gcoeff(res,i+1,j+1) = gmodulo(gmul(b,poleval(gcoeff(res,n+i-j+1,1),gel(auts,j))), pol);
    2315             :   }
    2316             : 
    2317       48860 :   for (i=0; i<n; i++)
    2318       37296 :     gcoeff(res,i+1,1) = gmodulo(gcoeff(res,i+1,1),pol);
    2319             : 
    2320       11564 :   return gerepilecopy(av, res);
    2321             : }
    2322             : 
    2323             : static GEN
    2324        1365 : elementmultable(GEN mt, GEN x)
    2325             : {
    2326        1365 :   pari_sp av = avma;
    2327        1365 :   long D = lg(mt)-1, i;
    2328        1365 :   GEN z = NULL;
    2329        7308 :   for (i=1; i<=D; i++)
    2330             :   {
    2331        5943 :     GEN c = gel(x,i);
    2332        5943 :     if (!gequal0(c))
    2333             :     {
    2334        2135 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
    2335        2135 :       z = z? RgM_add(z, M): M;
    2336             :     }
    2337             :   }
    2338        1365 :   if (!z) { set_avma(av); return zeromatcopy(D,D); }
    2339        1365 :   return gerepileupto(av, z);
    2340             : }
    2341             : /* mt a t_VEC of Flm modulo m */
    2342             : static GEN
    2343       24639 : algbasismultable_Flm(GEN mt, GEN x, ulong m)
    2344             : {
    2345       24639 :   pari_sp av = avma;
    2346       24639 :   long D = lg(gel(mt,1))-1, i;
    2347       24639 :   GEN z = NULL;
    2348      269001 :   for (i=1; i<=D; i++)
    2349             :   {
    2350      244362 :     ulong c = x[i];
    2351      244362 :     if (c)
    2352             :     {
    2353       33625 :       GEN M = Flm_Fl_mul(gel(mt,i),c, m);
    2354       33625 :       z = z? Flm_add(z, M, m): M;
    2355             :     }
    2356             :   }
    2357       24639 :   if (!z) { set_avma(av); return zero_Flm(D,D); }
    2358       24639 :   return gerepileupto(av, z);
    2359             : }
    2360             : static GEN
    2361      226640 : elementabsmultable_Z(GEN mt, GEN x)
    2362             : {
    2363      226640 :   long i, l = lg(x);
    2364      226640 :   GEN z = NULL;
    2365     2410471 :   for (i = 1; i < l; i++)
    2366             :   {
    2367     2183831 :     GEN c = gel(x,i);
    2368     2183831 :     if (signe(c))
    2369             :     {
    2370      852014 :       GEN M = ZM_Z_mul(gel(mt,i),c);
    2371      852014 :       z = z? ZM_add(z, M): M;
    2372             :     }
    2373             :   }
    2374      226640 :   return z;
    2375             : }
    2376             : static GEN
    2377      115040 : elementabsmultable(GEN mt, GEN x)
    2378             : {
    2379      115040 :   GEN d, z = elementabsmultable_Z(mt, Q_remove_denom(x,&d));
    2380      115040 :   return (z && d)? ZM_Z_div(z, d): z;
    2381             : }
    2382             : static GEN
    2383      111600 : elementabsmultable_Fp(GEN mt, GEN x, GEN p)
    2384             : {
    2385      111600 :   GEN z = elementabsmultable_Z(mt, x);
    2386      111600 :   return z? FpM_red(z, p): z;
    2387             : }
    2388             : static GEN
    2389      226640 : algbasismultable(GEN al, GEN x)
    2390             : {
    2391      226640 :   pari_sp av = avma;
    2392      226640 :   GEN z, p = alg_get_char(al), mt = alg_get_multable(al);
    2393      226640 :   z = signe(p)? elementabsmultable_Fp(mt, x, p): elementabsmultable(mt, x);
    2394      226640 :   if (!z)
    2395             :   {
    2396         710 :     long D = lg(mt)-1;
    2397         710 :     set_avma(av); return zeromat(D,D);
    2398             :   }
    2399      225930 :   return gerepileupto(av, z);
    2400             : }
    2401             : 
    2402             : static GEN
    2403        1365 : algalgmultable_csa(GEN al, GEN x)
    2404             : {
    2405        1365 :   GEN nf = alg_get_center(al), m;
    2406             :   long i,j;
    2407        1365 :   m = elementmultable(alg_get_relmultable(al), x);
    2408        7308 :   for (i=1; i<lg(m); i++)
    2409       37758 :     for(j=1; j<lg(m); j++)
    2410       31815 :       gcoeff(m,i,j) = basistoalg(nf,gcoeff(m,i,j));
    2411        1365 :   return m;
    2412             : }
    2413             : 
    2414             : /* assumes x in algebraic form */
    2415             : static GEN
    2416       12635 : algalgmultable(GEN al, GEN x)
    2417             : {
    2418       12635 :   switch(alg_type(al))
    2419             :   {
    2420       11564 :     case al_CYCLIC: return algalgmultable_cyc(al, x);
    2421        1071 :     case al_CSA: return algalgmultable_csa(al, x);
    2422             :   }
    2423             :   return NULL; /*LCOV_EXCL_LINE*/
    2424             : }
    2425             : 
    2426             : /* on the natural basis */
    2427             : /* assumes x in algebraic form */
    2428             : static GEN
    2429        9863 : algZmultable(GEN al, GEN x) {
    2430        9863 :   pari_sp av = avma;
    2431        9863 :   return gerepileupto(av, algmtK2Z(al,algalgmultable(al,x)));
    2432             : }
    2433             : 
    2434             : /* x integral */
    2435             : static GEN
    2436       36568 : algbasisrightmultable(GEN al, GEN x)
    2437             : {
    2438       36568 :   long N = alg_get_absdim(al), i,j,k;
    2439       36568 :   GEN res = zeromatcopy(N,N), c, mt = alg_get_multable(al), p = alg_get_char(al);
    2440       36568 :   if (gequal0(p)) p = NULL;
    2441      330925 :   for (i=1; i<=N; i++) {
    2442      294357 :     c = gel(x,i);
    2443      294357 :     if (!gequal0(c)) {
    2444      872263 :       for (j=1; j<=N; j++)
    2445     7418194 :       for(k=1; k<=N; k++) {
    2446     6640130 :         if (p) gcoeff(res,k,j) = Fp_add(gcoeff(res,k,j), Fp_mul(c, gcoeff(gel(mt,j),k,i), p), p);
    2447     5015262 :         else gcoeff(res,k,j) = addii(gcoeff(res,k,j), mulii(c, gcoeff(gel(mt,j),k,i)));
    2448             :       }
    2449             :     }
    2450             :   }
    2451       36568 :   return res;
    2452             : }
    2453             : 
    2454             : /* basis for matrices : 1, E_{i,j} for (i,j)!=(1,1) */
    2455             : /* index : ijk = ((i-1)*N+j-1)*n + k */
    2456             : /* square matrices only, coefficients in basis form, shallow function */
    2457             : static GEN
    2458       23961 : algmat2basis(GEN al, GEN M)
    2459             : {
    2460       23961 :   long n = alg_get_absdim(al), N = lg(M)-1, i, j, k, ij, ijk;
    2461             :   GEN res, x;
    2462       23961 :   res = zerocol(N*N*n);
    2463       75131 :   for (i=1; i<=N; i++) {
    2464      163310 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2465      112140 :       x = gcoeff(M,i,j);
    2466      819532 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2467      707392 :         gel(res, ijk) = gel(x, k);
    2468      707392 :         if (i>1 && i==j) gel(res, ijk) = gsub(gel(res,ijk), gel(res,k));
    2469             :       }
    2470             :     }
    2471             :   }
    2472             : 
    2473       23961 :   return res;
    2474             : }
    2475             : 
    2476             : static GEN
    2477         294 : algbasis2mat(GEN al, GEN M, long N)
    2478             : {
    2479         294 :   long n = alg_get_absdim(al), i, j, k, ij, ijk;
    2480             :   GEN res, x;
    2481         294 :   res = zeromatcopy(N,N);
    2482         882 :   for (i=1; i<=N; i++)
    2483        1764 :   for (j=1; j<=N; j++)
    2484        1176 :     gcoeff(res,i,j) = zerocol(n);
    2485             : 
    2486         882 :   for (i=1; i<=N; i++) {
    2487        1764 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2488        1176 :       x = gcoeff(res,i,j);
    2489        9240 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2490        8064 :         gel(x,k) = gel(M,ijk);
    2491        8064 :         if (i>1 && i==j) gel(x,k) = gadd(gel(x,k), gel(M,k));
    2492             :       }
    2493             :     }
    2494             :   }
    2495             : 
    2496         294 :   return res;
    2497             : }
    2498             : 
    2499             : static GEN
    2500       23884 : algmatbasis_ei(GEN al, long ijk, long N)
    2501             : {
    2502       23884 :   long n = alg_get_absdim(al), i, j, k, ij;
    2503             :   GEN res;
    2504             : 
    2505       23884 :   res = zeromatcopy(N,N);
    2506       74900 :   for (i=1; i<=N; i++)
    2507      162848 :   for (j=1; j<=N; j++)
    2508      111832 :     gcoeff(res,i,j) = zerocol(n);
    2509             : 
    2510       23884 :   k = ijk%n;
    2511       23884 :   if (k==0) k=n;
    2512       23884 :   ij = (ijk-k)/n+1;
    2513             : 
    2514       23884 :   if (ij==1) {
    2515       16947 :     for (i=1; i<=N; i++)
    2516       11410 :       gcoeff(res,i,i) = col_ei(n,k);
    2517        5537 :     return res;
    2518             :   }
    2519             : 
    2520       18347 :   j = ij%N;
    2521       18347 :   if (j==0) j=N;
    2522       18347 :   i = (ij-j)/N+1;
    2523             : 
    2524       18347 :   gcoeff(res,i,j) = col_ei(n,k);
    2525       18347 :   return res;
    2526             : }
    2527             : 
    2528             : /* FIXME lazy implementation! */
    2529             : static GEN
    2530         910 : algleftmultable_mat(GEN al, GEN M)
    2531             : {
    2532         910 :   long N = lg(M)-1, n = alg_get_absdim(al), D = N*N*n, j;
    2533             :   GEN res, x, Mx;
    2534         910 :   if (N == 0) return cgetg(1, t_MAT);
    2535         903 :   if (N != nbrows(M)) pari_err_DIM("algleftmultable_mat (nonsquare)");
    2536         882 :   res = cgetg(D+1, t_MAT);
    2537       24766 :   for (j=1; j<=D; j++) {
    2538       23884 :     x = algmatbasis_ei(al, j, N);
    2539       23884 :     Mx = algmul(al, M, x);
    2540       23884 :     gel(res, j) = algmat2basis(al, Mx);
    2541             :   }
    2542         882 :   return res;
    2543             : }
    2544             : 
    2545             : /* left multiplication table on integral basis */
    2546             : static GEN
    2547        6951 : algleftmultable(GEN al, GEN x)
    2548             : {
    2549        6951 :   pari_sp av = avma;
    2550             :   long tx;
    2551             :   GEN res;
    2552             : 
    2553        6951 :   checkalg(al);
    2554        6951 :   tx = alg_model(al,x);
    2555        6944 :   switch(tx) {
    2556          98 :     case al_TRIVIAL : res = mkmatcopy(mkcol(gel(x,1))); break;
    2557         196 :     case al_ALGEBRAIC : x = algalgtobasis(al,x);
    2558        6328 :     case al_BASIS : res = algbasismultable(al,x); break;
    2559         518 :     case al_MATRIX : res = algleftmultable_mat(al,x); break;
    2560             :     default : return NULL; /* LCOV_EXCL_LINE */
    2561             :   }
    2562        6937 :   return gerepileupto(av,res);
    2563             : }
    2564             : 
    2565             : static GEN
    2566        4102 : algbasissplittingmatrix_csa(GEN al, GEN x)
    2567             : {
    2568        4102 :   long d = alg_get_degree(al), i, j;
    2569        4102 :   GEN rnf = alg_get_splittingfield(al), splba = alg_get_splittingbasis(al), splbainv = alg_get_splittingbasisinv(al), M;
    2570        4102 :   M = algbasismultable(al,x);
    2571        4102 :   M = RgM_mul(M, splba); /* TODO best order ? big matrix /Q vs small matrix /nf */
    2572        4102 :   M = RgM_mul(splbainv, M);
    2573       12131 :   for (i=1; i<=d; i++)
    2574       23912 :   for (j=1; j<=d; j++)
    2575       15883 :     gcoeff(M,i,j) = rnfeltabstorel(rnf, gcoeff(M,i,j));
    2576        4102 :   return M;
    2577             : }
    2578             : 
    2579             : static GEN
    2580         728 : algmat_tomatrix(GEN al, GEN x) /* abs = 0 */
    2581             : {
    2582             :   GEN res;
    2583             :   long i,j;
    2584         728 :   if (lg(x) == 1) return cgetg(1, t_MAT);
    2585         700 :   res = zeromatcopy(nbrows(x),lg(x)-1);
    2586        2212 :   for (j=1; j<lg(x); j++)
    2587        4879 :   for (i=1; i<lgcols(x); i++)
    2588        3367 :     gcoeff(res,i,j) = algtomatrix(al,gcoeff(x,i,j),0);
    2589         700 :   return shallowmatconcat(res);
    2590             : }
    2591             : 
    2592             : static GEN
    2593          42 : R_tomatrix(GEN x)
    2594             : {
    2595          42 :   long t = H_model(x);
    2596          42 :   if (t == H_QUATERNION) pari_err_TYPE("R_tomatrix", x);
    2597          35 :   if (t == H_MATRIX) return x;
    2598          21 :   return mkmat(mkcol(x));
    2599             : }
    2600             : static GEN
    2601          84 : C_tomatrix(GEN z, long abs)
    2602             : {
    2603             :   GEN x,y;
    2604          84 :   long t = H_model(z), nrows, ncols;
    2605          84 :   if (t == H_QUATERNION) pari_err_TYPE("C_tomatrix", z);
    2606          77 :   if (!abs)
    2607             :   {
    2608          14 :     if (t == H_MATRIX) return z;
    2609           7 :     return mkmat(mkcol(z));
    2610             :   }
    2611          63 :   if (t == H_MATRIX)
    2612             :   {
    2613             :     /* Warning: this is not the same choice of basis as for other algebras */
    2614             :     GEN res, a, b;
    2615             :     long i,j;
    2616          56 :     RgM_dimensions(z,&nrows,&ncols);
    2617          56 :     res = zeromatcopy(2*nrows,2*ncols);
    2618         168 :     for (i=1; i<=nrows; i++)
    2619         336 :       for (j=1; j<=ncols; j++)
    2620             :       {
    2621         224 :         a = real_i(gcoeff(z,i,j));
    2622         224 :         b = imag_i(gcoeff(z,i,j));
    2623         224 :         gcoeff(res,2*i-1,2*j-1) = a;
    2624         224 :         gcoeff(res,2*i,2*j) = a;
    2625         224 :         gcoeff(res,2*i-1,2*j) = gneg(b);
    2626         224 :         gcoeff(res,2*i,2*j-1) = b;
    2627             :       }
    2628          56 :     return res;
    2629             :   }
    2630           7 :   x = real_i(z);
    2631           7 :   y = imag_i(z);
    2632           7 :   return mkmat22(x,gneg(y),y,x);
    2633             : }
    2634             : static GEN
    2635        2331 : H_tomatrix(GEN x, long abs)
    2636             : {
    2637        2331 :   long tx = H_model(x);
    2638        2324 :   GEN a = NULL, b =NULL, c = NULL, d = NULL, md = NULL, M = NULL;
    2639        2324 :   if (abs) {
    2640         287 :     if (tx == H_MATRIX) return algleftmultable_mat(NULL,x);
    2641         154 :     switch(tx)
    2642             :     {
    2643          35 :       case H_SCALAR:
    2644          35 :         a = real_i(x);
    2645          35 :         b = imag_i(x);
    2646          35 :         c = gen_0;
    2647          35 :         d = gen_0;
    2648          35 :         break;
    2649         119 :       case H_QUATERNION:
    2650         119 :         a = gel(x,1);
    2651         119 :         b = gel(x,2);
    2652         119 :         c = gel(x,3);
    2653         119 :         d = gel(x,4);
    2654         119 :         break;
    2655             :     }
    2656         154 :     M = scalarmat(a,4);
    2657         154 :     gcoeff(M,2,1) = gcoeff(M,4,3) = b;
    2658         154 :     gcoeff(M,1,2) = gcoeff(M,3,4) = gneg(b);
    2659         154 :     gcoeff(M,3,1) = gcoeff(M,2,4) = c;
    2660         154 :     gcoeff(M,4,2) = gcoeff(M,1,3) = gneg(c);
    2661         154 :     gcoeff(M,4,1) = gcoeff(M,3,2) = d;
    2662         154 :     gcoeff(M,2,3) = gcoeff(M,1,4) = gneg(d);
    2663             :   }
    2664             :   else /* abs == 0 */
    2665             :   {
    2666        2037 :     if (tx == H_MATRIX) return algmat_tomatrix(NULL,x);
    2667        1778 :     switch(tx)
    2668             :     {
    2669         273 :       case H_SCALAR:
    2670         273 :         M = mkmat22(
    2671             :             x,      gen_0,
    2672             :             gen_0,  conj_i(x)
    2673             :             );
    2674         273 :         break;
    2675        1505 :       case H_QUATERNION:
    2676        1505 :         a = gel(x,1);
    2677        1505 :         b = gel(x,2);
    2678        1505 :         c = gel(x,3);
    2679        1505 :         md = gneg(gel(x,4));
    2680        1505 :         M = mkmat22(
    2681             :             mkcomplex(a,b),     mkcomplex(gneg(c),md),
    2682             :             mkcomplex(c,md),    mkcomplex(a,gneg(b))
    2683             :             );
    2684             :     }
    2685        1932 :   }
    2686        1932 :   return M;
    2687             : }
    2688             : 
    2689             : GEN
    2690        9667 : algtomatrix(GEN al, GEN x, long abs)
    2691             : {
    2692        9667 :   pari_sp av = avma;
    2693        9667 :   GEN res = NULL;
    2694             :   long ta, tx;
    2695        9667 :   checkalg(al);
    2696        9667 :   ta = alg_type(al);
    2697        9667 :   if (ta==al_REAL)
    2698             :   {
    2699        2268 :     switch(alg_get_absdim(al)) {
    2700          42 :       case 1: res = R_tomatrix(x); break;
    2701          84 :       case 2: res = C_tomatrix(x,abs); break;
    2702        2135 :       case 4: res = H_tomatrix(x,abs); break;
    2703           7 :       default: pari_err_TYPE("algtomatrix [apply alginit]", al);
    2704             :     }
    2705        2240 :     return gerepilecopy(av, res);
    2706             :   }
    2707        7399 :   if (abs || ta==al_TABLE) return algleftmultable(al,x);
    2708        6622 :   tx = alg_model(al,x);
    2709        6622 :   if (tx == al_MATRIX) res = algmat_tomatrix(al,x);
    2710        6153 :   else switch (alg_type(al))
    2711             :   {
    2712        2051 :     case al_CYCLIC:
    2713        2051 :       if (tx==al_BASIS) x = algbasistoalg(al,x);
    2714        2051 :       res = algalgmultable(al,x);
    2715        2051 :       break;
    2716        4102 :     case al_CSA:
    2717        4102 :       if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2718        4102 :       res = algbasissplittingmatrix_csa(al,x);
    2719        4102 :       break;
    2720             :     default: return NULL; /*LCOV_EXCL_LINE*/
    2721             :   }
    2722        6622 :   return gerepilecopy(av,res);
    2723             : }
    2724             : 
    2725             : /*  x^(-1)*y, NULL if no solution */
    2726             : static GEN
    2727         112 : C_divl_i(GEN x, GEN y)
    2728             : {
    2729         112 :   long tx = H_model(x), ty = H_model(y);
    2730         112 :   if (tx != ty) pari_err_TYPE2("C_divl", x, y);
    2731         105 :   switch (tx) {
    2732          42 :     case H_SCALAR:
    2733          42 :       if (gequal0(x)) return gequal0(y) ? gen_0 : NULL;
    2734          14 :       else return gdiv(y,x);
    2735          56 :     case H_MATRIX:
    2736          56 :       if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2737           7 :         pari_err_DIM("C_divl (nonsquare)");
    2738          49 :       if (lg(x) != lg(y)) pari_err_DIM("C_divl");
    2739          42 :       if (lg(y) == 1) return cgetg(1, t_MAT);
    2740          42 :       return RgM_invimage(x, y);
    2741           7 :     default: pari_err_TYPE("C_divl", x); return NULL;
    2742             :   }
    2743             : }
    2744             : /* H^k -> C^2k */
    2745             : static GEN
    2746         140 : HC_to_CC(GEN v)
    2747             : {
    2748         140 :   long l = lg(v), i;
    2749         140 :   GEN w = cgetg(2*l-1, t_COL), a, b, c, d;
    2750         420 :   for (i=1; i<l; i++)
    2751             :   {
    2752         280 :     H_compo(gel(v,i),&a,&b,&c,&d);
    2753         280 :     gel(w,2*i-1) = mkcomplex(a,b);
    2754         280 :     gel(w,2*i) = mkcomplex(c,gneg(d));
    2755             :   }
    2756         140 :   return w;
    2757             : }
    2758             : /* C^2k -> H^k */
    2759             : static GEN
    2760          98 : CC_to_HC(GEN w)
    2761             : {
    2762          98 :   long l = lg(w), i, lv = (l+1)/2;
    2763          98 :   GEN v = cgetg(lv, t_COL), ab, cd;
    2764         294 :   for (i=1; i<lv; i++)
    2765             :   {
    2766         196 :     ab = gel(w,2*i-1);
    2767         196 :     cd = gel(w,2*i);
    2768         196 :     gel(v,i) = mkcol4(real_i(ab),imag_i(ab),real_i(cd),gneg(imag_i(cd)));
    2769             :   }
    2770          98 :   return v;
    2771             : }
    2772             : /* M_{k,n}(H) -> M_{2k,n}(C) */
    2773             : static GEN
    2774         210 : HM_to_CM(GEN x) pari_APPLY_same(HC_to_CC(gel(x,i)));
    2775             : /* M_{2k,n}(C) -> M_{k,n}(H) */
    2776             : static GEN
    2777         147 : CM_to_HM(GEN x) pari_APPLY_same(CC_to_HC(gel(x,i)));
    2778             : /*  x^(-1)*y, NULL if no solution */
    2779             : static GEN
    2780         203 : H_divl_i(GEN x, GEN y)
    2781             : {
    2782         203 :   pari_sp av = avma;
    2783         203 :   long tx = H_model(x), ty = H_model(y);
    2784         189 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_divl", x, y);
    2785         168 :   if (tx==H_MATRIX)
    2786             :   {
    2787             :     GEN mx, my, mxdivy;
    2788          98 :     if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2789          14 :       pari_err_DIM("H_divl (nonsquare)");
    2790          84 :     if (lg(x) != lg(y)) pari_err_DIM("H_divl");
    2791          77 :     if (lg(y) == 1) return cgetg(1, t_MAT);
    2792          70 :     mx = H_tomatrix(x,0);
    2793          70 :     my = HM_to_CM(y);
    2794          70 :     mxdivy = RgM_invimage(mx, my);
    2795          70 :     if (!mxdivy) return gc_NULL(av);
    2796          49 :     return gerepilecopy(av,CM_to_HM(mxdivy));
    2797             :   }
    2798          70 :   if (gequal0(y)) return gen_0;
    2799          56 :   if (gequal0(x)) return NULL;
    2800          42 :   return gerepilecopy(av,H_mul(H_inv(x),y));
    2801             : }
    2802             : /*  x^(-1)*y, NULL if no solution */
    2803             : static GEN
    2804        1715 : algdivl_i(GEN al, GEN x, GEN y, long tx, long ty) {
    2805        1715 :   pari_sp av = avma;
    2806        1715 :   GEN res, p = alg_get_char(al), mtx;
    2807        1715 :   if (tx != ty) {
    2808         343 :     if (tx==al_ALGEBRAIC) { x = algalgtobasis(al,x); tx=al_BASIS; }
    2809         343 :     if (ty==al_ALGEBRAIC) { y = algalgtobasis(al,y); ty=al_BASIS; }
    2810             :   }
    2811        1715 :   if (ty == al_MATRIX)
    2812             :   {
    2813          77 :     if (alg_type(al) != al_TABLE) y = algalgtobasis(al,y);
    2814          77 :     y = algmat2basis(al,y);
    2815             :   }
    2816        1715 :   if (signe(p)) res = FpM_FpC_invimage(algbasismultable(al,x),y,p);
    2817             :   else
    2818             :   {
    2819        1526 :     if (ty==al_ALGEBRAIC)   mtx = algalgmultable(al,x);
    2820         819 :     else                    mtx = algleftmultable(al,x);
    2821        1526 :     res = inverseimage(mtx,y);
    2822             :   }
    2823        1715 :   if (!res || lg(res)==1) return gc_NULL(av);
    2824        1687 :   if (tx == al_MATRIX) {
    2825         294 :     res = algbasis2mat(al, res, lg(x)-1);
    2826         294 :     return gerepilecopy(av,res);
    2827             :   }
    2828        1393 :   return gerepileupto(av,res);
    2829             : }
    2830             : static GEN
    2831        1001 : algdivl_i2(GEN al, GEN x, GEN y)
    2832             : {
    2833             :   long tx, ty;
    2834        1001 :   checkalg(al);
    2835        1001 :   if (alg_type(al)==al_REAL) switch(alg_get_absdim(al)) {
    2836         112 :     case 1: case 2: return C_divl_i(x,y);
    2837         147 :     case 4: return H_divl_i(x,y);
    2838             :   }
    2839         742 :   tx = alg_model(al,x);
    2840         735 :   ty = alg_model(al,y);
    2841         735 :   if (tx == al_MATRIX) {
    2842         140 :     if (ty != al_MATRIX) pari_err_TYPE2("\\", x, y);
    2843         133 :     if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2844          28 :       pari_err_DIM("algdivl (nonsquare)");
    2845         105 :     if (lg(x) != lg(y)) pari_err_DIM("algdivl");
    2846          84 :     if (lg(y) == 1) return cgetg(1, t_MAT);
    2847             :   }
    2848         672 :   return algdivl_i(al,x,y,tx,ty);
    2849             : }
    2850             : 
    2851         875 : GEN algdivl(GEN al, GEN x, GEN y)
    2852             : {
    2853             :   GEN z;
    2854         875 :   z = algdivl_i2(al,x,y);
    2855         728 :   if (!z) pari_err_INV("algdivl", x);
    2856         714 :   return z;
    2857             : }
    2858             : 
    2859             : int
    2860         126 : algisdivl(GEN al, GEN x, GEN y, GEN* ptz)
    2861             : {
    2862         126 :   pari_sp av = avma;
    2863         126 :   GEN z = algdivl_i2(al,x,y);
    2864         126 :   if (!z) return gc_bool(av,0);
    2865          84 :   if (ptz != NULL) *ptz = z;
    2866          84 :   return 1;
    2867             : }
    2868             : 
    2869             : static GEN
    2870         140 : C_inv(GEN x)
    2871             : {
    2872         140 :   switch (H_model(x))
    2873             :   {
    2874          63 :     case H_SCALAR: return gequal0(x) ? NULL : ginv(x);
    2875          70 :     case H_MATRIX: return RgM_inv(x);
    2876           7 :     default: pari_err_TYPE("alginv_i", x);
    2877             :   }
    2878             :   return NULL; /*LCOV_EXCL_LINE*/
    2879             : }
    2880             : static GEN
    2881         259 : H_inv(GEN x)
    2882             : {
    2883         259 :   pari_sp av = avma;
    2884             :   GEN nm, xi;
    2885             :   long i;
    2886         259 :   switch (H_model(x))
    2887             :   {
    2888          28 :     case H_SCALAR:
    2889          28 :       if (gequal0(x)) return NULL;
    2890          14 :       return ginv(x);
    2891         161 :     case H_QUATERNION:
    2892         161 :       if (gequal0(x)) return NULL;
    2893         154 :       nm = H_norm(x, 0);
    2894         154 :       xi = gdiv(x,nm);
    2895         616 :       for(i=2; i<=4; i++) gel(xi,i) = gneg(gel(xi,i));
    2896         154 :       return gerepilecopy(av,xi);
    2897          63 :     case H_MATRIX:
    2898          63 :       if (lg(x)==1) return cgetg(1,t_MAT);
    2899          56 :       return H_divl_i(x, matid(lg(x)-1));
    2900             :   }
    2901             :   return NULL; /*LCOV_EXCL_LINE*/
    2902             : }
    2903             : static GEN
    2904        1512 : alginv_i(GEN al, GEN x)
    2905             : {
    2906        1512 :   pari_sp av = avma;
    2907        1512 :   GEN res = NULL, p = alg_get_char(al);
    2908             :   long tx, n, ta;
    2909        1512 :   ta = alg_type(al);
    2910        1512 :   if (ta==al_REAL) switch(alg_get_absdim(al)) {
    2911         140 :     case 1: case 2: return C_inv(x);
    2912         217 :     case 4: return H_inv(x);
    2913           7 :     default: pari_err_TYPE("alginv_i [apply alginit]", al);
    2914             :   }
    2915        1148 :   tx = alg_model(al,x);
    2916        1127 :   switch(tx) {
    2917          63 :     case al_TRIVIAL :
    2918          63 :       if (signe(p)) { res = mkcol(Fp_inv(gel(x,1),p)); break; }
    2919          49 :       else          { res = mkcol(ginv(gel(x,1))); break; }
    2920         455 :     case al_ALGEBRAIC :
    2921             :       switch(ta) {
    2922         350 :         case al_CYCLIC: n = alg_get_degree(al); break;
    2923         105 :         case al_CSA: n = alg_get_dim(al); break;
    2924             :         default: return NULL; /* LCOV_EXCL_LINE */
    2925             :       }
    2926         455 :       res = algdivl_i(al, x, col_ei(n,1), tx, al_ALGEBRAIC); break;
    2927         371 :     case al_BASIS : res = algdivl_i(al, x, col_ei(alg_get_absdim(al),1), tx,
    2928         371 :                                                             al_BASIS); break;
    2929         238 :     case al_MATRIX :
    2930         238 :       n = lg(x)-1;
    2931         238 :       if (n==0) return cgetg(1, t_MAT);
    2932         224 :       if (n != nbrows(x)) pari_err_DIM("alginv_i (nonsquare)");
    2933         217 :       res = algdivl_i(al, x, col_ei(n*n*alg_get_absdim(al),1), tx, al_BASIS);
    2934             :         /* cheat on type because wrong dimension */
    2935             :   }
    2936        1106 :   if (!res) return gc_NULL(av);
    2937        1092 :   return gerepilecopy(av,res);
    2938             : }
    2939             : GEN
    2940        1323 : alginv(GEN al, GEN x)
    2941             : {
    2942             :   GEN z;
    2943        1323 :   checkalg(al);
    2944        1323 :   z = alginv_i(al,x);
    2945        1274 :   if (!z) pari_err_INV("alginv", x);
    2946        1239 :   return z;
    2947             : }
    2948             : 
    2949             : int
    2950         189 : algisinv(GEN al, GEN x, GEN* ptix)
    2951             : {
    2952         189 :   pari_sp av = avma;
    2953             :   GEN ix;
    2954         189 :   if (al) checkalg(al);
    2955         189 :   ix = alginv_i(al,x);
    2956         189 :   if (!ix) return gc_bool(av,0);
    2957         133 :   if (ptix != NULL) *ptix = ix;
    2958         133 :   return 1;
    2959             : }
    2960             : 
    2961             : /*  x*y^(-1)  */
    2962             : GEN
    2963         469 : algdivr(GEN al, GEN x, GEN y) { return algmul(al, x, alginv(al, y)); }
    2964             : 
    2965       26049 : static GEN _mul(void* data, GEN x, GEN y) { return algmul((GEN)data,x,y); }
    2966       49799 : static GEN _sqr(void* data, GEN x) { return algsqr((GEN)data,x); }
    2967             : 
    2968             : static GEN
    2969          21 : algmatid(GEN al, long N)
    2970             : {
    2971          21 :   long n = alg_get_absdim(al), i, j;
    2972             :   GEN res, one, zero;
    2973             : 
    2974          21 :   res = zeromatcopy(N,N);
    2975          21 :   one = col_ei(n,1);
    2976          21 :   zero = zerocol(n);
    2977          49 :   for (i=1; i<=N; i++)
    2978          84 :   for (j=1; j<=N; j++)
    2979          56 :     gcoeff(res,i,j) = i==j ? one : zero;
    2980          21 :   return res;
    2981             : }
    2982             : 
    2983             : GEN
    2984       12882 : algpow(GEN al, GEN x, GEN n)
    2985             : {
    2986       12882 :   pari_sp av = avma;
    2987             :   GEN res;
    2988       12882 :   long s = signe(n);
    2989       12882 :   checkalg(al);
    2990       12882 :   if (!s && alg_type(al)==al_REAL)
    2991             :   {
    2992          63 :     if (H_model(x) == H_MATRIX) return matid(lg(x)-1);
    2993          35 :     else                        return gen_1;
    2994             :   }
    2995       12819 :   switch (s) {
    2996          28 :     case 0:
    2997          28 :       if (alg_model(al,x) == al_MATRIX)
    2998          21 :         res = algmatid(al,lg(x)-1);
    2999             :       else
    3000           7 :         res = col_ei(alg_get_absdim(al),1);
    3001          28 :       return res;
    3002       12644 :     case 1:
    3003       12644 :       res = gen_pow_i(x, n, (void*)al, _sqr, _mul); break;
    3004         147 :     default: /* -1 */
    3005         147 :       res = gen_pow_i(alginv(al,x), gneg(n), (void*)al, _sqr, _mul);
    3006             :   }
    3007       12777 :   return gerepilecopy(av,res);
    3008             : }
    3009             : 
    3010             : static GEN
    3011         378 : algredcharpoly_i(GEN al, GEN x, long v)
    3012             : {
    3013         378 :   GEN rnf = alg_get_splittingfield(al);
    3014         378 :   GEN cp = charpoly(algtomatrix(al,x,0),v);
    3015         371 :   long i, m = lg(cp);
    3016        1540 :   for (i=2; i<m; i++) gel(cp,i) = rnfeltdown(rnf, gel(cp,i));
    3017         371 :   return cp;
    3018             : }
    3019             : 
    3020             : /* assumes al is CSA or CYCLIC */
    3021             : static GEN
    3022         385 : algredcharpoly(GEN al, GEN x, long v)
    3023             : {
    3024         385 :   pari_sp av = avma;
    3025         385 :   long w = gvar(rnf_get_pol(alg_get_center(al)));
    3026         385 :   if (varncmp(v,w)>=0) pari_err_PRIORITY("algredcharpoly",pol_x(v),">=",w);
    3027         378 :   switch(alg_type(al))
    3028             :   {
    3029         378 :     case al_CYCLIC:
    3030             :     case al_CSA:
    3031         378 :       return gerepileupto(av, algredcharpoly_i(al, x, v));
    3032             :   }
    3033             :   return NULL; /*LCOV_EXCL_LINE*/
    3034             : }
    3035             : 
    3036             : static GEN
    3037       21077 : algbasischarpoly(GEN al, GEN x, long v)
    3038             : {
    3039       21077 :   pari_sp av = avma;
    3040       21077 :   GEN p = alg_get_char(al), mx;
    3041       21077 :   if (alg_model(al,x) == al_MATRIX) mx = algleftmultable_mat(al,x);
    3042       20986 :   else                              mx = algbasismultable(al,x);
    3043       21070 :   if (signe(p)) {
    3044       19173 :     GEN res = FpM_charpoly(mx,p);
    3045       19173 :     setvarn(res,v);
    3046       19173 :     return gerepileupto(av, res);
    3047             :   }
    3048        1897 :   return gerepileupto(av, charpoly(mx,v));
    3049             : }
    3050             : 
    3051             : static GEN
    3052          35 : R_charpoly(GEN x, long v, long abs)
    3053             : {
    3054          35 :   pari_sp av = avma;
    3055          35 :   GEN res = NULL;
    3056          35 :   switch (H_model(x))
    3057             :   {
    3058          14 :     case H_SCALAR: res = mkpoln(2, gen_1, gneg(x)); break;
    3059          14 :     case H_MATRIX:
    3060          14 :       res = charpoly(x,v);
    3061          14 :       if (abs) res = gpowgs(res,nbrows(x));
    3062          14 :       break;
    3063           7 :     default: pari_err_TYPE("R_charpoly", x);
    3064             :   }
    3065          28 :   if (v) setvarn(res, v);
    3066          28 :   return gerepilecopy(av, res);
    3067             : }
    3068             : static GEN
    3069          35 : C_charpoly(GEN x, long v, long abs)
    3070             : {
    3071          35 :   pari_sp av = avma;
    3072          35 :   GEN res = NULL;
    3073          35 :   switch (H_model(x))
    3074             :   {
    3075          14 :     case H_SCALAR:
    3076          14 :       if (abs)  res = mkpoln(3, gen_1, gneg(gshift(real_i(x),1)), cxnorm(x));
    3077           7 :       else      res = mkpoln(2, gen_1, gneg(x));
    3078          14 :       break;
    3079          14 :     case H_MATRIX:
    3080          14 :       res = charpoly(x,v);
    3081          14 :       if (abs) res = gpowgs(real_i(gmul(res,gconj(res))),nbrows(x));
    3082          14 :       break;
    3083           7 :     default: pari_err_TYPE("C_charpoly", x);
    3084             :   }
    3085          28 :   if (v) setvarn(res, v);
    3086          28 :   return gerepilecopy(av, res);
    3087             : }
    3088             : static GEN
    3089          98 : H_charpoly(GEN x, long v, long abs)
    3090             : {
    3091          98 :   pari_sp av = avma;
    3092             :   GEN res;
    3093          98 :   if (H_model(x) == H_MATRIX) return greal(charpoly(H_tomatrix(x,abs),v));
    3094          70 :   res = mkpoln(3, gen_1, gneg(H_trace(x,0)), H_norm(x,0));
    3095          70 :   if (v) setvarn(res, v);
    3096          70 :   if (abs) res = gsqr(res);
    3097          70 :   return gerepilecopy(av, res);
    3098             : }
    3099             : 
    3100             : GEN
    3101       21280 : algcharpoly(GEN al, GEN x, long v, long abs)
    3102             : {
    3103             :   long ta;
    3104       21280 :   if (v<0) v=0;
    3105       21280 :   checkalg(al);
    3106       21280 :   ta = alg_type(al);
    3107       21280 :   if (ta == al_REAL) switch (alg_get_absdim(al)) {
    3108          35 :     case 1: return R_charpoly(x, v, abs);
    3109          35 :     case 2: return C_charpoly(x, v, abs);
    3110          98 :     case 4: return H_charpoly(x, v, abs);
    3111           7 :     default: pari_err_TYPE("algcharpoly [apply alginit]", al);
    3112             :   }
    3113             : 
    3114             :   /* gneg(x[1]) left on stack */
    3115       21105 :   if (alg_model(al,x) == al_TRIVIAL) {
    3116          56 :     GEN p = alg_get_char(al);
    3117          56 :     if (signe(p)) return deg1pol(gen_1,Fp_neg(gel(x,1),p),v);
    3118          42 :     return deg1pol(gen_1,gneg(gel(x,1)),v);
    3119             :   }
    3120             : 
    3121       21042 :   switch(ta) {
    3122         490 :     case al_CYCLIC: case al_CSA:
    3123         490 :       if (abs)
    3124             :       {
    3125         105 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3126             :       }
    3127         385 :       else return algredcharpoly(al,x,v);
    3128       20657 :     case al_TABLE: return algbasischarpoly(al,x,v);
    3129             :     default : return NULL; /* LCOV_EXCL_LINE */
    3130             :   }
    3131             : }
    3132             : 
    3133             : /* assumes x in basis form */
    3134             : static GEN
    3135      236620 : algabstrace(GEN al, GEN x)
    3136             : {
    3137      236620 :   pari_sp av = avma;
    3138      236620 :   GEN res = NULL, p = alg_get_char(al);
    3139      236620 :   if (signe(p)) return FpV_dotproduct(x, alg_get_tracebasis(al), p);
    3140       36435 :   switch(alg_model(al,x)) {
    3141         154 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    3142       36281 :     case al_BASIS: res = RgV_dotproduct(x, alg_get_tracebasis(al)); break;
    3143             :   }
    3144       36281 :   return gerepileupto(av,res);
    3145             : }
    3146             : 
    3147             : static GEN
    3148        1372 : algredtrace(GEN al, GEN x)
    3149             : {
    3150        1372 :   pari_sp av = avma;
    3151        1372 :   GEN res = NULL;
    3152        1372 :   switch(alg_model(al,x)) {
    3153          35 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    3154         490 :     case al_BASIS: return algredtrace(al, algbasistoalg(al,x));
    3155             :                    /* TODO precompute too? */
    3156         847 :     case al_ALGEBRAIC:
    3157         847 :       switch(alg_type(al))
    3158             :       {
    3159         553 :         case al_CYCLIC:
    3160         553 :           res = rnfelttrace(alg_get_splittingfield(al),gel(x,1));
    3161         553 :           break;
    3162         294 :         case al_CSA:
    3163         294 :           res = gtrace(algalgmultable_csa(al,x));
    3164         294 :           res = gdiv(res, stoi(alg_get_degree(al)));
    3165         294 :           break;
    3166             :         default: return NULL; /* LCOV_EXCL_LINE */
    3167             :       }
    3168         847 :   }
    3169         847 :   return gerepileupto(av,res);
    3170             : }
    3171             : 
    3172             : static GEN
    3173         469 : algtrace_mat(GEN al, GEN M, long abs) {
    3174         469 :   pari_sp av = avma;
    3175         469 :   long N = lg(M)-1, i;
    3176         469 :   GEN res, p = alg_get_char(al);
    3177         469 :   if (N == 0) return gen_0;
    3178         448 :   if (N != nbrows(M)) pari_err_DIM("algtrace_mat (nonsquare)");
    3179             : 
    3180         434 :   if (!signe(p)) p = NULL;
    3181         434 :   if (alg_type(al) == al_TABLE) abs = 1;
    3182         434 :   res = algtrace(al, gcoeff(M,1,1), abs);
    3183         896 :   for (i=2; i<=N; i++) {
    3184         462 :     if (p)  res = Fp_add(res, algtrace(al,gcoeff(M,i,i),abs), p);
    3185         455 :     else    res = gadd(res, algtrace(al,gcoeff(M,i,i),abs));
    3186             :   }
    3187         434 :   if (abs) res = gmulgu(res, N); /* absolute trace */
    3188         434 :   return gerepileupto(av, res);
    3189             : }
    3190             : 
    3191             : static GEN
    3192          35 : R_trace(GEN x, long abs)
    3193             : {
    3194          35 :   pari_sp av = avma;
    3195          35 :   GEN res = NULL;
    3196          35 :   switch (H_model(x))
    3197             :   {
    3198          14 :     case H_SCALAR: res = gcopy(x); break;
    3199          14 :     case H_MATRIX: res = abs? mulrs(gtrace(x),nbrows(x)) : gtrace(x); break;
    3200           7 :     default: pari_err_TYPE("R_trace", x);
    3201             :   }
    3202          28 :   return gerepilecopy(av, res);
    3203             : }
    3204             : static GEN
    3205          35 : C_trace(GEN x, long abs)
    3206             : {
    3207          35 :   pari_sp av = avma;
    3208          35 :   GEN res = NULL;
    3209          35 :   switch (H_model(x))
    3210             :   {
    3211          14 :     case H_SCALAR: res = abs ? gshift(real_i(x),1) : x; break;
    3212          14 :     case H_MATRIX:
    3213          14 :       res = abs ? mulrs(real_i(gtrace(x)),2*nbrows(x)) : gtrace(x); break;
    3214           7 :     default: pari_err_TYPE("C_trace", x);
    3215             :   }
    3216          28 :   return gerepilecopy(av, res);
    3217             : }
    3218             : static GEN
    3219         567 : H_trace(GEN x, long abs)
    3220             : {
    3221         567 :   long s = abs? 2 : 1;
    3222         567 :   switch (H_model(x))
    3223             :   {
    3224         154 :     case H_SCALAR: return gshift(real_i(x),s);
    3225         329 :     case H_QUATERNION: return gshift(gel(x,1),s);
    3226          77 :     case H_MATRIX:
    3227          77 :       return algtrace_mat(NULL, x, abs);
    3228             :   }
    3229             :   return NULL; /*LCOV_EXCL_LINE*/
    3230             : }
    3231             : 
    3232             : GEN
    3233        2632 : algtrace(GEN al, GEN x, long abs)
    3234             : {
    3235             :   long ta;
    3236        2632 :   checkalg(al);
    3237        2632 :   ta = alg_type(al);
    3238        2632 :   if (ta==al_REAL) switch (alg_get_absdim(al)) {
    3239          35 :     case 1: return R_trace(x,abs);
    3240          35 :     case 2: return C_trace(x,abs);
    3241         497 :     case 4: return H_trace(x,abs);
    3242           7 :     default: pari_err_TYPE("algtrace [apply alginit]", al);
    3243             :   }
    3244        2058 :   if (alg_model(al,x) == al_MATRIX) return algtrace_mat(al,x,abs);
    3245        1666 :   switch(ta) {
    3246        1526 :     case al_CYCLIC: case al_CSA:
    3247        1526 :       if (!abs) return algredtrace(al,x);
    3248         644 :       if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3249         784 :     case al_TABLE: return algabstrace(al,x);
    3250             :     default : return NULL; /* LCOV_EXCL_LINE */
    3251             :   }
    3252             : }
    3253             : 
    3254             : static GEN
    3255       42288 : ZM_trace(GEN x)
    3256             : {
    3257       42288 :   long i, lx = lg(x);
    3258             :   GEN t;
    3259       42288 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    3260       41483 :   t = gcoeff(x,1,1);
    3261      705373 :   for (i = 2; i < lx; i++) t = addii(t, gcoeff(x,i,i));
    3262       41483 :   return t;
    3263             : }
    3264             : static GEN
    3265      131590 : FpM_trace(GEN x, GEN p)
    3266             : {
    3267      131590 :   long i, lx = lg(x);
    3268             :   GEN t;
    3269      131590 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    3270      123674 :   t = gcoeff(x,1,1);
    3271      911260 :   for (i = 2; i < lx; i++) t = Fp_add(t, gcoeff(x,i,i), p);
    3272      123674 :   return t;
    3273             : }
    3274             : 
    3275             : static GEN
    3276       40224 : algtracebasis(GEN al)
    3277             : {
    3278       40224 :   pari_sp av = avma;
    3279       40224 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    3280       40224 :   long i, l = lg(mt);
    3281       40224 :   GEN v = cgetg(l, t_VEC);
    3282      171814 :   if (signe(p)) for (i=1; i < l; i++) gel(v,i) = FpM_trace(gel(mt,i), p);
    3283       47834 :   else          for (i=1; i < l; i++) gel(v,i) = ZM_trace(gel(mt,i));
    3284       40224 :   return gerepileupto(av,v);
    3285             : }
    3286             : 
    3287             : /* Assume: i > 0, expo := p^i <= absdim, x contained in I_{i-1} given by mult
    3288             :  * table modulo modu=p^(i+1). Return Tr(x^(p^i)) mod modu */
    3289             : static ulong
    3290       24639 : algtracei(GEN mt, ulong p, ulong expo, ulong modu)
    3291             : {
    3292       24639 :   pari_sp av = avma;
    3293       24639 :   long j, l = lg(mt);
    3294       24639 :   ulong tr = 0;
    3295       24639 :   mt = Flm_powu(mt,expo,modu);
    3296      269001 :   for (j=1; j<l; j++) tr += ucoeff(mt,j,j);
    3297       24639 :   return gc_ulong(av, (tr/expo) % p);
    3298             : }
    3299             : 
    3300             : static GEN
    3301          42 : R_norm(GEN x, long abs)
    3302             : {
    3303          42 :   pari_sp av = avma;
    3304          42 :   GEN res = NULL;
    3305          42 :   switch (H_model(x))
    3306             :   {
    3307          14 :     case H_SCALAR: res = gcopy(x); break;
    3308          21 :     case H_MATRIX: res = abs ? powrs(det(x),nbrows(x)) : det(x); break;
    3309           7 :     default: pari_err_TYPE("R_norm", x);
    3310             :   }
    3311          35 :   return gerepilecopy(av,res);
    3312             : }
    3313             : static GEN
    3314          42 : C_norm(GEN x, long abs)
    3315             : {
    3316          42 :   pari_sp av = avma;
    3317          42 :   GEN res = NULL;
    3318          42 :   switch (H_model(x))
    3319             :   {
    3320          14 :     case H_SCALAR: res = abs ? cxnorm(x) : x; break;
    3321          21 :     case H_MATRIX: res = abs ? powrs(cxnorm(det(x)),nbrows(x)) : det(x); break;
    3322           7 :     default: pari_err_TYPE("C_norm", x);
    3323             :   }
    3324          35 :   return gerepilecopy(av,res);
    3325             : }
    3326             : static GEN
    3327         434 : H_norm(GEN x, long abs)
    3328             : {
    3329         434 :   pari_sp av = avma;
    3330         434 :   switch (H_model(x))
    3331             :   {
    3332          42 :     case H_SCALAR:
    3333          42 :       if (abs)  return gerepilecopy(av,gsqr(gnorm(x)));
    3334          35 :       else      return gnorm(x);
    3335         322 :     case H_QUATERNION:
    3336         322 :       if (abs)  return gerepilecopy(av,gsqr(gnorml2(x)));
    3337         294 :       else      return gnorml2(x);
    3338          63 :     case H_MATRIX:
    3339          63 :       return gerepilecopy(av,real_i(det(H_tomatrix(x,abs))));
    3340             :   }
    3341             :   return NULL; /*LCOV_EXCL_LINE*/
    3342             : }
    3343             : 
    3344             : GEN
    3345        1253 : algnorm(GEN al, GEN x, long abs)
    3346             : {
    3347        1253 :   pari_sp av = avma;
    3348             :   long tx, ta;
    3349             :   GEN p, rnf, res, mx;
    3350        1253 :   checkalg(al);
    3351        1253 :   ta = alg_type(al);
    3352        1253 :   if (ta==al_REAL) switch (alg_get_absdim(al)) {
    3353          42 :     case 1: return R_norm(x,abs);
    3354          42 :     case 2: return C_norm(x,abs);
    3355         210 :     case 4: return H_norm(x,abs);
    3356           7 :     default: pari_err_TYPE("algnorm [apply alginit]", al);
    3357             :   }
    3358         952 :   p = alg_get_char(al);
    3359         952 :   tx = alg_model(al,x);
    3360         952 :   if (signe(p)) {
    3361          21 :     if (tx == al_MATRIX)    mx = algleftmultable_mat(al,x);
    3362          14 :     else                    mx = algbasismultable(al,x);
    3363          21 :     return gerepileupto(av, FpM_det(mx,p));
    3364             :   }
    3365         931 :   if (tx == al_TRIVIAL) return gcopy(gel(x,1));
    3366             : 
    3367         889 :   switch(ta) {
    3368         819 :     case al_CYCLIC: case al_CSA:
    3369         819 :       if (abs)
    3370             :       {
    3371         196 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3372             :       }
    3373             :       else
    3374             :       {
    3375         623 :         rnf = alg_get_splittingfield(al);
    3376         623 :         res = rnfeltdown(rnf, det(algtomatrix(al,x,0)));
    3377         616 :         break;
    3378             :       }
    3379             :     case al_TABLE:
    3380         266 :       if (tx == al_MATRIX)  mx = algleftmultable_mat(al,x);
    3381         105 :       else                  mx = algbasismultable(al,x);
    3382         259 :       res = det(mx);
    3383         259 :       break;
    3384             :     default: return NULL; /* LCOV_EXCL_LINE */
    3385             :   }
    3386         875 :   return gerepileupto(av, res);
    3387             : }
    3388             : 
    3389             : static GEN
    3390       49245 : algalgtonat_cyc(GEN al, GEN x)
    3391             : {
    3392       49245 :   pari_sp av = avma;
    3393       49245 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    3394       49245 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    3395       49245 :   res = zerocol(N*n);
    3396      150941 :   for (i=0; i<n; i++) {
    3397      101696 :     c = gel(x,i+1);
    3398      101696 :     c = rnfeltreltoabs(rnf,c);
    3399      101696 :     if (!gequal0(c)) {
    3400       76958 :       c = algtobasis(nf,c);
    3401      415870 :       for (i1=1; i1<=N; i1++) gel(res,i*N+i1) = gel(c,i1);
    3402             :     }
    3403             :   }
    3404       49245 :   return gerepilecopy(av, res);
    3405             : }
    3406             : 
    3407             : static GEN
    3408       11375 : algalgtonat_csa(GEN al, GEN x)
    3409             : {
    3410       11375 :   pari_sp av = avma;
    3411       11375 :   GEN nf = alg_get_center(al), res, c;
    3412       11375 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    3413       11375 :   res = zerocol(d2*n);
    3414       56644 :   for (i=0; i<d2; i++) {
    3415       45269 :     c = gel(x,i+1);
    3416       45269 :     if (!gequal0(c)) {
    3417       31563 :       c = algtobasis(nf,c);
    3418       95095 :       for (i1=1; i1<=n; i1++) gel(res,i*n+i1) = gel(c,i1);
    3419             :     }
    3420             :   }
    3421       11375 :   return gerepilecopy(av, res);
    3422             : }
    3423             : 
    3424             : /* assumes al CSA or CYCLIC */
    3425             : static GEN
    3426       60620 : algalgtonat(GEN al, GEN x)
    3427             : {
    3428       60620 :   switch(alg_type(al))
    3429             :   {
    3430       49245 :     case al_CYCLIC: return algalgtonat_cyc(al, x);
    3431       11375 :     case al_CSA: return algalgtonat_csa(al, x);
    3432             :   }
    3433             :   return NULL; /*LCOV_EXCL_LINE*/
    3434             : }
    3435             : 
    3436             : static GEN
    3437       11235 : algnattoalg_cyc(GEN al, GEN x)
    3438             : {
    3439       11235 :   pari_sp av = avma;
    3440       11235 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    3441       11235 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    3442       11235 :   res = zerocol(n);
    3443       11235 :   c = zerocol(N);
    3444       47726 :   for (i=0; i<n; i++) {
    3445      319585 :     for (i1=1; i1<=N; i1++) gel(c,i1) = gel(x,i*N+i1);
    3446       36491 :     gel(res,i+1) = rnfeltabstorel(rnf,basistoalg(nf,c));
    3447             :   }
    3448       11235 :   return gerepilecopy(av, res);
    3449             : }
    3450             : 
    3451             : static GEN
    3452        1281 : algnattoalg_csa(GEN al, GEN x)
    3453             : {
    3454        1281 :   pari_sp av = avma;
    3455        1281 :   GEN nf = alg_get_center(al), res, c;
    3456        1281 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    3457        1281 :   res = zerocol(d2);
    3458        1281 :   c = zerocol(n);
    3459        6888 :   for (i=0; i<d2; i++) {
    3460       19166 :     for (i1=1; i1<=n; i1++) gel(c,i1) = gel(x,i*n+i1);
    3461        5607 :     gel(res,i+1) = basistoalg(nf,c);
    3462             :   }
    3463        1281 :   return gerepilecopy(av, res);
    3464             : }
    3465             : 
    3466             : /* assumes al CSA or CYCLIC */
    3467             : static GEN
    3468       12516 : algnattoalg(GEN al, GEN x)
    3469             : {
    3470       12516 :   switch(alg_type(al))
    3471             :   {
    3472       11235 :     case al_CYCLIC: return algnattoalg_cyc(al, x);
    3473        1281 :     case al_CSA: return algnattoalg_csa(al, x);
    3474             :   }
    3475             :   return NULL; /*LCOV_EXCL_LINE*/
    3476             : }
    3477             : 
    3478             : static GEN
    3479         182 : algalgtobasis_mat(GEN al, GEN x) /* componentwise */
    3480             : {
    3481         182 :   pari_sp av = avma;
    3482             :   long lx, lxj, i, j;
    3483             :   GEN res;
    3484         182 :   lx = lg(x);
    3485         182 :   res = cgetg(lx, t_MAT);
    3486         546 :   for (j=1; j<lx; j++) {
    3487         364 :     lxj = lg(gel(x,j));
    3488         364 :     gel(res,j) = cgetg(lxj, t_COL);
    3489        1092 :     for (i=1; i<lxj; i++)
    3490         728 :       gcoeff(res,i,j) = algalgtobasis(al,gcoeff(x,i,j));
    3491             :   }
    3492         182 :   return gerepilecopy(av,res);
    3493             : }
    3494             : GEN
    3495       61082 : algalgtobasis(GEN al, GEN x)
    3496             : {
    3497             :   pari_sp av;
    3498             :   long tx, ta;
    3499       61082 :   checkalg(al);
    3500       61082 :   ta = alg_type(al);
    3501       61082 :   if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algalgtobasis [use alginit]", al);
    3502       61061 :   tx = alg_model(al,x);
    3503       61061 :   if (tx==al_BASIS) return gcopy(x);
    3504       60802 :   if (tx==al_MATRIX) return algalgtobasis_mat(al,x);
    3505       60620 :   av = avma;
    3506       60620 :   x = algalgtonat(al,x);
    3507       60620 :   x = RgM_RgC_mul(alg_get_invbasis(al),x);
    3508       60620 :   return gerepileupto(av, x);
    3509             : }
    3510             : 
    3511             : static GEN
    3512         119 : algbasistoalg_mat(GEN al, GEN x) /* componentwise */
    3513             : {
    3514         119 :   long j, lx = lg(x);
    3515         119 :   GEN res = cgetg(lx, t_MAT);
    3516         357 :   for (j=1; j<lx; j++) {
    3517         238 :     long i, lxj = lg(gel(x,j));
    3518         238 :     gel(res,j) = cgetg(lxj, t_COL);
    3519         714 :     for (i=1; i<lxj; i++) gcoeff(res,i,j) = algbasistoalg(al,gcoeff(x,i,j));
    3520             :   }
    3521         119 :   return res;
    3522             : }
    3523             : GEN
    3524        2926 : algbasistoalg(GEN al, GEN x)
    3525             : {
    3526             :   pari_sp av;
    3527             :   long tx, ta;
    3528        2926 :   checkalg(al);
    3529        2926 :   ta = alg_type(al);
    3530        2926 :   if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algbasistoalg [use alginit]", al);
    3531        2905 :   tx = alg_model(al,x);
    3532        2905 :   if (tx==al_ALGEBRAIC) return gcopy(x);
    3533        2772 :   if (tx==al_MATRIX) return algbasistoalg_mat(al,x);
    3534        2653 :   av = avma;
    3535        2653 :   x = RgM_RgC_mul(alg_get_basis(al),x);
    3536        2653 :   x = algnattoalg(al,x);
    3537        2653 :   return gerepileupto(av, x);
    3538             : }
    3539             : 
    3540             : static GEN
    3541        4466 : R_random(GEN b)
    3542             : {
    3543        4466 :   pari_sp av = avma;
    3544        4466 :   long prec = realprec(b);
    3545        4466 :   GEN z = randomr(prec); shiftr_inplace(z, 1);
    3546        4466 :   return gerepileuptoleaf(av, mulrr(b,addsr(-1, z)));
    3547             : }
    3548             : static GEN
    3549         182 : C_random(GEN b)
    3550             : {
    3551         182 :   retmkcomplex(R_random(b), R_random(b));
    3552             : }
    3553             : static GEN
    3554         980 : H_random(GEN b)
    3555             : {
    3556         980 :   GEN res = cgetg(5, t_COL);
    3557             :   long i;
    3558        4900 :   for (i=1; i<=4; i++) gel(res,i) = R_random(b);
    3559         980 :   return res;
    3560             : }
    3561             : GEN
    3562       19677 : algrandom(GEN al, GEN b)
    3563             : {
    3564       19677 :   GEN res = NULL, p, N;
    3565             :   long i, n;
    3566       19677 :   checkalg(al);
    3567       19663 :   if (alg_type(al)==al_REAL)
    3568             :   {
    3569        1365 :     if (typ(b) != t_REAL) pari_err_TYPE("algrandom",b);
    3570        1358 :     if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
    3571        1351 :     switch(alg_get_absdim(al))
    3572             :     {
    3573         182 :       case 1: res = R_random(b); break;
    3574         182 :       case 2: res = C_random(b); break;
    3575         980 :       case 4: res = H_random(b); break;
    3576           7 :       default: pari_err_TYPE("algrandom [apply alginit]", al);
    3577             :     }
    3578        1344 :     return res;
    3579             :   }
    3580       18298 :   if (typ(b) != t_INT) pari_err_TYPE("algrandom",b);
    3581       18291 :   if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
    3582       18284 :   n = alg_get_absdim(al);
    3583       18284 :   N = addiu(shifti(b,1), 1); /* left on stack */
    3584       18284 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
    3585       18284 :   res = cgetg(n+1,t_COL);
    3586      163828 :   for (i = 1; i <= n; i++)
    3587             :   {
    3588      145544 :     pari_sp av = avma;
    3589      145544 :     GEN t = subii(randomi(N),b);
    3590      145544 :     if (p) t = modii(t, p);
    3591      145544 :     gel(res,i) = gerepileuptoint(av, t);
    3592             :   }
    3593       18284 :   return res;
    3594             : }
    3595             : 
    3596             : static GEN
    3597          77 : H_poleval(GEN pol, GEN x)
    3598             : {
    3599          77 :   pari_sp av = avma;
    3600             :   GEN res;
    3601             :   long i;
    3602          77 :   switch (H_model(x))
    3603             :   {
    3604          21 :     case H_SCALAR: return RgX_cxeval(pol, x, NULL);
    3605          42 :     case H_QUATERNION: break;
    3606           7 :     default: pari_err_TYPE("H_poleval", x);
    3607             :   }
    3608             : 
    3609          42 :   res = zerocol(4);
    3610         189 :   for (i=lg(pol)-1; i>1; i--)
    3611             :   {
    3612         147 :     gel(res,1) = gadd(gel(res,1), gel(pol,i));
    3613         147 :     if (i>2) res = H_mul(x, res);
    3614             :   }
    3615             : 
    3616          42 :   return gerepilecopy(av,res);
    3617             : }
    3618             : 
    3619             : /* Assumes pol has coefficients in the same ring as the COL x; x either
    3620             :  * in basis or algebraic form or [x,mx] where mx is the mult. table of x.
    3621             :  TODO more general version: pol with coeffs in center and x in basis form */
    3622             : GEN
    3623       17276 : algpoleval(GEN al, GEN pol, GEN x)
    3624             : {
    3625       17276 :   pari_sp av = avma;
    3626       17276 :   GEN p, mx = NULL, res;
    3627             :   long i;
    3628       17276 :   if (typ(pol) != t_POL) pari_err_TYPE("algpoleval", pol);
    3629       17262 :   checkalg(al);
    3630       17262 :   if (alg_type(al)==al_REAL) return H_poleval(pol,x);
    3631       17185 :   p = alg_get_char(al);
    3632       17185 :   if (typ(x) == t_VEC)
    3633             :   {
    3634        6097 :     if (lg(x) != 3) pari_err_TYPE("algpoleval [vector must be of length 2]", x);
    3635        6090 :     mx = gel(x,2);
    3636        6090 :     x = gel(x,1);
    3637        6090 :     if (typ(mx)!=t_MAT || !gequal(x,gel(mx,1)))
    3638          21 :       pari_err_TYPE("algpoleval [mx must be the multiplication table of x]", mx);
    3639             :   }
    3640             :   else
    3641             :   {
    3642       11088 :     switch(alg_model(al,x))
    3643             :     {
    3644          14 :       case al_ALGEBRAIC: mx = algalgmultable(al,x); break;
    3645       11046 :       case al_BASIS: if (!RgX_is_QX(pol))
    3646           7 :         pari_err_IMPL("algpoleval with x in basis form and pol not in Q[x]");
    3647       11053 :       case al_TRIVIAL: mx = algbasismultable(al,x); break;
    3648           7 :       default: pari_err_TYPE("algpoleval", x);
    3649             :     }
    3650             :   }
    3651       17136 :   res = zerocol(lg(mx)-1);
    3652       17136 :   if (signe(p)) {
    3653       64335 :     for (i=lg(pol)-1; i>1; i--)
    3654             :     {
    3655       47997 :       gel(res,1) = Fp_add(gel(res,1), gel(pol,i), p);
    3656       47997 :       if (i>2) res = FpM_FpC_mul(mx, res, p);
    3657             :     }
    3658             :   }
    3659             :   else {
    3660        4879 :     for (i=lg(pol)-1; i>1; i--)
    3661             :     {
    3662        4081 :       gel(res,1) = gadd(gel(res,1), gel(pol,i));
    3663        4081 :       if (i>2) res = RgM_RgC_mul(mx, res);
    3664             :     }
    3665             :   }
    3666       17136 :   return gerepileupto(av, res);
    3667             : }
    3668             : 
    3669             : /** GRUNWALD-WANG **/
    3670             : /*
    3671             : Song Wang's PhD thesis (pdf pages)
    3672             : p.25 definition of chi_b. K^Ker(chi_b) = K(b^(1/m))
    3673             : p.26 bound on the conductor (also Cohen adv. GTM 193 p.166)
    3674             : p.21 & p.34 description special case, also on wikipedia:
    3675             : http://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem#Special_fields
    3676             : p.77 Kummer case
    3677             : */
    3678             : 
    3679             : /* n > 0. Is n = 2^k ? */
    3680             : static int
    3681         329 : uispow2(ulong n) { return !(n &(n-1)); }
    3682             : 
    3683             : static GEN
    3684         378 : get_phi0(GEN bnr, GEN Lpr, GEN Ld, GEN pl, long *pr, long *pn)
    3685             : {
    3686         378 :   const long NTRY = 10; /* FIXME: magic constant */
    3687         378 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3688         378 :   GEN S = bnr_get_cyc(bnr);
    3689             :   GEN Sst, G, globGmod, loc, X, Rglob, Rloc, H, U, Lconj;
    3690             :   long i, j, r, nbfrob, nbloc, nz, t;
    3691             : 
    3692         378 :   *pn = n;
    3693         378 :   *pr = r = lg(S)-1;
    3694         378 :   if (!r) return NULL;
    3695         329 :   Sst = cgetg(r+1, t_VECSMALL); /* Z/n-dual */
    3696        1589 :   for (i=1; i<=r; i++) Sst[i] = ugcdiu(gel(S,i), n);
    3697         329 :   if (Sst[1] != n) return NULL;
    3698         329 :   Lconj = NULL;
    3699         329 :   nbloc = nbfrob = lg(Lpr)-1;
    3700         329 :   if (uispow2(n))
    3701             :   {
    3702         259 :     long l = lg(pl), k = 0;
    3703         259 :     GEN real = cgetg(l, t_VECSMALL);
    3704         973 :     for (i = 1; i < l; i++)
    3705         714 :       if (pl[i] == -1) real[++k] = i;
    3706         259 :     if (k)
    3707             :     {
    3708         259 :       GEN nf = bnr_get_nf(bnr), I = bid_get_fact(bnr_get_bid(bnr));
    3709         259 :       GEN v, y, C = idealchineseinit(bnr, I);
    3710         259 :       long r1 = nf_get_r1(nf), n = nbrows(I);
    3711         259 :       nbloc += k;
    3712         259 :       Lconj = cgetg(k+1, t_VEC);
    3713         259 :       v = const_vecsmall(r1, 1);
    3714         259 :       y = const_vec(n, gen_1);
    3715         707 :       for (i = 1; i <= k; i++)
    3716             :       {
    3717         448 :         v[real[i]] = -1; gel(Lconj,i) = idealchinese(nf, mkvec2(C,v), y);
    3718         448 :         v[real[i]] = 1;
    3719             :       }
    3720             :     }
    3721             :   }
    3722         329 :   globGmod = cgetg(r+1,t_MAT);
    3723         329 :   G = cgetg(r+1,t_VECSMALL);
    3724        1589 :   for (i = 1; i <= r; i++)
    3725             :   {
    3726        1260 :     G[i] = n / Sst[i]; /* pairing between S and Sst */
    3727        1260 :     gel(globGmod,i) = cgetg(nbloc+1,t_VECSMALL);
    3728             :   }
    3729             : 
    3730             :   /* compute images of Frobenius elements (and complex conjugation) */
    3731         329 :   loc = cgetg(nbloc+1,t_VECSMALL);
    3732         700 :   for (i = 1; i <= nbloc; i++)
    3733             :   {
    3734             :     long L;
    3735         539 :     if (i <= nbfrob)
    3736             :     {
    3737         224 :       X = gel(Lpr, i);
    3738         224 :       L = Ld[i];
    3739             :     }
    3740             :     else
    3741             :     { /* X = 1 (mod f), sigma_i(x) < 0, positive at all other real places */
    3742         315 :       X = gel(Lconj, i-nbfrob);
    3743         315 :       L = 2;
    3744             :     }
    3745         539 :     X = ZV_to_Flv(isprincipalray(bnr,X), n);
    3746        2275 :     for (nz=0,j=1; j<=r; j++)
    3747             :     {
    3748        1736 :       ulong c = (X[j] * G[j]) % L;
    3749        1736 :       ucoeff(globGmod,i,j) = c;
    3750        1736 :       if (c) nz = 1;
    3751             :     }
    3752         539 :     if (!nz) return NULL;
    3753         371 :     loc[i] = L;
    3754             :   }
    3755             : 
    3756             :   /* try some random elements in the dual */
    3757         161 :   Rglob = cgetg(r+1,t_VECSMALL);
    3758         466 :   for (t=0; t<NTRY; t++) {
    3759        1783 :     for (j = 1; j <= r; j++) Rglob[j] = random_Fl(Sst[j]);
    3760         459 :     Rloc = zm_zc_mul(globGmod,Rglob);
    3761         996 :     for (i = 1; i <= nbloc; i++)
    3762         842 :       if (Rloc[i] % loc[i] == 0) break;
    3763         459 :     if (i > nbloc) return zv_to_ZV(Rglob);
    3764             :   }
    3765             : 
    3766             :   /* try to realize some random elements of the product of the local duals */
    3767           7 :   H = ZM_hnfall_i(shallowconcat(zm_to_ZM(globGmod),
    3768             :                                 diagonal_shallow(zv_to_ZV(loc))), &U, 2);
    3769             :   /* H,U nbloc x nbloc */
    3770           7 :   Rloc = cgetg(nbloc+1,t_COL);
    3771          77 :   for (t = 0; t < NTRY; t++)
    3772             :   { /* nonzero random coordinate */ /* TODO add special case ? */
    3773         560 :     for (i = 1; i <= nbloc; i++) gel(Rloc,i) = stoi(1 + random_Fl(loc[i]-1));
    3774          70 :     Rglob = hnf_invimage(H, Rloc);
    3775          70 :     if (Rglob)
    3776             :     {
    3777           0 :       Rglob = ZM_ZC_mul(U,Rglob);
    3778           0 :       return vecslice(Rglob,1,r);
    3779             :     }
    3780             :   }
    3781           7 :   return NULL;
    3782             : }
    3783             : 
    3784             : static GEN
    3785         378 : bnrgwsearch(GEN bnr, GEN Lpr, GEN Ld, GEN pl)
    3786             : {
    3787         378 :   pari_sp av = avma;
    3788             :   long n, r;
    3789         378 :   GEN phi0 = get_phi0(bnr,Lpr,Ld,pl, &r,&n), gn, v, H,U;
    3790         378 :   if (!phi0) return gc_const(av, gen_0);
    3791         154 :   gn = stoi(n);
    3792             :   /* compute kernel of phi0 */
    3793         154 :   v = ZV_extgcd(vec_append(phi0, gn));
    3794         154 :   U = vecslice(gel(v,2), 1,r);
    3795         154 :   H = ZM_hnfmodid(rowslice(U, 1,r), gn);
    3796         154 :   return gerepileupto(av, H);
    3797             : }
    3798             : 
    3799             : GEN
    3800         154 : bnfgwgeneric(GEN bnf, GEN Lpr, GEN Ld, GEN pl, long var)
    3801             : {
    3802         154 :   pari_sp av = avma;
    3803         154 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3804             :   forprime_t S;
    3805         154 :   GEN bnr = NULL, ideal = gen_1, nf, dec, H = gen_0, finf, pol;
    3806             :   ulong ell, p;
    3807             :   long deg, i, degell;
    3808         154 :   (void)uisprimepower(n, &ell);
    3809         154 :   nf = bnf_get_nf(bnf);
    3810         154 :   deg = nf_get_degree(nf);
    3811         154 :   degell = ugcd(deg,ell-1);
    3812         154 :   finf = cgetg(lg(pl),t_VEC);
    3813         427 :   for (i=1; i<lg(pl); i++) gel(finf,i) = pl[i]==-1 ? gen_1 : gen_0;
    3814             : 
    3815         154 :   u_forprime_init(&S, 2, ULONG_MAX);
    3816         679 :   while ((p = u_forprime_next(&S))) {
    3817         679 :     if (Fl_powu(p % ell, degell, ell) != 1) continue; /* ell | p^deg-1 ? */
    3818         364 :     dec = idealprimedec(nf, utoipos(p));
    3819         700 :     for (i=1; i<lg(dec); i++) {
    3820         490 :       GEN pp = gel(dec,i);
    3821         490 :       if (RgV_isin(Lpr,pp)) continue;
    3822             :         /* TODO also accept the prime ideals at which there is a condition
    3823             :          * (use local Artin)? */
    3824         434 :       if (smodis(idealnorm(nf,pp),ell) != 1) continue; /* ell | N(pp)-1 ? */
    3825         378 :       ideal = idealmul(bnf,ideal,pp);
    3826             :       /* TODO: give factorization ? */
    3827         378 :       bnr = Buchray(bnf, mkvec2(ideal,finf), nf_INIT);
    3828         378 :       H = bnrgwsearch(bnr,Lpr,Ld,pl);
    3829         378 :       if (H != gen_0)
    3830             :       {
    3831         154 :         pol = rnfkummer(bnr,H,nf_get_prec(nf));
    3832         154 :         setvarn(pol, var);
    3833         154 :         return gerepileupto(av,pol);
    3834             :       }
    3835             :     }
    3836             :   }
    3837             :   pari_err_BUG("bnfgwgeneric (no suitable p)"); /*LCOV_EXCL_LINE*/
    3838             :   return NULL;/*LCOV_EXCL_LINE*/
    3839             : }
    3840             : 
    3841             : /* pr.p != ell */
    3842             : static GEN
    3843         168 : localextdeg(GEN nf, GEN pr, long d, ulong ell, long n)
    3844             : {
    3845             :   GEN modpr, T, p, gen, k;
    3846         168 :   if (d == 1) return gen_1;
    3847         154 :   k = powuu(ell, Z_lval(subiu(pr_norm(pr),1), ell));
    3848         154 :   k = divis(k, n / d);
    3849         154 :   modpr = nf_to_Fq_init(nf, &pr, &T, &p);
    3850         154 :   (void)Fq_sqrtn(gen_1, k, T, p, &gen);
    3851         154 :   return Fq_to_nf(gen, modpr);
    3852             : }
    3853             : /* pr.p = ell */
    3854             : static GEN
    3855         119 : localextdegell(GEN nf, GEN pr, GEN E, long d, long n)
    3856             : {
    3857             :   GEN x;
    3858         119 :   if (d == 1) return gen_1;
    3859         112 :   x = nfadd(nf, gen_1, pr_get_gen(pr));
    3860         112 :   return nfpowmodideal(nf, x, stoi(n / d), idealpow(nf, pr, E));
    3861             : }
    3862             : 
    3863             : /* Ld[i] must be nontrivial powers of the same prime ell */
    3864             : /* pl : -1 at real places at which the extension must ramify, 0 elsewhere */
    3865             : GEN
    3866         196 : nfgwkummer(GEN nf, GEN Lpr, GEN Ld, GEN pl, long var)
    3867             : {
    3868         196 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3869             :   ulong ell;
    3870         196 :   long i, l = lg(Lpr), v = uisprimepower(n, &ell);
    3871         196 :   GEN E = cgetg(l, t_COL), y = cgetg(l, t_VEC);
    3872             : 
    3873         483 :   for (i = 1; i < l; i++)
    3874             :   {
    3875         287 :     GEN pr = gel(Lpr,i), p = pr_get_p(pr);
    3876         287 :     if (!absequalui(ell, p))
    3877             :     {
    3878         168 :       gel(E, i) = gen_1;
    3879         168 :       gel(y, i) = localextdeg(nf, pr, Ld[i], ell, n);
    3880             :     }
    3881             :     else
    3882             :     {
    3883         119 :       long e = pr_get_e(pr);
    3884         119 :       gel(E, i) = addui(1 + v*e, divsi(e, subiu(p,1)));
    3885         119 :       gel(y, i) = localextdegell(nf, pr, gel(E,i), Ld[i], n);
    3886             :     }
    3887             :   }
    3888             :   /* TODO use a factoredextchinese to ease computations afterwards ? */
    3889         196 :   y = idealchinese(nf, mkvec2(mkmat2(Lpr,E), pl), y);
    3890         196 :   return gsub(gpowgs(pol_x(var),n), basistoalg(nf, y));
    3891             : }
    3892             : 
    3893             : static GEN
    3894         805 : get_vecsmall(GEN v)
    3895             : {
    3896         805 :   switch(typ(v))
    3897             :   {
    3898         679 :     case t_VECSMALL: return v;
    3899         119 :     case t_VEC: if (RgV_is_ZV(v)) return ZV_to_zv(v);
    3900             :   }
    3901           7 :   pari_err_TYPE("nfgrunwaldwang",v);
    3902             :   return NULL;/*LCOV_EXCL_LINE*/
    3903             : }
    3904             : GEN
    3905         448 : nfgrunwaldwang(GEN nf0, GEN Lpr, GEN Ld, GEN pl, long var)
    3906             : {
    3907             :   ulong n, ell, ell2;
    3908         448 :   pari_sp av = avma;
    3909             :   GEN nf, bnf;
    3910             :   long t, w, i, vnf;
    3911             : 
    3912         448 :   if (var < 0) var = 0;
    3913         448 :   nf = get_nf(nf0,&t);
    3914         448 :   if (!nf) pari_err_TYPE("nfgrunwaldwang",nf0);
    3915         448 :   vnf = nf_get_varn(nf);
    3916         448 :   if (varncmp(var, vnf) >= 0)
    3917           7 :     pari_err_PRIORITY("nfgrunwaldwang", pol_x(var), ">=", vnf);
    3918         441 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("nfgrunwaldwang",Lpr);
    3919         427 :   if (lg(Lpr) != lg(Ld)) pari_err_DIM("nfgrunwaldwang [#Lpr != #Ld]");
    3920         420 :   if (nf_get_degree(nf)==1) Lpr = shallowcopy(Lpr);
    3921         945 :   for (i=1; i<lg(Lpr); i++) {
    3922         532 :     GEN pr = gel(Lpr,i);
    3923         532 :     if (nf_get_degree(nf)==1 && typ(pr)==t_INT)
    3924          77 :       gel(Lpr,i) = gel(idealprimedec(nf,pr), 1);
    3925         455 :     else checkprid(pr);
    3926             :   }
    3927         413 :   if (lg(pl)-1 != nf_get_r1(nf))
    3928           7 :     pari_err_DOMAIN("nfgrunwaldwang [pl should have r1 components]", "#pl",
    3929           7 :         "!=", stoi(nf_get_r1(nf)), stoi(lg(pl)-1));
    3930             : 
    3931         406 :   Ld = get_vecsmall(Ld);
    3932         399 :   pl = get_vecsmall(pl);
    3933         399 :   bnf = get_bnf(nf0,&t);
    3934         399 :   n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3935             : 
    3936         399 :   if (!uisprimepower(n, &ell))
    3937           7 :     pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (a)");
    3938         882 :   for (i=1; i<lg(Ld); i++)
    3939         497 :     if (Ld[i]!=1 && (!uisprimepower(Ld[i],&ell2) || ell2!=ell))
    3940           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (b)");
    3941        1015 :   for (i=1; i<lg(pl); i++)
    3942         637 :     if (pl[i]==-1 && ell%2)
    3943           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (c)");
    3944             : 
    3945         378 :   w = bnf? bnf_get_tuN(bnf): itos(gel(nfrootsof1(nf),1));
    3946             : 
    3947             :   /* TODO choice between kummer and generic ? Let user choose between speed
    3948             :    * and size */
    3949         378 :   if (w%n==0 && lg(Ld)>1)
    3950         196 :     return gerepileupto(av, nfgwkummer(nf,Lpr,Ld,pl,var));
    3951         182 :   if (ell==n)
    3952             :   {
    3953         154 :     if (!bnf) bnf = Buchall(nf, nf_FORCE, 0);
    3954         154 :     return gerepileupto(av, bnfgwgeneric(bnf,Lpr,Ld,pl,var));
    3955             :   }
    3956          28 :   pari_err_IMPL("nfgrunwaldwang for nonprime degree");
    3957             :   return NULL; /*LCOV_EXCL_LINE*/
    3958             : }
    3959             : 
    3960             : /** HASSE INVARIANTS **/
    3961             : 
    3962             : /* TODO long -> ulong + uel */
    3963             : static GEN
    3964        1022 : hasseconvert(GEN H, long n)
    3965             : {
    3966             :   GEN h, c;
    3967             :   long i, l;
    3968        1022 :   switch(typ(H)) {
    3969         952 :     case t_VEC:
    3970         952 :       l = lg(H); h = cgetg(l,t_VECSMALL);
    3971         952 :       if (l == 1) return h;
    3972         840 :       c = gel(H,1);
    3973         840 :       if (typ(c) == t_VEC && l == 3)
    3974         322 :         return mkvec2(gel(H,1),hasseconvert(gel(H,2),n));
    3975        1449 :       for (i=1; i<l; i++)
    3976             :       {
    3977         959 :         c = gel(H,i);
    3978         959 :         switch(typ(c)) {
    3979         714 :           case t_INT:  break;
    3980           7 :           case t_INTMOD:
    3981           7 :             c = gel(c,2); break;
    3982         217 :           case t_FRAC :
    3983         217 :             c = gmulgs(c,n);
    3984         217 :             if (typ(c) == t_INT) break;
    3985           7 :             pari_err_DOMAIN("hasseconvert [degree should be a denominator of the invariant]", "denom(h)", "ndiv", stoi(n), Q_denom(gel(H,i)));
    3986          21 :           default : pari_err_TYPE("Hasse invariant", c);
    3987             :         }
    3988         931 :         h[i] = smodis(c,n);
    3989             :       }
    3990         490 :       return h;
    3991          63 :     case t_VECSMALL: return H;
    3992             :   }
    3993           7 :   pari_err_TYPE("Hasse invariant", H);
    3994             :   return NULL;/*LCOV_EXCL_LINE*/
    3995             : }
    3996             : 
    3997             : /* assume f >= 2 */
    3998             : static long
    3999         420 : cyclicrelfrob0(GEN nf, GEN aut, GEN pr, GEN q, long f, long g)
    4000             : {
    4001         420 :   GEN T, p, a, b, modpr = nf_to_Fq_init(nf,&pr,&T,&p);
    4002             :   long s;
    4003             : 
    4004         420 :   a = pol_x(nf_get_varn(nf));
    4005         420 :   b = galoisapply(nf, aut, modpr_genFq(modpr));
    4006         420 :   b = nf_to_Fq(nf, b, modpr);
    4007        1323 :   for (s = 0; !ZX_equal(a, b); s++) a = Fq_pow(a, q, T, p);
    4008         420 :   return g * Fl_inv(s, f); /* < n */
    4009             : }
    4010             : 
    4011             : static long
    4012        1008 : cyclicrelfrob(GEN rnf, GEN auts, GEN pr)
    4013             : {
    4014        1008 :   pari_sp av = avma;
    4015        1008 :   long f,g,frob, n = rnf_get_degree(rnf);
    4016        1008 :   GEN P = rnfidealprimedec(rnf, pr);
    4017             : 
    4018        1008 :   if (pr_get_e(gel(P,1)) > pr_get_e(pr))
    4019           0 :     pari_err_DOMAIN("cyclicrelfrob","e(PR/pr)",">",gen_1,pr);
    4020        1008 :   g = lg(P) - 1;
    4021        1008 :   f = n / g;
    4022             : 
    4023        1008 :   if (f <= 2) frob = g % n;
    4024             :   else {
    4025         420 :     GEN nf2, PR = gel(P,1);
    4026         420 :     GEN autabs = rnfeltreltoabs(rnf,gel(auts,g));
    4027         420 :     nf2 = obj_check(rnf,rnf_NFABS);
    4028         420 :     autabs = nfadd(nf2, autabs, gmul(rnf_get_k(rnf), rnf_get_alpha(rnf)));
    4029         420 :     frob = cyclicrelfrob0(nf2, autabs, PR, pr_norm(pr), f, g);
    4030             :   }
    4031        1008 :   return gc_long(av, frob);
    4032             : }
    4033             : 
    4034             : static long
    4035         581 : localhasse(GEN rnf, GEN cnd, GEN pl, GEN auts, GEN b, long k)
    4036             : {
    4037         581 :   pari_sp av = avma;
    4038             :   long v, m, h, lfa, frob, n, i;
    4039             :   GEN previous, y, pr, nf, q, fa;
    4040         581 :   nf = rnf_get_nf(rnf);
    4041         581 :   n = rnf_get_degree(rnf);
    4042         581 :   pr = gcoeff(cnd,k,1);
    4043         581 :   v = nfval(nf, b, pr);
    4044         581 :   m = lg(cnd)>1 ? nbrows(cnd) : 0;
    4045             : 
    4046             :   /* add the valuation of b to the conductor... */
    4047         581 :   previous = gcoeff(cnd,k,2);
    4048         581 :   gcoeff(cnd,k,2) = addis(previous, v);
    4049             : 
    4050         581 :   y = const_vec(m, gen_1);
    4051         581 :   gel(y,k) = b;
    4052             :   /* find a factored element y congruent to b mod pr^(vpr(b)+vpr(cnd)) and to 1 mod the conductor. */
    4053         581 :   y = factoredextchinese(nf, cnd, y, pl, &fa);
    4054         581 :   h = 0;
    4055         581 :   lfa = nbrows(fa);
    4056             :   /* sum of all Hasse invariants of (rnf/nf,aut,y) is 0, Hasse invariants at q!=pr are easy, Hasse invariant at pr is the same as for al=(rnf/nf,aut,b). */
    4057        1120 :   for (i=1; i<=lfa; i++) {
    4058         539 :     q = gcoeff(fa,i,1);
    4059         539 :     if (cmp_prime_ideal(pr,q)) {
    4060         504 :       frob = cyclicrelfrob(rnf, auts, q);
    4061         504 :       frob = Fl_mul(frob,umodiu(gcoeff(fa,i,2),n),n);
    4062         504 :       h = Fl_add(h,frob,n);
    4063             :     }
    4064             :   }
    4065             :   /* ...then restore it. */
    4066         581 :   gcoeff(cnd,k,2) = previous;
    4067         581 :   return gc_long(av, Fl_neg(h,n));
    4068             : }
    4069             : 
    4070             : static GEN
    4071         749 : allauts(GEN rnf, GEN aut)
    4072             : {
    4073         749 :   long n = rnf_get_degree(rnf), i;
    4074         749 :   GEN pol = rnf_get_pol(rnf), vaut;
    4075         749 :   if (n==1) n=2;
    4076         749 :   vaut = cgetg(n,t_VEC);
    4077         749 :   aut = lift_shallow(rnfbasistoalg(rnf,aut));
    4078         749 :   if (typ(aut) != t_POL || varn(pol) != varn(aut))
    4079           0 :     pari_err_TYPE("alg_cyclic", aut);
    4080         749 :   gel(vaut,1) = aut;
    4081        1064 :   for (i=1; i<n-1; i++)
    4082         315 :     gel(vaut,i+1) = RgX_rem(poleval(gel(vaut,i), aut), pol);
    4083         749 :   return vaut;
    4084             : }
    4085             : 
    4086             : static GEN
    4087         259 : clean_factor(GEN fa)
    4088             : {
    4089         259 :   GEN P2,E2, P = gel(fa,1), E = gel(fa,2);
    4090         259 :   long l = lg(P), i, j = 1;
    4091         259 :   P2 = cgetg(l, t_COL);
    4092         259 :   E2 = cgetg(l, t_COL);
    4093         902 :   for (i = 1;i < l; i++)
    4094         643 :     if (signe(gel(E,i))) {
    4095         489 :       gel(P2,j) = gel(P,i);
    4096         489 :       gel(E2,j) = gel(E,i); j++;
    4097             :     }
    4098         259 :   setlg(P2,j);
    4099         259 :   setlg(E2,j); return mkmat2(P2,E2);
    4100             : }
    4101             : 
    4102             : /* shallow concat x[1],...x[nx],y[1], ... y[ny], returning a t_COL. To be
    4103             :  * used when we do not know whether x,y are t_VEC or t_COL */
    4104             : static GEN
    4105         518 : colconcat(GEN x, GEN y)
    4106             : {
    4107         518 :   long i, lx = lg(x), ly = lg(y);
    4108         518 :   GEN z=cgetg(lx+ly-1, t_COL);
    4109         868 :   for (i=1; i<lx; i++) z[i]     = x[i];
    4110        1454 :   for (i=1; i<ly; i++) z[lx+i-1]= y[i];
    4111         518 :   return z;
    4112             : }
    4113             : 
    4114             : /* return v(x) at all primes in listpr, replace x by cofactor */
    4115             : static GEN
    4116        1008 : nfmakecoprime(GEN nf, GEN *px, GEN listpr)
    4117             : {
    4118        1008 :   long j, l = lg(listpr);
    4119        1008 :   GEN x1, x = *px, L = cgetg(l, t_COL);
    4120             : 
    4121        1008 :   if (typ(x) != t_MAT)
    4122             :   { /* scalar, divide at the end (fast valuation) */
    4123         868 :     x1 = NULL;
    4124        2015 :     for (j=1; j<l; j++)
    4125             :     {
    4126        1147 :       GEN pr = gel(listpr,j), e;
    4127        1147 :       long v = nfval(nf, x, pr);
    4128        1147 :       e = stoi(v); gel(L,j) = e;
    4129        1336 :       if (v) x1 = x1? idealmulpowprime(nf, x1, pr, e)
    4130         189 :                     : idealpow(nf, pr, e);
    4131             :     }
    4132         868 :     if (x1) x = idealdivexact(nf, idealhnf(nf,x), x1);
    4133             :   }
    4134             :   else
    4135             :   { /* HNF, divide as we proceed (reduce size) */
    4136         196 :     for (j=1; j<l; j++)
    4137             :     {
    4138          56 :       GEN pr = gel(listpr,j);
    4139          56 :       long v = idealval(nf, x, pr);
    4140          56 :       gel(L,j) = stoi(v);
    4141          56 :       if (v) x = idealmulpowprime(nf, x, pr, stoi(-v));
    4142             :     }
    4143             :   }
    4144        1008 :   *px = x; return L;
    4145             : }
    4146             : 
    4147             : /* Caveat: factorizations are not sorted wrt cmp_prime_ideal: Lpr comes first */
    4148             : static GEN
    4149         259 : computecnd(GEN rnf, GEN Lpr)
    4150             : {
    4151             :   GEN id, nf, fa, Le, P,E;
    4152         259 :   long n = rnf_get_degree(rnf);
    4153             : 
    4154         259 :   nf = rnf_get_nf(rnf);
    4155         259 :   id = rnf_get_idealdisc(rnf);
    4156         259 :   Le = nfmakecoprime(nf, &id, Lpr);
    4157         259 :   fa = idealfactor(nf, id); /* part of D_{L/K} coprime with Lpr */
    4158         259 :   P =  colconcat(Lpr,gel(fa,1));
    4159         259 :   E =  colconcat(Le, gel(fa,2));
    4160         259 :   fa = mkmat2(P, gdiventgs(E, eulerphiu(n)));
    4161         259 :   return mkvec2(fa, clean_factor(fa));
    4162             : }
    4163             : 
    4164             : /* h >= 0 */
    4165             : static void
    4166          21 : nextgen(GEN gene, long h, GEN* gens, GEN* hgens, long* ngens, long* curgcd) {
    4167          21 :   long nextgcd = ugcd(h,*curgcd);
    4168          21 :   if (nextgcd == *curgcd) return;
    4169          21 :   (*ngens)++;
    4170          21 :   gel(*gens,*ngens) = gene;
    4171          21 :   gel(*hgens,*ngens) = utoi(h);
    4172          21 :   *curgcd = nextgcd;
    4173          21 :   return;
    4174             : }
    4175             : 
    4176             : static int
    4177          35 : dividesmod(long d, long h, long n) { return !(h%cgcd(d,n)); }
    4178             : 
    4179             : /* ramified prime with nontrivial Hasse invariant */
    4180             : static GEN
    4181          21 : localcomplete(GEN rnf, GEN pl, GEN cnd, GEN auts, long j, long n, long h, long* v)
    4182             : {
    4183             :   GEN nf, gens, hgens, pr, modpr, T, p, sol, U, b, gene, randg, pu;
    4184             :   long ngens, i, d, np, d1, d2, hg, dnf, vcnd, curgcd;
    4185          21 :   nf = rnf_get_nf(rnf);
    4186          21 :   pr = gcoeff(cnd,j,1);
    4187          21 :   np = umodiu(pr_norm(pr), n);
    4188          21 :   dnf = nf_get_degree(nf);
    4189          21 :   vcnd = itos(gcoeff(cnd,j,2));
    4190          21 :   ngens = 13+dnf;
    4191          21 :   gens = zerovec(ngens);
    4192          21 :   hgens = zerovec(ngens);
    4193          21 :   *v = 0;
    4194          21 :   curgcd = 0;
    4195          21 :   ngens = 0;
    4196             : 
    4197          21 :   if (!uisprime(n)) {
    4198           0 :     gene =  pr_get_gen(pr);
    4199           0 :     hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4200           0 :     nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4201             :   }
    4202             : 
    4203          21 :   if (ugcd(np,n) != 1) { /* GCD(Np,n) != 1 */
    4204          21 :     pu = idealprincipalunits(nf,pr,vcnd);
    4205          21 :     pu = abgrp_get_gen(pu);
    4206          42 :     for (i=1; i<lg(pu) && !dividesmod(curgcd,h,n); i++) {
    4207          21 :       gene = gel(pu,i);
    4208          21 :       hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4209          21 :       nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4210             :     }
    4211             :   }
    4212             : 
    4213          21 :   d = ugcd(np-1,n);
    4214          21 :   if (d != 1) { /* GCD(Np-1,n) != 1 */
    4215           7 :     modpr = nf_to_Fq_init(nf, &pr, &T, &p);
    4216           7 :     while (!dividesmod(curgcd,h,n)) { /* TODO gener_FpXQ_local */
    4217           0 :       if (T==NULL) randg = randomi(p);
    4218           0 :       else randg = random_FpX(degpol(T), varn(T),p);
    4219             : 
    4220           0 :       if (!gequal0(randg) && !gequal1(randg)) {
    4221           0 :         gene = Fq_to_nf(randg, modpr);
    4222           0 :         hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4223           0 :         nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4224             :       }
    4225             :     }
    4226             :   }
    4227             : 
    4228          21 :   setlg(gens,ngens+1);
    4229          21 :   setlg(hgens,ngens+1);
    4230             : 
    4231          21 :   sol = ZV_extgcd(hgens);
    4232          21 :   U = ZV_to_Flv(gmael(sol,2,ngens), n);
    4233          21 :   d = itou(gel(sol,1));
    4234          21 :   d1 = ugcd(d, n);
    4235          21 :   d2 = d / d1;
    4236          21 :   d = Fl_mul(h / d1, Fl_inv(d2,n), n);
    4237          21 :   if (d != 1) U = Flv_Fl_mul(U, d, n);
    4238          42 :   for (i = 1, b = gen_1; i <= ngens; i++)
    4239          21 :     if (U[i]) b = nfmul(nf, b, nfpow_u(nf, gel(gens,i), U[i]));
    4240          21 :   *v = U[1]; return b;
    4241             : }
    4242             : 
    4243             : static int
    4244         381 : testsplits(GEN data, GEN fa)
    4245             : {
    4246         381 :   GEN rnf = gel(data,1), forbid = gel(data,2), P = gel(fa,1), E = gel(fa,2);
    4247         381 :   long i, n, l = lg(P);
    4248             : 
    4249         777 :   for (i = 1; i < l; i++)
    4250             :   {
    4251         415 :     GEN pr = gel(P,i);
    4252         415 :     if (tablesearch(forbid, pr, &cmp_prime_ideal)) return 0;
    4253             :   }
    4254         362 :   n = rnf_get_degree(rnf);
    4255         507 :   for (i = 1; i < l; i++)
    4256             :   {
    4257         248 :     long e = itos(gel(E,i)) % n;
    4258         248 :     if (e)
    4259             :     {
    4260         234 :       GEN L = rnfidealprimedec(rnf, gel(P,i));
    4261         234 :       long g = lg(L) - 1;
    4262         234 :       if ((e * g) % n) return 0;
    4263             :     }
    4264             :   }
    4265         259 :   return 1;
    4266             : }
    4267             : 
    4268             : /* remove entries with Hasse invariant 0 */
    4269             : static GEN
    4270         546 : hassereduce(GEN hf)
    4271             : {
    4272         546 :   GEN pr,h, PR = gel(hf,1), H = gel(hf,2);
    4273         546 :   long i, j, l = lg(PR);
    4274             : 
    4275         546 :   pr= cgetg(l, t_VEC);
    4276         546 :   h = cgetg(l, t_VECSMALL);
    4277        1253 :   for (i = j = 1; i < l; i++)
    4278         707 :     if (H[i]) {
    4279         378 :       gel(pr,j) = gel(PR,i);
    4280         378 :       h[j] = H[i]; j++;
    4281             :     }
    4282         546 :   setlg(pr,j);
    4283         546 :   setlg(h,j); return mkvec2(pr,h);
    4284             : }
    4285             : 
    4286             : /* rnf complete */
    4287             : static GEN
    4288         259 : alg_complete0(GEN rnf, GEN aut, GEN hf, GEN hi, long maxord)
    4289             : {
    4290         259 :   pari_sp av = avma;
    4291             :   GEN nf, pl, pl2, cnd, prcnd, cnds, y, Lpr, auts, b, fa, data, hfe;
    4292             :   GEN forbid, al, ind;
    4293             :   long D, n, d, i, j, l;
    4294         259 :   nf = rnf_get_nf(rnf);
    4295         259 :   n = rnf_get_degree(rnf);
    4296         259 :   d = nf_get_degree(nf);
    4297         259 :   D = d*n*n;
    4298         259 :   checkhasse(nf,hf,hi,n);
    4299         259 :   hf = hassereduce(hf);
    4300         259 :   Lpr = gel(hf,1);
    4301         259 :   hfe = gel(hf,2);
    4302             : 
    4303         259 :   auts = allauts(rnf,aut);
    4304             : 
    4305         259 :   pl = leafcopy(hi); /* conditions on the final b */
    4306         259 :   pl2 = leafcopy(hi); /* conditions for computing local Hasse invariants */
    4307         259 :   l = lg(pl); ind = cgetg(l, t_VECSMALL);
    4308         658 :   for (i = j = 1; i < l; i++)
    4309         399 :     if (hi[i]) { pl[i] = -1; pl2[i] = 1; } else ind[j++] = i;
    4310         259 :   setlg(ind, j);
    4311         259 :   y = nfpolsturm(nf, rnf_get_pol(rnf), ind);
    4312         483 :   for (i = 1; i < j; i++)
    4313         224 :     if (!signe(gel(y,i))) { pl[ind[i]] = 1; pl2[ind[i]] = 1; }
    4314             : 
    4315         259 :   cnds = computecnd(rnf,Lpr);
    4316         259 :   prcnd = gel(cnds,1);
    4317         259 :   cnd = gel(cnds,2);
    4318         259 :   y = cgetg(lgcols(prcnd),t_VEC);
    4319         259 :   forbid = vectrunc_init(lg(Lpr));
    4320         434 :   for (i=j=1; i<lg(Lpr); i++)
    4321             :   {
    4322         175 :     GEN pr = gcoeff(prcnd,i,1), yi;
    4323         175 :     long v, e = itou( gcoeff(prcnd,i,2) );
    4324         175 :     if (!e) {
    4325         154 :       long frob = cyclicrelfrob(rnf,auts,pr), f1 = ugcd(frob,n);
    4326         154 :       vectrunc_append(forbid, pr);
    4327         154 :       yi = gen_0;
    4328         154 :       v = ((hfe[i]/f1) * Fl_inv(frob/f1,n)) % n;
    4329             :     }
    4330             :     else
    4331          21 :       yi = localcomplete(rnf, pl2, cnd, auts, j++, n, hfe[i], &v);
    4332         175 :     gel(y,i) = yi;
    4333         175 :     gcoeff(prcnd,i,2) = stoi(e + v);
    4334             :   }
    4335         727 :   for (; i<lgcols(prcnd); i++) gel(y,i) = gen_1;
    4336         259 :   gen_sort_inplace(forbid, (void*)&cmp_prime_ideal, &cmp_nodata, NULL);
    4337         259 :   data = mkvec2(rnf,forbid);
    4338         259 :   b = factoredextchinesetest(nf,prcnd,y,pl,&fa,data,testsplits);
    4339             : 
    4340         259 :   al = cgetg(12, t_VEC);
    4341         259 :   gel(al,10)= gen_0; /* must be set first */
    4342         259 :   gel(al,1) = rnf;
    4343         259 :   gel(al,2) = auts;
    4344         259 :   gel(al,3) = basistoalg(nf,b);
    4345         259 :   gel(al,4) = hi;
    4346             :   /* add primes | disc or b with trivial Hasse invariant to hf */
    4347         259 :   Lpr = gel(prcnd,1); y = b;
    4348         259 :   (void)nfmakecoprime(nf, &y, Lpr);
    4349         259 :   Lpr = shallowconcat(Lpr, gel(idealfactor(nf,y), 1));
    4350         259 :   settyp(Lpr,t_VEC);
    4351         259 :   hf = mkvec2(Lpr, shallowconcat(hfe, const_vecsmall(lg(Lpr)-lg(hfe), 0)));
    4352         259 :   gel(al,5) = hf;
    4353         259 :   gel(al,6) = gen_0;
    4354         259 :   gel(al,7) = matid(D);
    4355         259 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    4356         259 :   gel(al,9) = algnatmultable(al,D);
    4357         259 :   gel(al,11)= algtracebasis(al);
    4358         259 :   if (maxord) al = alg_maximal_primes(al, prV_primes(Lpr));
    4359         259 :   return gerepilecopy(av, al);
    4360             : }
    4361             : 
    4362             : GEN
    4363           0 : alg_complete(GEN rnf, GEN aut, GEN hf, GEN hi, long maxord)
    4364             : {
    4365           0 :   long n = rnf_get_degree(rnf);
    4366           0 :   rnfcomplete(rnf);
    4367           0 :   return alg_complete0(rnf,aut,hasseconvert(hf,n),hasseconvert(hi,n), maxord);
    4368             : }
    4369             : 
    4370             : void
    4371        1358 : checkhasse(GEN nf, GEN hf, GEN hi, long n)
    4372             : {
    4373             :   GEN Lpr, Lh;
    4374             :   long i, sum;
    4375        1358 :   if (typ(hf) != t_VEC || lg(hf) != 3) pari_err_TYPE("checkhasse [hf]", hf);
    4376        1351 :   Lpr = gel(hf,1);
    4377        1351 :   Lh = gel(hf,2);
    4378        1351 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("checkhasse [Lpr]", Lpr);
    4379        1351 :   if (typ(Lh) != t_VECSMALL) pari_err_TYPE("checkhasse [Lh]", Lh);
    4380        1351 :   if (typ(hi) != t_VECSMALL) pari_err_TYPE("checkhasse [hi]", hi);
    4381        1351 :   if ((nf && lg(hi) != nf_get_r1(nf)+1))
    4382           7 :     pari_err_DOMAIN("checkhasse [hi should have r1 components]","#hi","!=",stoi(nf_get_r1(nf)),stoi(lg(hi)-1));
    4383        1344 :   if (lg(Lpr) != lg(Lh))
    4384           7 :     pari_err_DIM("checkhasse [Lpr and Lh should have same length]");
    4385        3150 :   for (i=1; i<lg(Lpr); i++) checkprid(gel(Lpr,i));
    4386        1337 :   if (lg(gen_sort_uniq(Lpr, (void*)cmp_prime_ideal, cmp_nodata)) < lg(Lpr))
    4387           7 :     pari_err(e_MISC, "error in checkhasse [duplicate prime ideal]");
    4388        1330 :   sum = 0;
    4389        3129 :   for (i=1; i<lg(Lh); i++) sum = (sum+Lh[i])%n;
    4390        3122 :   for (i=1; i<lg(hi); i++) {
    4391        1806 :       if (hi[i] && 2*hi[i] != n) pari_err_DOMAIN("checkhasse", "Hasse invariant at real place [must be 0 or 1/2]", "!=", n%2? gen_0 : stoi(n/2), stoi(hi[i]));
    4392        1792 :       sum = (sum+hi[i])%n;
    4393             :   }
    4394        1316 :   if (sum<0) sum = n+sum;
    4395        1316 :   if (sum != 0)
    4396           7 :     pari_err_DOMAIN("checkhasse","sum(Hasse invariants)","!=",gen_0,Lh);
    4397        1309 : }
    4398             : 
    4399             : static GEN
    4400         357 : hassecoprime(GEN hf, GEN hi, long n)
    4401             : {
    4402         357 :   pari_sp av = avma;
    4403             :   long l, i, j, lk, inv;
    4404             :   GEN fa, P,E, res, hil, hfl;
    4405         357 :   hi = hasseconvert(hi, n);
    4406         343 :   hf = hasseconvert(hf, n);
    4407         322 :   checkhasse(NULL,hf,hi,n);
    4408         280 :   fa = factoru(n);
    4409         280 :   P = gel(fa,1); l = lg(P);
    4410         280 :   E = gel(fa,2);
    4411         280 :   res = cgetg(l,t_VEC);
    4412         567 :   for (i=1; i<l; i++) {
    4413         287 :     lk = upowuu(P[i],E[i]);
    4414         287 :     inv = Fl_invsafe((n/lk)%lk, lk);
    4415         287 :     hil = gcopy(hi);
    4416         287 :     hfl = gcopy(hf);
    4417             : 
    4418         287 :     if (P[i] == 2)
    4419         623 :       for (j=1; j<lg(hil); j++) hil[j] = hi[j]==0 ? 0 : lk/2;
    4420             :     else
    4421          98 :       for (j=1; j<lg(hil); j++) hil[j] = 0;
    4422         819 :     for (j=1; j<lgcols(hfl); j++) gel(hfl,2)[j] = (gel(hf,2)[j]*inv)%lk;
    4423         287 :     hfl = hassereduce(hfl);
    4424         287 :     gel(res,i) = mkvec3(hfl,hil,utoi(lk));
    4425             :   }
    4426             : 
    4427         280 :   return gerepilecopy(av, res);
    4428             : }
    4429             : 
    4430             : /* no garbage collection */
    4431             : static GEN
    4432          77 : genefrob(GEN nf, GEN gal, GEN r)
    4433             : {
    4434             :   long i;
    4435          77 :   GEN g = identity_perm(nf_get_degree(nf)), fa = Z_factor(r), p, pr, frob;
    4436         126 :   for (i=1; i<lgcols(fa); i++) {
    4437          49 :     p = gcoeff(fa,i,1);
    4438          49 :     pr = idealprimedec(nf, p);
    4439          49 :     pr = gel(pr,1);
    4440          49 :     frob = idealfrobenius(nf, gal, pr);
    4441          49 :     g = perm_mul(g, perm_pow(frob, gcoeff(fa,i,2)));
    4442             :   }
    4443          77 :   return g;
    4444             : }
    4445             : 
    4446             : static GEN
    4447         259 : rnfcycaut(GEN rnf)
    4448             : {
    4449         259 :   GEN nf2 = obj_check(rnf, rnf_NFABS);
    4450             :   GEN L, alpha, pol, salpha, s, sj, polabs, k, X, pol0, nf;
    4451             :   long i, d, j;
    4452         259 :   d = rnf_get_degree(rnf);
    4453         259 :   L = galoisconj(nf2,NULL);
    4454         259 :   alpha = lift_shallow(rnf_get_alpha(rnf));
    4455         259 :   pol = rnf_get_pol(rnf);
    4456         259 :   k = rnf_get_k(rnf);
    4457         259 :   polabs = rnf_get_polabs(rnf);
    4458         259 :   nf = rnf_get_nf(rnf);
    4459         259 :   pol0 = nf_get_pol(nf);
    4460         259 :   X = RgX_rem(pol_x(varn(pol0)), pol0);
    4461             : 
    4462             :   /* TODO check mod prime of degree 1 */
    4463         364 :   for (i=1; i<lg(L); i++) {
    4464         364 :     s = gel(L,i);
    4465         364 :     salpha = RgX_RgXQ_eval(alpha,s,polabs);
    4466         364 :     if (!gequal(alpha,salpha)) continue;
    4467             : 
    4468         322 :     s = lift_shallow(rnfeltabstorel(rnf,s));
    4469         322 :     sj = s = gsub(s, gmul(k,X));
    4470         623 :     for (j=1; !gequal0(gsub(sj,pol_x(varn(s)))); j++)
    4471         301 :       sj = RgX_RgXQ_eval(sj,s,pol);
    4472         322 :     if (j<d) continue;
    4473         259 :     return s;
    4474             :   }
    4475             :   return NULL; /*LCOV_EXCL_LINE*/
    4476             : }
    4477             : 
    4478             : /* returns the smallest prime not in P */
    4479             : static GEN
    4480          84 : extraprime(GEN P)
    4481             : {
    4482             :   forprime_t T;
    4483             :   GEN p;
    4484          84 :   forprime_init(&T, gen_2, NULL);
    4485          98 :   while ((p = forprime_next(&T))) if (!ZV_search(P, p)) break;
    4486          84 :   return p;
    4487             : }
    4488             : 
    4489             : /* true nf */
    4490             : GEN
    4491         371 : alg_hasse(GEN nf, long n, GEN hf, GEN hi, long var, long maxord)
    4492             : {
    4493         371 :   pari_sp av = avma;
    4494         371 :   GEN primary, al = gen_0, al2, rnf, hil, hfl, Ld, pl, pol, Lpr, aut, Lpr2, Ld2;
    4495             :   long i, lk, j, maxdeg;
    4496         371 :   dbg_printf(1)("alg_hasse\n");
    4497         371 :   if (n<=1) pari_err_DOMAIN("alg_hasse", "degree", "<=", gen_1, stoi(n));
    4498         357 :   primary = hassecoprime(hf, hi, n);
    4499         546 :   for (i=1; i<lg(primary); i++) {
    4500         287 :     lk = itos(gmael(primary,i,3));
    4501         287 :     hfl = gmael(primary,i,1);
    4502         287 :     hil = gmael(primary,i,2);
    4503         287 :     checkhasse(nf, hfl, hil, lk);
    4504         280 :     dbg_printf(1)("alg_hasse: i=%d hf=%Ps hi=%Ps lk=%d\n", i, hfl, hil, lk);
    4505             : 
    4506         280 :     if (lg(gel(hfl,1))>1 || lk%2==0) {
    4507         273 :       maxdeg = 1;
    4508         273 :       Lpr = gel(hfl,1);
    4509         273 :       Ld = gcopy(gel(hfl,2));
    4510         462 :       for (j=1; j<lg(Ld); j++)
    4511             :       {
    4512         189 :         Ld[j] = lk/ugcd(lk,Ld[j]);
    4513         189 :         maxdeg = maxss(Ld[j],maxdeg);
    4514             :       }
    4515         273 :       pl = leafcopy(hil);
    4516         686 :       for (j=1; j<lg(pl); j++) if(pl[j])
    4517             :       {
    4518         175 :         pl[j] = -1;
    4519         175 :         maxdeg = maxss(maxdeg,2);
    4520             :       }
    4521             : 
    4522         273 :       Lpr2 = Lpr;
    4523         273 :       Ld2 = Ld;
    4524         273 :       if (maxdeg<lk)
    4525             :       {
    4526         154 :         if (maxdeg==1 && lk==2 && lg(pl)>1) pl[1] = -1;
    4527             :         else
    4528             :         {
    4529          84 :           GEN p = extraprime(prV_primes(Lpr));
    4530          84 :           Lpr2 = vec_append(Lpr2, idealprimedec_galois(nf, p));
    4531          84 :           Ld2 = vecsmall_append(Ld2, lk);
    4532             :         }
    4533             :       }
    4534             : 
    4535         273 :       dbg_printf(2)("alg_hasse: calling nfgrunwaldwang Lpr=%Ps Pd=%Ps pl=%Ps\n",
    4536             :           Lpr, Ld, pl);
    4537         273 :       pol = nfgrunwaldwang(nf, Lpr2, Ld2, pl, var);
    4538         259 :       dbg_printf(2)("alg_hasse: calling rnfinit(%Ps)\n", pol);
    4539         259 :       rnf = rnfinit0(nf,pol,1);
    4540         259 :       dbg_printf(2)("alg_hasse: computing automorphism\n");
    4541         259 :       aut = rnfcycaut(rnf);
    4542         259 :       dbg_printf(2)("alg_hasse: calling alg_complete\n");
    4543         259 :       al2 = alg_complete0(rnf,aut,hfl,hil,maxord);
    4544             :     }
    4545           7 :     else al2 = alg_matrix(nf, lk, var, maxord);
    4546             : 
    4547         266 :     if (i==1) al = al2;
    4548           7 :     else      al = algtensor(al,al2,maxord);
    4549             :   }
    4550         259 :   return gerepilecopy(av,al);
    4551             : }
    4552             : 
    4553             : /** CYCLIC ALGEBRA WITH GIVEN HASSE INVARIANTS **/
    4554             : 
    4555             : /* no garbage collection */
    4556             : static GEN
    4557          77 : subcycloindep(GEN nf, long n, long v, GEN *pr)
    4558             : {
    4559             :   pari_sp av;
    4560             :   forprime_t S;
    4561             :   ulong p;
    4562          77 :   u_forprime_arith_init(&S, 1, ULONG_MAX, 1, n);
    4563          77 :   av = avma;
    4564          84 :   while ((p = u_forprime_next(&S)))
    4565             :   {
    4566          84 :     ulong r = pgener_Fl(p);
    4567          84 :     GEN pol = galoissubcyclo(utoipos(p), utoipos(Fl_powu(r,n,p)), 0, v);
    4568          84 :     GEN fa = nffactor(nf, pol);
    4569          84 :     if (lgcols(fa) == 2) { *pr = utoipos(r); return pol; }
    4570           7 :     set_avma(av);
    4571             :   }
    4572             :   pari_err_BUG("subcycloindep (no suitable prime = 1(mod n))"); /*LCOV_EXCL_LINE*/
    4573             :   *pr = NULL; return NULL; /*LCOV_EXCL_LINE*/
    4574             : }
    4575             : 
    4576             : GEN
    4577          84 : alg_matrix(GEN nf, long n, long v, long maxord)
    4578             : {
    4579          84 :   pari_sp av = avma;
    4580             :   GEN pol, gal, rnf, cyclo, g, r, aut;
    4581          84 :   dbg_printf(1)("alg_matrix\n");
    4582          84 :   if (n<=0) pari_err_DOMAIN("alg_matrix", "n", "<=", gen_0, stoi(n));
    4583          77 :   pol = subcycloindep(nf, n, v, &r);
    4584          77 :   rnf = rnfinit(nf, pol);
    4585          77 :   cyclo = nfinit(pol, nf_get_prec(nf));
    4586          77 :   gal = galoisinit(cyclo, NULL);
    4587          77 :   g = genefrob(cyclo,gal,r);
    4588          77 :   aut = galoispermtopol(gal,g);
    4589          77 :   return gerepileupto(av, alg_cyclic(rnf, aut, gen_1, maxord));
    4590             : }
    4591             : 
    4592             : GEN
    4593         280 : alg_hilbert(GEN nf, GEN a, GEN b, long v, long maxord)
    4594             : {
    4595         280 :   pari_sp av = avma;
    4596             :   GEN rnf, aut;
    4597         280 :   dbg_printf(1)("alg_hilbert\n");
    4598         280 :   if (!isint1(Q_denom(a)))
    4599           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(a)", "!=", gen_1,a);
    4600         273 :   if (!isint1(Q_denom(b)))
    4601           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(b)", "!=", gen_1,b);
    4602             : 
    4603         266 :   if (v < 0) v = 0;
    4604         266 :   rnf = rnfinit(nf, deg2pol_shallow(gen_1, gen_0, gneg(a), v));
    4605         259 :   aut = gneg(pol_x(v));
    4606         259 :   return gerepileupto(av, alg_cyclic(rnf, aut, b, maxord));
    4607             : }
    4608             : 
    4609             : /* return a structure representing the algebra of real numbers */
    4610             : static GEN
    4611          14 : mk_R()
    4612             : {
    4613          14 :   pari_sp av = avma;
    4614             :   GEN al;
    4615          14 :   al = zerovec(11);
    4616          14 :   gel(al,1) = stor(1,3);
    4617          14 :   gel(al,2) = mkvec(gel(al,1));
    4618          14 :   gel(al,3) = gen_1;
    4619          14 :   gel(al,4) = mkvecsmall(0);
    4620          14 :   gel(al,8) = gel(al,7) = matid(1);
    4621          14 :   gel(al,9) = mkvec(matid(1));
    4622          14 :   return gerepilecopy(av,al);
    4623             : }
    4624             : /* return a structure representing the algebra of complex numbers */
    4625             : static GEN
    4626          14 : mk_C()
    4627             : {
    4628          14 :   pari_sp av = avma;
    4629             :   GEN al, I;
    4630          14 :   al = zerovec(11);
    4631          14 :   I = gen_I();
    4632          14 :   gel(al,1) = I;
    4633          14 :   gel(al,2) = mkvec(I);
    4634          14 :   gel(al,3) = gen_1;
    4635          14 :   gel(al,4) = cgetg(1,t_VECSMALL);
    4636          14 :   gel(al,8) = gel(al,7) = matid(2);
    4637          14 :   gel(al,9) = mkvec2(
    4638             :     matid(2),
    4639             :     mkmat22(gen_0,gen_m1,gen_1,gen_0)
    4640             :   );
    4641          14 :   return gerepilecopy(av,al);
    4642             : }
    4643             : /* return a structure representing the Hamilton quaternion algebra */
    4644             : static GEN
    4645          14 : mk_H()
    4646             : {
    4647          14 :   pari_sp av = avma;
    4648             :   GEN al, I;
    4649          14 :   al = zerovec(11);
    4650          14 :   I = gen_I();
    4651          14 :   gel(al,1) = I;
    4652          14 :   gel(al,2) = mkvec(gconj(I));
    4653          14 :   gel(al,3) = gen_m1;
    4654          14 :   gel(al,4) = mkvecsmall(1);
    4655          14 :   gel(al,8) = gel(al,7) = matid(4);
    4656          14 :   gel(al,9) = mkvec4(
    4657             :     matid(4),
    4658             :     H_tomatrix(I,1),
    4659             :     H_tomatrix(mkcol4(gen_0,gen_0,gen_1,gen_0),1),
    4660             :     H_tomatrix(mkcol4(gen_0,gen_0,gen_0,gen_1),1)
    4661             :   );
    4662          14 :   return gerepilecopy(av,al);
    4663             : }
    4664             : 
    4665             : GEN
    4666        1162 : alginit(GEN A, GEN B, long v, long maxord)
    4667             : {
    4668             :   long w;
    4669        1162 :   if (typ(A) == t_COMPLEX) return mk_C();
    4670        1148 :   if (typ(A) == t_REAL)
    4671             :   {
    4672          35 :     if (is_scalar_t(typ(B)) && gequal0(B)) return mk_R();
    4673          21 :     if (typ(B) == t_FRAC && gequal(B, mkfrac(gen_1,gen_2))) return mk_H();
    4674           7 :     pari_err_DOMAIN("alginit", "real Hasse invariant [must be 0 or 1/2]", "", NULL, B);
    4675             :   }
    4676        1113 :   switch(nftyp(A))
    4677             :   {
    4678         931 :     case typ_NF:
    4679         931 :       if (v<0) v=0;
    4680         931 :       w = gvar(nf_get_pol(A));
    4681         931 :       if (varncmp(v,w)>=0) pari_err_PRIORITY("alginit", pol_x(v), ">=", w);
    4682         917 :       switch(typ(B))
    4683             :       {
    4684             :         long nB;
    4685          77 :         case t_INT: return alg_matrix(A, itos(B), v, maxord);
    4686         833 :         case t_VEC:
    4687         833 :           nB = lg(B)-1;
    4688         833 :           if (nB && typ(gel(B,1)) == t_MAT) return alg_csa_table(A,B,v,maxord);
    4689             :           switch(nB)
    4690             :           {
    4691         280 :             case 2: return alg_hilbert(A, gel(B,1), gel(B,2), v, maxord);
    4692         378 :             case 3:
    4693         378 :               if (typ(gel(B,1))!=t_INT)
    4694           7 :                   pari_err_TYPE("alginit [degree should be an integer]", gel(B,1));
    4695         371 :               return alg_hasse(A, itos(gel(B,1)), gel(B,2), gel(B,3), v,
    4696             :                                                                       maxord);
    4697             :           }
    4698             :       }
    4699          14 :       pari_err_TYPE("alginit", B); break;
    4700             : 
    4701         168 :     case typ_RNF:
    4702         168 :       if (typ(B) != t_VEC || lg(B) != 3) pari_err_TYPE("alginit", B);
    4703         154 :       return alg_cyclic(A, gel(B,1), gel(B,2), maxord);
    4704             :   }
    4705          14 :   pari_err_TYPE("alginit", A);
    4706             :   return NULL;/*LCOV_EXCL_LINE*/
    4707             : }
    4708             : 
    4709             : /* assumes al CSA or CYCLIC */
    4710             : static GEN
    4711         889 : algnatmultable(GEN al, long D)
    4712             : {
    4713             :   GEN res, x;
    4714             :   long i;
    4715         889 :   res = cgetg(D+1,t_VEC);
    4716       10752 :   for (i=1; i<=D; i++) {
    4717        9863 :     x = algnattoalg(al,col_ei(D,i));
    4718        9863 :     gel(res,i) = algZmultable(al,x);
    4719             :   }
    4720         889 :   return res;
    4721             : }
    4722             : 
    4723             : /* no garbage collection */
    4724             : static void
    4725         490 : algcomputehasse(GEN al)
    4726             : {
    4727             :   long r1, k, n, m, m1, m2, m3, i, m23, m123;
    4728             :   GEN rnf, nf, b, fab, disc2, cnd, fad, auts, pr, pl, perm, y, hi, PH, H, L;
    4729             : 
    4730         490 :   rnf = alg_get_splittingfield(al);
    4731         490 :   n = rnf_get_degree(rnf);
    4732         490 :   nf = rnf_get_nf(rnf);
    4733         490 :   b = alg_get_b(al);
    4734         490 :   r1 = nf_get_r1(nf);
    4735         490 :   auts = alg_get_auts(al);
    4736         490 :   (void)alg_get_abssplitting(al);
    4737             : 
    4738         490 :   y = nfpolsturm(nf, rnf_get_pol(rnf), NULL);
    4739         490 :   pl = cgetg(r1+1, t_VECSMALL);
    4740             :   /* real places where rnf/nf ramifies */
    4741        1029 :   for (k = 1; k <= r1; k++) pl[k] = !signe(gel(y,k));
    4742             : 
    4743             :   /* infinite Hasse invariants */
    4744         490 :   if (odd(n)) hi = const_vecsmall(r1, 0);
    4745             :   else
    4746             :   {
    4747         413 :     GEN s = nfsign(nf, b);
    4748         413 :     hi = cgetg(r1+1, t_VECSMALL);
    4749         896 :     for (k = 1; k<=r1; k++) hi[k] = (s[k] && pl[k]) ? (n/2) : 0;
    4750             :   }
    4751             : 
    4752         490 :   fab = idealfactor(nf, b);
    4753         490 :   disc2 = rnf_get_idealdisc(rnf);
    4754         490 :   L = nfmakecoprime(nf, &disc2, gel(fab,1));
    4755         490 :   m = lg(L)-1;
    4756             :   /* m1 = #{pr|b: pr \nmid disc}, m3 = #{pr|b: pr | disc} */
    4757         490 :   perm = cgetg(m+1, t_VECSMALL);
    4758         875 :   for (i=1, m1=m, k=1; k<=m; k++)
    4759         385 :     if (signe(gel(L,k))) perm[m1--] = k; else perm[i++] = k;
    4760         490 :   m3 = m - m1;
    4761             : 
    4762             :   /* disc2 : factor of disc coprime to b */
    4763         490 :   fad = idealfactor(nf, disc2);
    4764             :   /* m2 : number of prime factors of disc not dividing b */
    4765         490 :   m2 = nbrows(fad);
    4766         490 :   m23 = m2+m3;
    4767         490 :   m123 = m1+m2+m3;
    4768             : 
    4769             :   /* initialize the possibly ramified primes (hasse) and the factored conductor of rnf/nf (cnd) */
    4770         490 :   cnd = zeromatcopy(m23,2);
    4771         490 :   PH = cgetg(m123+1, t_VEC); /* ramified primes */
    4772         490 :   H = cgetg(m123+1, t_VECSMALL); /* Hasse invariant */
    4773             :   /* compute Hasse invariant at primes that are unramified in rnf/nf */
    4774         840 :   for (k=1; k<=m1; k++) {/* pr | b, pr \nmid disc */
    4775         350 :     long frob, e, j = perm[k];
    4776         350 :     pr = gcoeff(fab,j,1);
    4777         350 :     e = itos(gcoeff(fab,j,2));
    4778         350 :     frob = cyclicrelfrob(rnf, auts, pr);
    4779         350 :     gel(PH,k) = pr;
    4780         350 :     H[k] = Fl_mul(frob, e, n);
    4781             :   }
    4782             :   /* compute Hasse invariant at primes that are ramified in rnf/nf */
    4783        1015 :   for (k=1; k<=m2; k++) {/* pr \nmid b, pr | disc */
    4784         525 :     pr = gcoeff(fad,k,1);
    4785         525 :     gel(PH,k+m1) = pr;
    4786         525 :     gcoeff(cnd,k,1) = pr;
    4787         525 :     gcoeff(cnd,k,2) = gcoeff(fad,k,2);
    4788             :   }
    4789         525 :   for (k=1; k<=m3; k++) { /* pr | (b, disc) */
    4790          35 :     long j = perm[k+m1];
    4791          35 :     pr = gcoeff(fab,j,1);
    4792          35 :     gel(PH,k+m1+m2) = pr;
    4793          35 :     gcoeff(cnd,k+m2,1) = pr;
    4794          35 :     gcoeff(cnd,k+m2,2) = gel(L,j);
    4795             :   }
    4796         490 :   gel(cnd,2) = gdiventgs(gel(cnd,2), eulerphiu(n));
    4797        1050 :   for (k=1; k<=m23; k++) H[k+m1] = localhasse(rnf, cnd, pl, auts, b, k);
    4798         490 :   gel(al,4) = hi;
    4799         490 :   perm = gen_indexsort(PH, (void*)&cmp_prime_ideal, &cmp_nodata);
    4800         490 :   gel(al,5) = mkvec2(vecpermute(PH,perm),vecsmallpermute(H,perm));
    4801         490 :   checkhasse(nf,alg_get_hasse_f(al),alg_get_hasse_i(al),n);
    4802         490 : }
    4803             : 
    4804             : static GEN
    4805         777 : alg_maximal_primes(GEN al, GEN P)
    4806             : {
    4807         777 :   pari_sp av = avma;
    4808         777 :   long l = lg(P), i;
    4809        2091 :   for (i=1; i<l; i++)
    4810             :   {
    4811        1314 :     if (i != 1) al = gerepilecopy(av, al);
    4812        1314 :     al = alg_pmaximal(al,gel(P,i));
    4813             :   }
    4814         777 :   return al;
    4815             : }
    4816             : 
    4817             : GEN
    4818         504 : alg_cyclic(GEN rnf, GEN aut, GEN b, long maxord)
    4819             : {
    4820         504 :   pari_sp av = avma;
    4821             :   GEN al, nf;
    4822             :   long D, n, d;
    4823         504 :   dbg_printf(1)("alg_cyclic\n");
    4824         504 :   checkrnf(rnf); nf = rnf_get_nf(rnf);
    4825         504 :   b = nf_to_scalar_or_basis(nf, b);
    4826         497 :   if (typ(b) == t_FRAC || (typ(b) == t_COL && !RgV_is_ZV(b)))
    4827           7 :     pari_err_DOMAIN("alg_cyclic", "denominator(b)", "!=", gen_1,b);
    4828             : 
    4829         490 :   n = rnf_get_degree(rnf);
    4830         490 :   d = nf_get_degree(nf);
    4831         490 :   D = d*n*n;
    4832             : 
    4833         490 :   al = cgetg(12,t_VEC);
    4834         490 :   gel(al,10)= gen_0; /* must be set first */
    4835         490 :   gel(al,1) = rnf;
    4836         490 :   gel(al,2) = allauts(rnf, aut);
    4837         490 :   gel(al,3) = basistoalg(nf,b);
    4838         490 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4839         490 :   gel(al,6) = gen_0;
    4840         490 :   gel(al,7) = matid(D);
    4841         490 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    4842         490 :   gel(al,9) = algnatmultable(al,D);
    4843         490 :   gel(al,11)= algtracebasis(al);
    4844             : 
    4845         490 :   algcomputehasse(al);
    4846             : 
    4847         490 :   if (maxord) {
    4848         427 :     GEN hf = alg_get_hasse_f(al), pr = gel(hf,1);
    4849         427 :     al = alg_maximal_primes(al, prV_primes(pr));
    4850             :   }
    4851         490 :   return gerepilecopy(av, al);
    4852             : }
    4853             : 
    4854             : static int
    4855         420 : ismaximalsubfield(GEN al, GEN x, GEN d, long v, GEN *pt_minpol)
    4856             : {
    4857         420 :   GEN cp = algbasischarpoly(al, x, v), lead;
    4858         420 :   if (!ispower(cp, d, pt_minpol)) return 0;
    4859         420 :   lead = leading_coeff(*pt_minpol);
    4860         420 :   if (isintm1(lead)) *pt_minpol = gneg(*pt_minpol);
    4861         420 :   return ZX_is_irred(*pt_minpol);
    4862             : }
    4863             : 
    4864             : static GEN
    4865         140 : findmaximalsubfield(GEN al, GEN d, long v)
    4866             : {
    4867         140 :   long count, nb=2, i, N = alg_get_absdim(al), n = nf_get_degree(alg_get_center(al));
    4868         140 :   GEN x, minpol, maxc = gen_1;
    4869             : 
    4870         231 :   for (i=n+1; i<=N; i+=n) {
    4871         392 :     for (count=0; count<2 && i+count<=N; count++) {
    4872         301 :       x = col_ei(N,i+count);
    4873         301 :       if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4874             :     }
    4875             :   }
    4876             : 
    4877             :   while(1) {
    4878         119 :     x = zerocol(N);
    4879         504 :     for (count=0; count<nb; count++)
    4880             :     {
    4881         385 :       i = random_Fl(N)+1;
    4882         385 :       gel(x,i) = addiu(randomi(maxc),1);
    4883         385 :       if (random_bits(1)) gel(x,i) = negi(gel(x,i));
    4884             :     }
    4885         119 :     if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4886          56 :     if (!random_bits(3)) maxc = addiu(maxc,1);
    4887          56 :     if (nb<N) nb++;
    4888             :   }
    4889             : 
    4890             :   return NULL; /* LCOV_EXCL_LINE */
    4891             : }
    4892             : 
    4893             : static GEN
    4894         140 : frobeniusform(GEN al, GEN x)
    4895             : {
    4896             :   GEN M, FP, P, Pi;
    4897             : 
    4898             :   /* /!\ has to be the *right* multiplication table */
    4899         140 :   M = algbasisrightmultable(al, x);
    4900             : 
    4901         140 :   FP = matfrobenius(M,2,0); /* M = P^(-1)*F*P */
    4902         140 :   P = gel(FP,2);
    4903         140 :   Pi = RgM_inv(P);
    4904         140 :   return mkvec2(P, Pi);
    4905             : }
    4906             : 
    4907             : static void
    4908         140 : computesplitting(GEN al, long d, long v)
    4909             : {
    4910         140 :   GEN subf, x, pol, polabs, basis, P, Pi, nf = alg_get_center(al), rnf, Lbasis, Lbasisinv, Q, pows;
    4911         140 :   long i, n = nf_get_degree(nf), nd = n*d, N = alg_get_absdim(al), j, j2;
    4912             : 
    4913         140 :   subf = findmaximalsubfield(al, utoipos(d), v);
    4914         140 :   x = gel(subf, 1);
    4915         140 :   polabs = gel(subf, 2);
    4916             : 
    4917             :   /* Frobenius form to obtain L-vector space structure */
    4918         140 :   basis = frobeniusform(al, x);
    4919         140 :   P = gel(basis, 1);
    4920         140 :   Pi = gel(basis, 2);
    4921             : 
    4922             :   /* construct rnf of splitting field */
    4923         140 :   pol = nffactor(nf,polabs);
    4924         140 :   pol = gcoeff(pol,1,1);
    4925         140 :   gel(al,1) = rnf = rnfinit(nf, pol);
    4926             :   /* since pol is irreducible over Q, we have k=0 in rnf. */
    4927         140 :   if (!gequal0(rnf_get_k(rnf)))
    4928             :     pari_err_BUG("computesplitting (k!=0)"); /*LCOV_EXCL_LINE*/
    4929         140 :   gel(al,6) = gen_0;
    4930         140 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4931             : 
    4932             :   /* construct splitting data */
    4933         140 :   Lbasis = cgetg(d+1, t_MAT);
    4934         378 :   for (j=j2=1; j<=d; j++, j2+=nd)
    4935         238 :     gel(Lbasis,j) = gel(Pi,j2);
    4936             : 
    4937         140 :   Q = zeromatcopy(d,N);
    4938         140 :   pows = pol_x_powers(nd,v);
    4939         378 :   for (i=j=1; j<=N; j+=nd, i++)
    4940        1155 :   for (j2=0; j2<nd; j2++)
    4941         917 :     gcoeff(Q,i,j+j2) = mkpolmod(gel(pows,j2+1),polabs);
    4942         140 :   Lbasisinv = RgM_mul(Q,P);
    4943             : 
    4944         140 :   gel(al,3) = mkvec3(x,Lbasis,Lbasisinv);
    4945         140 : }
    4946             : 
    4947             : /* assumes that mt defines a central simple algebra over nf */
    4948             : GEN
    4949         168 : alg_csa_table(GEN nf, GEN mt0, long v, long maxord)
    4950             : {
    4951         168 :   pari_sp av = avma;
    4952             :   GEN al, mt;
    4953         168 :   long n, D, d2 = lg(mt0)-1, d = usqrt(d2);
    4954         168 :   dbg_printf(1)("alg_csa_table\n");
    4955             : 
    4956         168 :   mt = check_relmt(nf,mt0);
    4957         154 :   if (!mt) pari_err_TYPE("alg_csa_table", mt0);
    4958         147 :   n = nf_get_degree(nf);
    4959         147 :   D = n*d2;
    4960         147 :   if (d*d != d2)
    4961           7 :     pari_err_DOMAIN("alg_csa_table","(nonsquare) dimension","!=",stoi(d*d),mt);
    4962             : 
    4963         140 :   al = cgetg(12, t_VEC);
    4964         140 :   gel(al,10) = gen_0; /* must be set first */
    4965         140 :   gel(al,1) = zerovec(12); gmael(al,1,10) = nf;
    4966         140 :   gmael(al,1,1) = gpowgs(pol_x(0), d); /* placeholder before splitting field */
    4967         140 :   gel(al,2) = mt;
    4968         140 :   gel(al,3) = gen_0; /* placeholder */
    4969         140 :   gel(al,4) = gel(al,5) = gen_0; /* TODO Hasse invariants */
    4970         140 :   gel(al,5) = gel(al,6) = gen_0; /* placeholder */
    4971         140 :   gel(al,7) = matid(D);
    4972         140 :   gel(al,8) = matid(D);
    4973         140 :   gel(al,9) = algnatmultable(al,D);
    4974         140 :   gel(al,11)= algtracebasis(al);
    4975         140 :   if (maxord) al = alg_maximal(al);
    4976         140 :   computesplitting(al, d, v);
    4977         140 :   return gerepilecopy(av, al);
    4978             : }
    4979             : 
    4980             : static GEN
    4981       37583 : algtableinit_i(GEN mt0, GEN p)
    4982             : {
    4983             :   GEN al, mt;
    4984             :   long i, n;
    4985             : 
    4986       37583 :   if (p && !signe(p)) p = NULL;
    4987       37583 :   mt = check_mt(mt0,p);
    4988       37583 :   if (!mt) pari_err_TYPE("algtableinit", mt0);
    4989       37576 :   if (!p && !isint1(Q_denom(mt0)))
    4990           7 :     pari_err_DOMAIN("algtableinit", "denominator(mt)", "!=", gen_1, mt0);
    4991       37569 :   n = lg(mt)-1;
    4992       37569 :   al = cgetg(12, t_VEC);
    4993      262983 :   for (i=1; i<=6; i++) gel(al,i) = gen_0;
    4994       37569 :   gel(al,7) = matid(n);
    4995       37569 :   gel(al,8) = matid(n);
    4996       37569 :   gel(al,9) = mt;
    4997       37569 :   gel(al,10) = p? p: gen_0;
    4998       37569 :   gel(al,11)= algtracebasis(al);
    4999       37569 :   return al;
    5000             : }
    5001             : GEN
    5002        4200 : algtableinit(GEN mt0, GEN p)
    5003             : {
    5004        4200 :   pari_sp av = avma;
    5005        4200 :   if (p)
    5006             :   {
    5007        4074 :     if (typ(p) != t_INT) pari_err_TYPE("algtableinit",p);
    5008        4067 :     if (signe(p) && !BPSW_psp(p)) pari_err_PRIME("algtableinit",p);
    5009             :   }
    5010        4179 :   return gerepilecopy(av, algtableinit_i(mt0, p));
    5011             : }
    5012             : 
    5013             : /** REPRESENTATIONS OF GROUPS **/
    5014             : 
    5015             : static GEN
    5016         294 : list_to_regular_rep(GEN elts, long n)
    5017             : {
    5018             :   GEN reg, elts2, g;
    5019             :   long i,j;
    5020         294 :   elts = shallowcopy(elts);
    5021         294 :   gen_sort_inplace(elts, (void*)&vecsmall_lexcmp, &cmp_nodata, NULL);
    5022         294 :   reg = cgetg(n+1, t_VEC);
    5023         294 :   gel(reg,1) = identity_perm(n);
    5024        3857 :   for (i=2; i<=n; i++) {
    5025        3563 :     g = perm_inv(gel(elts,i));
    5026        3563 :     elts2 = cgetg(n+1, t_VEC);
    5027       74543 :     for (j=1; j<=n; j++) gel(elts2,j) = perm_mul(g,gel(elts,j));
    5028        3563 :     gen_sort_inplace(elts2, (void*)&vecsmall_lexcmp, &cmp_nodata, &gel(reg,i));
    5029             :   }
    5030         294 :   return reg;
    5031             : }
    5032             : 
    5033             : static GEN
    5034        3857 : matrix_perm(GEN perm, long n)
    5035             : {
    5036             :   GEN m;
    5037             :   long j;
    5038        3857 :   m = cgetg(n+1, t_MAT);
    5039       78694 :   for (j=1; j<=n; j++) {
    5040       74837 :     gel(m,j) = col_ei(n,perm[j]);
    5041             :   }
    5042        3857 :   return m;
    5043             : }
    5044             : 
    5045             : GEN
    5046         847 : conjclasses_algcenter(GEN cc, GEN p)
    5047             : {
    5048         847 :   GEN mt, elts = gel(cc,1), conjclass = gel(cc,2), rep = gel(cc,3), card;
    5049         847 :   long i, nbcl = lg(rep)-1, n = lg(elts)-1;
    5050             :   pari_sp av;
    5051             : 
    5052         847 :   card = zero_Flv(nbcl);
    5053       14819 :   for (i=1; i<=n; i++) card[conjclass[i]]++;
    5054             : 
    5055             :   /* multiplication table of the center of Z[G] (class functions) */
    5056         847 :   mt = cgetg(nbcl+1,t_VEC);
    5057        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = zero_Flm_copy(nbcl,nbcl);
    5058         847 :   av = avma;
    5059        7217 :   for (i=1;i<=nbcl;i++)
    5060             :   {
    5061        6370 :     GEN xi = gel(elts,rep[i]), mi = gel(mt,i);
    5062             :     long j,k;
    5063      132244 :     for (j=1;j<=n;j++)
    5064             :     {
    5065      125874 :       GEN xj = gel(elts,j);
    5066      125874 :       k = vecsearch(elts, perm_mul(xi,xj), NULL);
    5067      125874 :       ucoeff(mi, conjclass[k], conjclass[j])++;
    5068             :     }
    5069       70238 :     for (k=1; k<=nbcl; k++)
    5070      852362 :       for (j=1; j<=nbcl; j++)
    5071             :       {
    5072      788494 :         ucoeff(mi,k,j) *= card[i];
    5073      788494 :         ucoeff(mi,k,j) /= card[k];
    5074             :       }
    5075        6370 :     set_avma(av);
    5076             :   }
    5077        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = Flm_to_ZM(gel(mt,i));
    5078         847 :   return algtableinit_i(mt,p);
    5079             : }
    5080             : 
    5081             : GEN
    5082         329 : alggroupcenter(GEN G, GEN p, GEN *pcc)
    5083             : {
    5084         329 :   pari_sp av = avma;
    5085         329 :   GEN cc = group_to_cc(G), al = conjclasses_algcenter(cc, p);
    5086         315 :   if (!pcc) return gerepilecopy(av,al);
    5087           7 :   *pcc = cc; return gc_all(av, 2, &al, pcc);
    5088             : }
    5089             : 
    5090             : static GEN
    5091         294 : groupelts_algebra(GEN elts, GEN p)
    5092             : {
    5093         294 :   pari_sp av = avma;
    5094             :   GEN mt;
    5095         294 :   long i, n = lg(elts)-1;
    5096         294 :   elts = list_to_regular_rep(elts,n);
    5097         294 :   mt = cgetg(n+1, t_VEC);
    5098        4151 :   for (i=1; i<=n; i++) gel(mt,i) = matrix_perm(gel(elts,i),n);
    5099         294 :   return gerepilecopy(av, algtableinit_i(mt,p));
    5100             : }
    5101             : 
    5102             : GEN
    5103         329 : alggroup(GEN gal, GEN p)
    5104             : {
    5105         329 :   GEN elts = checkgroupelts(gal);
    5106         294 :   return groupelts_algebra(elts, p);
    5107             : }
    5108             : 
    5109             : /** MAXIMAL ORDER **/
    5110             : 
    5111             : static GEN
    5112       55305 : mattocol(GEN M, long n)
    5113             : {
    5114       55305 :   GEN C = cgetg(n*n+1, t_COL);
    5115             :   long i,j,ic;
    5116       55305 :   ic = 1;
    5117     1121740 :   for (i=1; i<=n; i++)
    5118    27481792 :   for (j=1; j<=n; j++, ic++) gel(C,ic) = gcoeff(M,i,j);
    5119       55305 :   return C;
    5120             : }
    5121             : 
    5122             : /* Ip is a lift of a left O/pO-ideal where O is the integral basis of al */
    5123             : static GEN
    5124        4196 : algleftordermodp(GEN al, GEN Ip, GEN p)
    5125             : {
    5126        4196 :   pari_sp av = avma;
    5127             :   GEN I, Ii, M, mt, K, imi, p2;
    5128             :   long n, i;
    5129        4196 :   n = alg_get_absdim(al);
    5130        4196 :   mt = alg_get_multable(al);
    5131        4196 :   p2 = sqri(p);
    5132             : 
    5133        4196 :   I = ZM_hnfmodid(Ip, p);
    5134        4196 :   Ii = ZM_inv(I,NULL);
    5135             : 
    5136        4196 :   M = cgetg(n+1, t_MAT);
    5137       59501 :   for (i=1; i<=n; i++) {
    5138       55305 :     imi = FpM_mul(Ii, FpM_mul(gel(mt,i), I, p2), p2);
    5139       55305 :     imi = ZM_Z_divexact(imi, p);
    5140       55305 :     gel(M,i) = mattocol(imi, n);
    5141             :   }
    5142        4196 :   K = FpM_ker(M, p);
    5143        4196 :   if (lg(K)==1) { set_avma(av); return matid(n); }
    5144        1766 :   K = ZM_hnfmodid(K,p);
    5145             : 
    5146        1766 :   return gerepileupto(av, ZM_Z_div(K,p));
    5147             : }
    5148             : 
    5149             : static GEN
    5150        5240 : alg_ordermodp(GEN al, GEN p)
    5151             : {
    5152             :   GEN alp;
    5153        5240 :   long i, N = alg_get_absdim(al);
    5154        5240 :   alp = cgetg(12, t_VEC);
    5155       47160 :   for (i=1; i<=8; i++) gel(alp,i) = gen_0;
    5156        5240 :   gel(alp,9) = cgetg(N+1, t_VEC);
    5157       63118 :   for (i=1; i<=N; i++) gmael(alp,9,i) = FpM_red(gmael(al,9,i), p);
    5158        5240 :   gel(alp,10) = p;
    5159        5240 :   gel(alp,11) = cgetg(N+1, t_VEC);
    5160       63118 :   for (i=1; i<=N; i++) gmael(alp,11,i) = Fp_red(gmael(al,11,i), p);
    5161             : 
    5162        5240 :   return alp;
    5163             : }
    5164             : 
    5165             : static GEN
    5166        3080 : algpradical_i(GEN al, GEN p, GEN zprad, GEN projs)
    5167             : {
    5168        3080 :   pari_sp av = avma;
    5169        3080 :   GEN alp = alg_ordermodp(al, p), liftrad, projrad, alq, alrad, res, Lalp, radq;
    5170             :   long i;
    5171        3080 :   if (lg(zprad)==1) {
    5172        2057 :     liftrad = NULL;
    5173        2057 :     projrad = NULL;
    5174             :   }
    5175             :   else {
    5176        1023 :     alq = alg_quotient(alp, zprad, 1);
    5177        1023 :     alp = gel(alq,1);
    5178        1023 :     projrad = gel(alq,2);
    5179        1023 :     liftrad = gel(alq,3);
    5180             :   }
    5181             : 
    5182        3080 :   if (projs) {
    5183         606 :     if (projrad) {
    5184          40 :       projs = gcopy(projs);
    5185         120 :       for (i=1; i<lg(projs); i++)
    5186          80 :         gel(projs,i) = FpM_FpC_mul(projrad, gel(projs,i), p);
    5187             :     }
    5188         606 :     Lalp = alg_centralproj(alp, projs, 1);
    5189             : 
    5190         606 :     alrad = cgetg(lg(Lalp),t_VEC);
    5191        2346 :     for (i=1; i<lg(Lalp); i++) {
    5192        1740 :       alq = gel(Lalp,i);
    5193        1740 :       radq = algradical(gel(alq,1));
    5194        1740 :       if (gequal0(radq))
    5195        1096 :         gel(alrad,i) = cgetg(1,t_MAT);
    5196             :       else {
    5197         644 :         radq = FpM_mul(gel(alq,3),radq,p);
    5198         644 :         gel(alrad,i) = radq;
    5199             :       }
    5200             :     }
    5201         606 :     alrad = shallowmatconcat(alrad);
    5202         606 :     alrad = FpM_image(alrad,p);
    5203             :   }
    5204        2474 :   else alrad = algradical(alp);
    5205             : 
    5206        3080 :   if (!gequal0(alrad)) {
    5207        2305 :     if (liftrad) alrad = FpM_mul(liftrad, alrad, p);
    5208        2305 :     res = shallowmatconcat(mkvec2(alrad, zprad));
    5209        2305 :     res = FpM_image(res,p);
    5210             :   }
    5211         775 :   else res = lg(zprad)==1 ? gen_0 : zprad;
    5212        3080 :   return gerepilecopy(av, res);
    5213             : }
    5214             : 
    5215             : static GEN
    5216        2160 : algpdecompose0(GEN al, GEN prad, GEN p, GEN projs)
    5217             : {
    5218        2160 :   pari_sp av = avma;
    5219        2160 :   GEN alp, quo, ss, liftm = NULL, projm = NULL, dec, res, I, Lss, deci;
    5220             :   long i, j;
    5221             : 
    5222        2160 :   alp = alg_ordermodp(al, p);
    5223        2160 :   if (!gequal0(prad)) {
    5224        1664 :     quo = alg_quotient(alp, prad, 1);
    5225        1664 :     ss = gel(quo,1);
    5226        1664 :     projm = gel(quo,2);
    5227        1664 :     liftm = gel(quo,3);
    5228             :   }
    5229         496 :   else ss = alp;
    5230             : 
    5231        2160 :   if (projs) {
    5232         536 :     if (projm) {
    5233        1395 :       for (i=1; i<lg(projs); i++)
    5234        1026 :         gel(projs,i) = FpM_FpC_mul(projm, gel(projs,i), p);
    5235             :     }
    5236         536 :     Lss = alg_centralproj(ss, projs, 1);
    5237             : 
    5238         536 :     dec = cgetg(lg(Lss),t_VEC);
    5239        2115 :     for (i=1; i<lg(Lss); i++) {
    5240        1579 :       gel(dec,i) = algsimpledec_ss(gmael(Lss,i,1), 1);
    5241        1579 :       deci = gel(dec,i);
    5242        3522 :       for (j=1; j<lg(deci); j++)
    5243        1943 :        gmael(deci,j,3) = FpM_mul(gmael(Lss,i,3), gmael(deci,j,3), p);
    5244             :     }
    5245         536 :     dec = shallowconcat1(dec);
    5246             :   }
    5247        1624 :   else dec = algsimpledec_ss(ss,1);
    5248             : 
    5249        2160 :   res = cgetg(lg(dec),t_VEC);
    5250        6434 :   for (i=1; i<lg(dec); i++) {
    5251        4274 :     I = gmael(dec,i,3);
    5252        4274 :     if (liftm) I = FpM_mul(liftm,I,p);
    5253        4274 :     I = shallowmatconcat(mkvec2(I,prad));
    5254        4274 :     gel(res,i) = I;
    5255             :   }
    5256             : 
    5257        2160 :   return gerepilecopy(av, res);
    5258             : }
    5259             : 
    5260             : /* finds a nontrivial ideal of O/prad or gen_0 if there is none. */
    5261             : static GEN
    5262         846 : algpdecompose_i(GEN al, GEN p, GEN zprad, GEN projs)
    5263             : {
    5264         846 :   pari_sp av = avma;
    5265         846 :   GEN prad = algpradical_i(al,p,zprad,projs);
    5266         846 :   return gerepileupto(av, algpdecompose0(al, prad, p, projs));
    5267             : }
    5268             : 
    5269             : /* ord is assumed to be in hnf wrt the integral basis of al. */
    5270             : /* assumes that alg_get_invbasis(al) is integral. */
    5271             : static GEN
    5272        1766 : alg_change_overorder_shallow(GEN al, GEN ord)
    5273             : {
    5274             :   GEN al2, mt, iord, mtx, den, den2, div;
    5275             :   long i, n;
    5276        1766 :   n = alg_get_absdim(al);
    5277             : 
    5278        1766 :   iord = QM_inv(ord);
    5279        1766 :   al2 = shallowcopy(al);
    5280        1766 :   ord = Q_remove_denom(ord,&den);
    5281             : 
    5282        1766 :   gel(al2,7) = Q_remove_denom(gel(al,7), &den2);
    5283        1766 :   if (den2) div = mulii(den,den2);
    5284         665 :   else      div = den;
    5285        1766 :   gel(al2,7) = ZM_Z_div(ZM_mul(gel(al2,7), ord), div);
    5286             : 
    5287        1766 :   gel(al2,8) = ZM_mul(iord, gel(al,8));
    5288             : 
    5289        1766 :   mt = cgetg(n+1,t_VEC);
    5290        1766 :   gel(mt,1) = matid(n);
    5291        1766 :   div = sqri(den);
    5292       19832 :   for (i=2; i<=n; i++) {
    5293       18066 :     mtx = algbasismultable(al,gel(ord,i));
    5294       18066 :     gel(mt,i) = ZM_mul(iord, ZM_mul(mtx, ord));
    5295       18066 :     gel(mt,i) = ZM_Z_divexact(gel(mt,i), div);
    5296             :   }
    5297        1766 :   gel(al2,9) = mt;
    5298             : 
    5299        1766 :   gel(al2,11) = algtracebasis(al2);
    5300             : 
    5301        1766 :   return al2;
    5302             : }
    5303             : 
    5304             : static GEN
    5305       11011 : algfromcenter(GEN al, GEN x)
    5306             : {
    5307       11011 :   GEN nf = alg_get_center(al);
    5308             :   long n;
    5309       11011 :   switch(alg_type(al)) {
    5310        9877 :     case al_CYCLIC:
    5311        9877 :       n = alg_get_degree(al);
    5312        9877 :       break;
    5313        1134 :     case al_CSA:
    5314        1134 :       n = alg_get_dim(al);
    5315        1134 :       break;
    5316             :     default: return NULL; /*LCOV_EXCL_LINE*/
    5317             :   }
    5318       11011 :   return algalgtobasis(al, scalarcol(basistoalg(nf, x), n));
    5319             : }
    5320             : 
    5321             : /* x is an ideal of the center in hnf form */
    5322             : static GEN
    5323        3080 : algfromcenterhnf(GEN al, GEN x)
    5324             : {
    5325             :   GEN res;
    5326             :   long i;
    5327        3080 :   res = cgetg(lg(x), t_MAT);
    5328        9877 :   for (i=1; i<lg(x); i++) gel(res,i) = algfromcenter(al, gel(x,i));
    5329        3080 :   return res;
    5330             : }
    5331             : 
    5332             : /* assumes al is CSA or CYCLIC */
    5333             : static GEN
    5334        1314 : algcenter_precompute(GEN al, GEN p)
    5335             : {
    5336        1314 :   GEN fa, pdec, nfprad, projs, nf = alg_get_center(al);
    5337             :   long i, np;
    5338             : 
    5339        1314 :   pdec = idealprimedec(nf, p);
    5340        1314 :   settyp(pdec, t_COL);
    5341        1314 :   np = lg(pdec)-1;
    5342        1314 :   fa = mkmat2(pdec, const_col(np, gen_1));
    5343        1314 :   if (dvdii(nf_get_disc(nf), p))
    5344         342 :     nfprad = idealprodprime(nf, pdec);
    5345             :   else
    5346         972 :     nfprad = scalarmat_shallow(p, nf_get_degree(nf));
    5347        1314 :   fa = idealchineseinit(nf, fa);
    5348        1314 :   projs = cgetg(np+1, t_VEC);
    5349        3160 :   for (i=1; i<=np; i++) gel(projs, i) = idealchinese(nf, fa, vec_ei(np,i));
    5350        1314 :   return mkvec2(nfprad, projs);
    5351             : }
    5352             : 
    5353             : static GEN
    5354        3080 : algcenter_prad(GEN al, GEN p, GEN pre)
    5355             : {
    5356             :   GEN nfprad, zprad, mtprad;
    5357             :   long i;
    5358        3080 :   nfprad = gel(pre,1);
    5359        3080 :   zprad = algfromcenterhnf(al, nfprad);
    5360        3080 :   zprad = FpM_image(zprad, p);
    5361        3080 :   mtprad = cgetg(lg(zprad), t_VEC);
    5362        4672 :   for (i=1; i<lg(zprad); i++) gel(mtprad, i) = algbasismultable(al, gel(zprad,i));
    5363        3080 :   mtprad = shallowmatconcat(mtprad);
    5364        3080 :   zprad = FpM_image(mtprad, p);
    5365        3080 :   return zprad;
    5366             : }
    5367             : 
    5368             : static GEN
    5369        3080 : algcenter_p_projs(GEN al, GEN p, GEN pre)
    5370             : {
    5371             :   GEN projs, zprojs;
    5372             :   long i;
    5373        3080 :   projs = gel(pre,2);
    5374        3080 :   zprojs = cgetg(lg(projs), t_VEC);
    5375        7294 :   for (i=1; i<lg(projs); i++) gel(zprojs,i) = FpC_red(algfromcenter(al, gel(projs,i)),p);
    5376        3080 :   return zprojs;
    5377             : }
    5378             : 
    5379             : /* al is assumed to be simple */
    5380             : static GEN
    5381        1314 : alg_pmaximal(GEN al, GEN p)
    5382             : {
    5383             :   pari_sp av;
    5384        1314 :   long n = alg_get_absdim(al);
    5385        1314 :   GEN id = matid(n), al2 = al, prad, lord = gen_0, dec, zprad, projs, pre;
    5386             : 
    5387        1314 :   dbg_printf(0)("Round 2 (noncommutative) at p=%Ps, dim=%d\n", p, n);
    5388        1314 :   pre = algcenter_precompute(al,p); av = avma;
    5389             :   while (1) {
    5390        2234 :     zprad = algcenter_prad(al2, p, pre);
    5391        2234 :     projs = algcenter_p_projs(al2, p, pre);
    5392        2234 :     if (lg(projs) == 2) projs = NULL;
    5393        2234 :     prad = algpradical_i(al2,p,zprad,projs);
    5394        2234 :     if (typ(prad) == t_INT) break;
    5395        2213 :     lord = algleftordermodp(al2,prad,p);
    5396        2213 :     if (!cmp_universal(lord,id)) break;
    5397         920 :     al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
    5398             :   }
    5399             : 
    5400        1314 :   dec = algpdecompose0(al2,prad,p,projs); av = avma;
    5401        2160 :   while (lg(dec) > 2) {
    5402             :     long i;
    5403        2282 :     for (i = 1; i < lg(dec); i++) {
    5404        1983 :       GEN I = gel(dec,i);
    5405        1983 :       lord = algleftordermodp(al2,I,p);
    5406        1983 :       if (cmp_universal(lord,id)) break;
    5407             :     }
    5408        1145 :     if (i==lg(dec)) break;
    5409         846 :     al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
    5410         846 :     zprad = algcenter_prad(al2, p, pre);
    5411         846 :     projs = algcenter_p_projs(al2, p, pre);
    5412         846 :     if (lg(projs) == 2) projs = NULL;
    5413         846 :     dec = algpdecompose_i(al2,p,zprad,projs);
    5414             :   }
    5415        1314 :   return al2;
    5416             : }
    5417             : 
    5418             : static GEN
    5419        5796 : algtracematrix(GEN al)
    5420             : {
    5421             :   GEN M, mt;
    5422             :   long n, i, j;
    5423        5796 :   n = alg_get_absdim(al);
    5424        5796 :   mt = alg_get_multable(al);
    5425        5796 :   M = cgetg(n+1, t_MAT);
    5426       44795 :   for (i=1; i<=n; i++)
    5427             :   {
    5428       38999 :     gel(M,i) = cgetg(n+1,t_MAT);
    5429      274835 :     for (j=1; j<=i; j++)
    5430      235836 :       gcoeff(M,j,i) = gcoeff(M,i,j) = algabstrace(al,gmael(mt,i,j));
    5431             :   }
    5432        5796 :   return M;
    5433             : }
    5434             : static GEN
    5435         140 : algdisc_i(GEN al) { return ZM_det(algtracematrix(al)); }
    5436             : GEN
    5437          28 : algdisc(GEN al)
    5438             : {
    5439          28 :   pari_sp av = avma;
    5440          28 :   checkalg(al);
    5441          28 :   if (alg_type(al) == al_REAL) pari_err_TYPE("algdisc [real algebra]", al);
    5442           7 :   return gerepileuptoint(av, algdisc_i(al));
    5443             : }
    5444             : static GEN
    5445         133 : alg_maximal(GEN al)
    5446             : {
    5447         133 :   GEN fa = absZ_factor(algdisc_i(al));
    5448         133 :   return alg_maximal_primes(al, gel(fa,1));
    5449             : }
    5450             : 
    5451             : /** LATTICES **/
    5452             : 
    5453             : /*
    5454             :  Convention: lattice = [I,t] representing t*I, where
    5455             :  - I integral nonsingular upper-triangular matrix representing a lattice over
    5456             :    the integral basis of the algebra, and
    5457             :  - t>0 either an integer or a rational number.
    5458             : 
    5459             :  Recommended and returned by the functions below:
    5460             :  - I HNF and primitive
    5461             : */
    5462             : 
    5463             : /* TODO use hnfmodid whenever possible using a*O <= I <= O
    5464             :  * for instance a = ZM_det_triangular(I) */
    5465             : 
    5466             : static GEN
    5467       63343 : primlat(GEN lat)
    5468             : {
    5469             :   GEN m, t, c;
    5470       63343 :   m = alglat_get_primbasis(lat);
    5471       63343 :   t = alglat_get_scalar(lat);
    5472       63343 :   m = Q_primitive_part(m,&c);
    5473       63343 :   if (c) return mkvec2(m,gmul(t,c));
    5474       53760 :   return lat;
    5475             : }
    5476             : 
    5477             : /* assumes the lattice contains d * integral basis, d=0 allowed */
    5478             : GEN
    5479       51072 : alglathnf(GEN al, GEN m, GEN d)
    5480             : {
    5481       51072 :   pari_sp av = avma;
    5482             :   long N,i,j;
    5483             :   GEN m2, c;
    5484       51072 :   checkalg(al);
    5485       51072 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglathnf [real algebra]", al);
    5486       51065 :   N = alg_get_absdim(al);
    5487       51065 :   if (!d) d = gen_0;
    5488       51065 :   if (typ(m) == t_VEC) m = matconcat(m);
    5489       51065 :   if (typ(m) == t_COL) m = algleftmultable(al,m);
    5490       51065 :   if (typ(m) != t_MAT) pari_err_TYPE("alglathnf",m);
    5491       51058 :   if (typ(d) != t_FRAC && typ(d) != t_INT) pari_err_TYPE("alglathnf",d);
    5492       51058 :   if (lg(m)-1 < N || lg(gel(m,1))-1 != N) pari_err_DIM("alglathnf");
    5493      459242 :   for (i=1; i<=N; i++)
    5494     6820758 :     for (j=1; j<lg(m); j++)
    5495     6412546 :       if (typ(gcoeff(m,i,j)) != t_FRAC && typ(gcoeff(m,i,j)) != t_INT)
    5496           7 :         pari_err_TYPE("alglathnf", gcoeff(m,i,j));
    5497       51023 :   m2 = Q_primitive_part(m,&c);
    5498       51023 :   if (!c) c = gen_1;
    5499       51023 :   if (!signe(d)) d = detint(m2);
    5500       45593 :   else           d = gdiv(d,c); /* should be an integer */
    5501       51023 :   if (!signe(d)) pari_err_INV("alglathnf [m does not have full rank]", m2);
    5502       51009 :   m2 = ZM_hnfmodid(m2,d);
    5503       51009 :   return gerepilecopy(av, mkvec2(m2,c));
    5504             : }
    5505             : 
    5506             : static GEN
    5507       10689 : prepare_multipliers(GEN *a, GEN *b)
    5508             : {
    5509             :   GEN na, nb, da, db, d;
    5510       10689 :   na = numer_i(*a); da = denom_i(*a);
    5511       10689 :   nb = numer_i(*b); db = denom_i(*b);
    5512       10689 :   na = mulii(na,db);
    5513       10689 :   nb = mulii(nb,da);
    5514       10689 :   d = gcdii(na,nb);
    5515       10689 :   *a = diviiexact(na,d);
    5516       10689 :   *b = diviiexact(nb,d);
    5517       10689 :   return gdiv(d, mulii(da,db));
    5518             : }
    5519             : 
    5520             : static GEN
    5521       10689 : prepare_lat(GEN m1, GEN t1, GEN m2, GEN t2)
    5522             : {
    5523       10689 :   GEN d = prepare_multipliers(&t1, &t2);
    5524       10689 :   m1 = ZM_Z_mul(m1,t1);
    5525       10689 :   m2 = ZM_Z_mul(m2,t2);
    5526       10689 :   return mkvec3(m1,m2,d);
    5527             : }
    5528             : 
    5529             : static GEN
    5530       10703 : alglataddinter(GEN al, GEN lat1, GEN lat2, GEN *sum, GEN *inter)
    5531             : {
    5532             :   GEN d, m1, m2, t1, t2, M, prep, d1, d2, ds, di, K;
    5533       10703 :   checkalg(al);
    5534       10703 :   if (alg_type(al) == al_REAL)
    5535          14 :     pari_err_TYPE("alglataddinter [real algebra]", al);
    5536       10689 :   checklat(al,lat1);
    5537       10689 :   checklat(al,lat2);
    5538             : 
    5539       10689 :   m1 = alglat_get_primbasis(lat1);
    5540       10689 :   t1 = alglat_get_scalar(lat1);
    5541       10689 :   m2 = alglat_get_primbasis(lat2);
    5542       10689 :   t2 = alglat_get_scalar(lat2);
    5543       10689 :   prep = prepare_lat(m1, t1, m2, t2);
    5544       10689 :   m1 = gel(prep,1);
    5545       10689 :   m2 = gel(prep,2);
    5546       10689 :   d = gel(prep,3);
    5547       10689 :   M = matconcat(mkvec2(m1,m2));
    5548       10689 :   d1 = ZM_det_triangular(m1);
    5549       10689 :   d2 = ZM_det_triangular(m2);
    5550       10689 :   ds = gcdii(d1,d2);
    5551       10689 :   if (inter)
    5552             :   {
    5553        7112 :     di = diviiexact(mulii(d1,d2),ds);
    5554        7112 :     K = matkermod(M,di,sum);
    5555        7112 :     K = rowslice(K,1,lg(m1));
    5556        7112 :     *inter = hnfmodid(FpM_mul(m1,K,di),di);
    5557        7112 :     if (sum) *sum = hnfmodid(*sum,ds);
    5558             :   }
    5559        3577 :   else *sum = hnfmodid(M,ds);
    5560       10689 :   return d;
    5561             : }
    5562             : 
    5563             : GEN
    5564        3605 : alglatinter(GEN al, GEN lat1, GEN lat2, GEN* psum)
    5565             : {
    5566        3605 :   pari_sp av = avma;
    5567             :   GEN inter, d;
    5568        3605 :   d = alglataddinter(al, lat1, lat2, psum, &inter);
    5569        3598 :   inter = primlat(mkvec2(inter, d));
    5570        3598 :   if (!psum) return gerepilecopy(av, inter);
    5571          14 :   *psum = primlat(mkvec2(*psum,d));
    5572          14 :   return gc_all(av, 2, &inter, psum);
    5573             : }
    5574             : 
    5575             : GEN
    5576        7098 : alglatadd(GEN al, GEN lat1, GEN lat2, GEN* pinter)
    5577             : {
    5578        7098 :   pari_sp av = avma;
    5579             :   GEN sum, d;
    5580        7098 :   d = alglataddinter(al, lat1, lat2, &sum, pinter);
    5581        7091 :   sum = primlat(mkvec2(sum, d));
    5582        7091 :   if (!pinter) return gerepilecopy(av, sum);
    5583        3514 :   *pinter = primlat(mkvec2(*pinter,d));
    5584        3514 :   return gc_all(av, 2, &sum, pinter);
    5585             : }
    5586             : 
    5587             : /* TODO version that returns the quotient as abelian group? */
    5588             : /* return matrices to convert coordinates from one to other? */
    5589             : int
    5590       31556 : alglatsubset(GEN al, GEN lat1, GEN lat2, GEN* pindex)
    5591             : {
    5592       31556 :   pari_sp av = avma;
    5593             :   int res;
    5594             :   GEN m1, m2, m2i, m, t;
    5595       31556 :   checkalg(al);
    5596       31556 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatsubset [real algebra]", al);
    5597       31549 :   checklat(al,lat1);
    5598       31549 :   checklat(al,lat2);
    5599       31549 :   m1 = alglat_get_primbasis(lat1);
    5600       31549 :   m2 = alglat_get_primbasis(lat2);
    5601       31549 :   m2i = RgM_inv_upper(m2);
    5602       31549 :   t = gdiv(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    5603       31549 :   m = RgM_Rg_mul(RgM_mul(m2i,m1), t);
    5604       31549 :   res = RgM_is_ZM(m);
    5605       31549 :   if (!res || !pindex) return gc_int(av, res);
    5606        1757 :   *pindex = gerepileuptoint(av, mpabs(ZM_det_triangular(m)));
    5607        1757 :   return 1;
    5608             : }
    5609             : 
    5610             : GEN
    5611        5271 : alglatindex(GEN al, GEN lat1, GEN lat2)
    5612             : {
    5613        5271 :   pari_sp av = avma;
    5614             :   long N;
    5615             :   GEN res;
    5616        5271 :   checkalg(al);
    5617        5271 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatindex [real algebra]", al);
    5618        5264 :   checklat(al,lat1);
    5619        5264 :   checklat(al,lat2);
    5620        5264 :   N = alg_get_absdim(al);
    5621        5264 :   res = alglat_get_scalar(lat1);
    5622        5264 :   res = gdiv(res, alglat_get_scalar(lat2));
    5623        5264 :   res = gpowgs(res, N);
    5624        5264 :   res = gmul(res,RgM_det_triangular(alglat_get_primbasis(lat1)));
    5625        5264 :   res = gdiv(res, RgM_det_triangular(alglat_get_primbasis(lat2)));
    5626        5264 :   res = gabs(res,0);
    5627        5264 :   return gerepilecopy(av, res);
    5628             : }
    5629             : 
    5630             : GEN
    5631       45612 : alglatmul(GEN al, GEN lat1, GEN lat2)
    5632             : {
    5633       45612 :   pari_sp av = avma;
    5634             :   long N,i;
    5635             :   GEN m1, m2, m, V, lat, t, d, dp;
    5636       45612 :   checkalg(al);
    5637       45612 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatmul [real algebra]", al);
    5638       45605 :   if (typ(lat1)==t_COL)
    5639             :   {
    5640       19292 :     if (typ(lat2)==t_COL)
    5641           7 :       pari_err_TYPE("alglatmul [one of lat1, lat2 has to be a lattice]", lat2);
    5642       19285 :     checklat(al,lat2);
    5643       19285 :     lat1 = Q_remove_denom(lat1,&d);
    5644       19285 :     m = algbasismultable(al,lat1);
    5645       19285 :     m2 = alglat_get_primbasis(lat2);
    5646       19285 :     dp = mulii(detint(m),ZM_det_triangular(m2));
    5647       19285 :     m = ZM_mul(m,m2);
    5648       19285 :     t = alglat_get_scalar(lat2);
    5649       19285 :     if (d) t = gdiv(t,d);
    5650             :   }
    5651             :   else /* typ(lat1)!=t_COL */
    5652             :   {
    5653       26313 :     checklat(al,lat1);
    5654       26313 :     if (typ(lat2)==t_COL)
    5655             :     {
    5656       19285 :       lat2 = Q_remove_denom(lat2,&d);
    5657       19285 :       m = algbasisrightmultable(al,lat2);
    5658       19285 :       m1 = alglat_get_primbasis(lat1);
    5659       19285 :       dp = mulii(detint(m),ZM_det_triangular(m1));
    5660       19285 :       m = ZM_mul(m,m1);
    5661       19285 :       t = alglat_get_scalar(lat1);
    5662       19285 :       if (d) t = gdiv(t,d);
    5663             :     }
    5664             :     else /* typ(lat2)!=t_COL */
    5665             :     {
    5666        7028 :       checklat(al,lat2);
    5667        7021 :       N = alg_get_absdim(al);
    5668        7021 :       m1 = alglat_get_primbasis(lat1);
    5669        7021 :       m2 = alglat_get_primbasis(lat2);
    5670        7021 :       dp = mulii(ZM_det_triangular(m1), ZM_det_triangular(m2));
    5671        7021 :       V = cgetg(N+1,t_VEC);
    5672       63189 :       for (i=1; i<=N; i++) {
    5673       56168 :         gel(V,i) = algbasismultable(al,gel(m1,i));
    5674       56168 :         gel(V,i) = ZM_mul(gel(V,i),m2);
    5675             :       }
    5676        7021 :       m = matconcat(V);
    5677        7021 :       t = gmul(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    5678             :     }
    5679             :   }
    5680             : 
    5681       45591 :   lat = alglathnf(al,m,dp);
    5682       45591 :   gel(lat,2) = gmul(alglat_get_scalar(lat), t);
    5683       45591 :   lat = primlat(lat);
    5684       45591 :   return gerepilecopy(av, lat);
    5685             : }
    5686             : 
    5687             : int
    5688       17528 : alglatcontains(GEN al, GEN lat, GEN x, GEN *ptc)
    5689             : {
    5690       17528 :   pari_sp av = avma;
    5691             :   GEN m, t, sol;
    5692       17528 :   checkalg(al);
    5693       17528 :   if (alg_type(al) == al_REAL)
    5694           7 :     pari_err_TYPE("alglatcontains [real algebra]", al);
    5695       17521 :   checklat(al,lat);
    5696       17521 :   m = alglat_get_primbasis(lat);
    5697       17521 :   t = alglat_get_scalar(lat);
    5698       17521 :   x = RgC_Rg_div(x,t);
    5699       17521 :   if (!RgV_is_ZV(x)) return gc_bool(av,0);
    5700       17521 :   sol = hnf_solve(m,x);
    5701       17521 :   if (!sol) return gc_bool(av,0);
    5702        8771 :   if (!ptc) return gc_bool(av,1);
    5703        8764 :   *ptc = gerepilecopy(av, sol); return 1;
    5704             : }
    5705             : 
    5706             : GEN
    5707        8778 : alglatelement(GEN al, GEN lat, GEN c)
    5708             : {
    5709        8778 :   pari_sp av = avma;
    5710             :   GEN res;
    5711        8778 :   checkalg(al);
    5712        8778 :   if (alg_type(al) == al_REAL)
    5713           7 :     pari_err_TYPE("alglatelement [real algebra]", al);
    5714        8771 :   checklat(al,lat);
    5715        8771 :   if (typ(c)!=t_COL) pari_err_TYPE("alglatelement", c);
    5716        8764 :   res = ZM_ZC_mul(alglat_get_primbasis(lat),c);
    5717        8764 :   res = RgC_Rg_mul(res, alglat_get_scalar(lat));
    5718        8764 :   return gerepilecopy(av,res);
    5719             : }
    5720             : 
    5721             : /* idem QM_invimZ, knowing result is contained in 1/c*Z^n */
    5722             : static GEN
    5723        3535 : QM_invimZ_mod(GEN m, GEN c)
    5724             : {
    5725             :   GEN d, m0, K;
    5726        3535 :   m0 = Q_remove_denom(m, &d);
    5727        3535 :   if (d)    d = mulii(d,c);
    5728          21 :   else      d = c;
    5729        3535 :   K = matkermod(m0, d, NULL);
    5730        3535 :   if (lg(K)==1) K = scalarmat(d, lg(m)-1);
    5731        3521 :   else          K = hnfmodid(K, d);
    5732        3535 :   return RgM_Rg_div(K,c);
    5733             : }
    5734             : 
    5735             : /* If m is injective, computes a Z-basis of the submodule of elements whose
    5736             :  * image under m is integral */
    5737             : static GEN
    5738          14 : QM_invimZ(GEN m)
    5739             : {
    5740          14 :   return RgM_invimage(m, QM_ImQ_hnf(m));
    5741             : }
    5742             : 
    5743             : /* An isomorphism of R-modules M_{m,n}(R) -> R^{m*n} */
    5744             : static GEN
    5745       28322 : mat2col(GEN M, long m, long n)
    5746             : {
    5747             :   long i,j,k,p;
    5748             :   GEN C;
    5749       28322 :   p = m*n;
    5750       28322 :   C = cgetg(p+1,t_COL);
    5751      254702 :   for (i=1,k=1;i<=m;i++)
    5752     2036804 :     for (j=1;j<=n;j++,k++)
    5753     1810424 :       gel(C,k) = gcoeff(M,i,j);
    5754       28322 :   return C;
    5755             : }
    5756             : 
    5757             : static GEN
    5758        3535 : alglattransporter_i(GEN al, GEN lat1, GEN lat2, long right)
    5759             : {
    5760             :   GEN m1, m2, m2i, M, MT, mt, t1, t2, T, c;
    5761             :   long N, i;
    5762        3535 :   N = alg_get_absdim(al);
    5763        3535 :   m1 = alglat_get_primbasis(lat1);
    5764        3535 :   m2 = alglat_get_primbasis(lat2);
    5765        3535 :   m2i = RgM_inv_upper(m2);
    5766        3535 :   c = detint(m1);
    5767        3535 :   t1 = alglat_get_scalar(lat1);
    5768        3535 :   m1 = RgM_Rg_mul(m1,t1);
    5769        3535 :   t2 = alglat_get_scalar(lat2);
    5770        3535 :   m2i = RgM_Rg_div(m2i,t2);
    5771             : 
    5772        3535 :   MT = right? NULL: alg_get_multable(al);
    5773        3535 :   M = cgetg(N+1, t_MAT);
    5774       31815 :   for (i=1; i<=N; i++) {
    5775       28280 :     if (right) mt = algbasisrightmultable(al, vec_ei(N,i));
    5776       14168 :     else       mt = gel(MT,i);
    5777       28280 :     mt = RgM_mul(m2i,mt);
    5778       28280 :     mt = RgM_mul(mt,m1);
    5779       28280 :     gel(M,i) = mat2col(mt, N, N);
    5780             :   }
    5781             : 
    5782        3535 :   c = gdiv(t2,gmul(c,t1));
    5783        3535 :   c = denom_i(c);
    5784        3535 :   T = QM_invimZ_mod(M,c);
    5785        3535 :   return primlat(mkvec2(T,gen_1));
    5786             : }
    5787             : 
    5788             : /*
    5789             :    { x in al | x*lat1 subset lat2}
    5790             : */
    5791             : GEN
    5792        1778 : alglatlefttransporter(GEN al, GEN lat1, GEN lat2)
    5793             : {
    5794        1778 :   pari_sp av = avma;
    5795        1778 :   checkalg(al);
    5796        1778 :   if (alg_type(al) == al_REAL)
    5797           7 :     pari_err_TYPE("alglatlefttransporter [real algebra]", al);
    5798        1771 :   checklat(al,lat1);
    5799        1771 :   checklat(al,lat2);
    5800        1771 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,0));
    5801             : }
    5802             : 
    5803             : /*
    5804             :    { x in al | lat1*x subset lat2}
    5805             : */
    5806             : GEN
    5807        1771 : alglatrighttransporter(GEN al, GEN lat1, GEN lat2)
    5808             : {
    5809        1771 :   pari_sp av = avma;
    5810        1771 :   checkalg(al);
    5811        1771 :   if (alg_type(al) == al_REAL)
    5812           7 :     pari_err_TYPE("alglatrighttransporter [real algebra]", al);
    5813        1764 :   checklat(al,lat1);
    5814        1764 :   checklat(al,lat2);
    5815        1764 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,1));
    5816             : }
    5817             : 
    5818             : GEN
    5819          42 : algmakeintegral(GEN mt0, long maps)
    5820             : {
    5821          42 :   pari_sp av = avma;
    5822             :   long n,i;
    5823             :   GEN m,P,Pi,mt2,mt;
    5824          42 :   n = lg(mt0)-1;
    5825          42 :   mt = check_mt(mt0,NULL);
    5826          42 :   if (!mt) pari_err_TYPE("algmakeintegral", mt0);
    5827          21 :   if (isint1(Q_denom(mt0))) {
    5828           7 :     if (maps) mt = mkvec3(mt,matid(n),matid(n));
    5829           7 :     return gerepilecopy(av,mt);
    5830             :   }
    5831          14 :   dbg_printf(2)(" algmakeintegral: dim=%d, denom=%Ps\n", n, Q_denom(mt0));
    5832          14 :   m = cgetg(n+1,t_MAT);
    5833          56 :   for (i=1;i<=n;i++)
    5834          42 :     gel(m,i) = mat2col(gel(mt,i),n,n);
    5835          14 :   dbg_printf(2)(" computing order, dims m = %d x %d...\n", nbrows(m), lg(m)-1);
    5836          14 :   P = QM_invimZ(m);
    5837          14 :   dbg_printf(2)(" ...done.\n");
    5838          14 :   P = shallowmatconcat(mkvec2(col_ei(n,1),P));
    5839          14 :   P = hnf(P);
    5840          14 :   Pi = RgM_inv(P);
    5841          14 :   mt2 = change_Rgmultable(mt,P,Pi);
    5842          14 :   if (maps) mt2 = mkvec3(mt2,Pi,P); /* mt2, mt->mt2, mt2->mt */
    5843          14 :   return gerepilecopy(av,mt2);
    5844             : }
    5845             : 
    5846             : /** ORDERS **/
    5847             : 
    5848             : /** IDEALS **/
    5849             : 

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