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 - basemath - base4.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.2 lcov report (development 29115-f22e516b23) Lines: 1643 1794 91.6 %
Date: 2024-02-22 08:05:55 Functions: 164 179 91.6 %
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             : 
      15             : /*******************************************************************/
      16             : /*                                                                 */
      17             : /*                       BASIC NF OPERATIONS                       */
      18             : /*                           (continued)                           */
      19             : /*                                                                 */
      20             : /*******************************************************************/
      21             : #include "pari.h"
      22             : #include "paripriv.h"
      23             : 
      24             : #define DEBUGLEVEL DEBUGLEVEL_nf
      25             : 
      26             : /*******************************************************************/
      27             : /*                                                                 */
      28             : /*                     IDEAL OPERATIONS                            */
      29             : /*                                                                 */
      30             : /*******************************************************************/
      31             : 
      32             : /* A valid ideal is either principal (valid nf_element), or prime, or a matrix
      33             :  * on the integer basis in HNF.
      34             :  * A prime ideal is of the form [p,a,e,f,b], where the ideal is p.Z_K+a.Z_K,
      35             :  * p is a rational prime, a belongs to Z_K, e=e(P/p), f=f(P/p), and b
      36             :  * is Lenstra's constant, such that p.P^(-1)= p Z_K + b Z_K.
      37             :  *
      38             :  * An extended ideal is a couple [I,F] where I is an ideal and F is either an
      39             :  * algebraic number, or a factorization matrix attached to an algebraic number.
      40             :  * All routines work with either extended ideals or ideals (an omitted F is
      41             :  * assumed to be factor(1)). All ideals are output in HNF form. */
      42             : 
      43             : /* types and conversions */
      44             : 
      45             : long
      46     8747985 : idealtyp(GEN *ideal, GEN *arch)
      47             : {
      48     8747985 :   GEN x = *ideal;
      49     8747985 :   long t,lx,tx = typ(x);
      50             : 
      51     8747985 :   if (tx!=t_VEC || lg(x)!=3) { if (arch) *arch = NULL; }
      52             :   else
      53             :   {
      54     1233325 :     GEN a = gel(x,2);
      55     1233325 :     if (typ(a) == t_MAT && lg(a) != 3)
      56             :     { /* allow [;] */
      57          14 :       if (lg(a) != 1) pari_err_TYPE("idealtyp [extended ideal]",x);
      58           7 :       if (arch) *arch = trivial_fact();
      59             :     }
      60             :     else
      61     1233311 :       if (arch) *arch = a;
      62     1233318 :     x = gel(x,1); tx = typ(x);
      63             :   }
      64     8747978 :   switch(tx)
      65             :   {
      66     4890951 :     case t_MAT: lx = lg(x);
      67     4890951 :       if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
      68     4890790 :       if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
      69     4890760 :       t = id_MAT;
      70     4890760 :       break;
      71             : 
      72     2831284 :     case t_VEC: if (lg(x)!=6) pari_err_TYPE("idealtyp",x);
      73     2831243 :       t = id_PRIME; break;
      74             : 
      75     1025888 :     case t_POL: case t_POLMOD: case t_COL:
      76             :     case t_INT: case t_FRAC:
      77     1025888 :       t = id_PRINCIPAL; break;
      78           0 :     default:
      79           0 :       pari_err_TYPE("idealtyp",x);
      80             :       return 0; /*LCOV_EXCL_LINE*/
      81             :   }
      82     8748052 :   *ideal = x; return t;
      83             : }
      84             : 
      85             : /* true nf; v = [a,x,...], a in Z. Return (a,x) */
      86             : GEN
      87      695860 : idealhnf_two(GEN nf, GEN v)
      88             : {
      89      695860 :   GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
      90      695857 :   if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
      91      622771 :   return ZM_hnfmodid(m, p);
      92             : }
      93             : /* true nf */
      94             : GEN
      95     3603635 : pr_hnf(GEN nf, GEN pr)
      96             : {
      97     3603635 :   GEN p = pr_get_p(pr), m;
      98     3603604 :   if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
      99     3168244 :   m = zk_scalar_or_multable(nf, pr_get_gen(pr));
     100     3167964 :   return ZM_hnfmodprime(m, p);
     101             : }
     102             : 
     103             : GEN
     104     1363146 : idealhnf_principal(GEN nf, GEN x)
     105             : {
     106             :   GEN cx;
     107     1363146 :   x = nf_to_scalar_or_basis(nf, x);
     108     1363147 :   switch(typ(x))
     109             :   {
     110     1051005 :     case t_COL: break;
     111      279848 :     case t_INT:  if (!signe(x)) return cgetg(1,t_MAT);
     112      278217 :       return scalarmat(absi_shallow(x), nf_get_degree(nf));
     113       32294 :     case t_FRAC:
     114       32294 :       return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
     115           0 :     default: pari_err_TYPE("idealhnf",x);
     116             :   }
     117     1051005 :   x = Q_primitive_part(x, &cx);
     118     1050999 :   RgV_check_ZV(x, "idealhnf");
     119     1050999 :   x = zk_multable(nf, x);
     120     1050996 :   x = ZM_hnfmodid(x, zkmultable_capZ(x));
     121     1051006 :   return cx? ZM_Q_mul(x,cx): x;
     122             : }
     123             : 
     124             : /* x integral ideal in t_MAT form, nx columns */
     125             : static GEN
     126           7 : vec_mulid(GEN nf, GEN x, long nx, long N)
     127             : {
     128           7 :   GEN m = cgetg(nx*N + 1, t_MAT);
     129             :   long i, j, k;
     130          21 :   for (i=k=1; i<=nx; i++)
     131          56 :     for (j=1; j<=N; j++) gel(m, k++) = zk_ei_mul(nf, gel(x,i),j);
     132           7 :   return m;
     133             : }
     134             : /* true nf */
     135             : GEN
     136     1741043 : idealhnf_shallow(GEN nf, GEN x)
     137             : {
     138     1741043 :   long tx = typ(x), lx = lg(x), N;
     139             : 
     140             :   /* cannot use idealtyp because here we allow nonsquare matrices */
     141     1741043 :   if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
     142     1741043 :   if (tx == t_VEC && lx == 6) return pr_hnf(nf,x); /* PRIME */
     143     1257226 :   switch(tx)
     144             :   {
     145      100865 :     case t_MAT:
     146             :     {
     147             :       GEN cx;
     148      100865 :       long nx = lx-1;
     149      100865 :       N = nf_get_degree(nf);
     150      100865 :       if (nx == 0) return cgetg(1, t_MAT);
     151      100844 :       if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
     152      100837 :       if (nx == 1) return idealhnf_principal(nf, gel(x,1));
     153             : 
     154       84528 :       if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
     155       49259 :       x = Q_primitive_part(x, &cx);
     156       49259 :       if (nx < N) x = vec_mulid(nf, x, nx, N);
     157       49259 :       x = ZM_hnfmod(x, ZM_detmult(x));
     158       49259 :       return cx? ZM_Q_mul(x,cx): x;
     159             :     }
     160          14 :     case t_QFB:
     161             :     {
     162          14 :       pari_sp av = avma;
     163          14 :       GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
     164          14 :       GEN A = gel(x,1), B = gel(x,2);
     165          14 :       N = nf_get_degree(nf);
     166          14 :       if (N != 2)
     167           0 :         pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
     168          14 :       if (!equalii(qfb_disc(x), D))
     169           7 :         pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
     170             :       /* x -> A Z + (-B + sqrt(D)) / 2 Z
     171             :          K = Q[t]/T(t), t^2 + ut + v = 0,  u^2 - 4v = Df^2
     172             :          => t = (-u + sqrt(D) f)/2
     173             :          => sqrt(D)/2 = (t + u/2)/f */
     174           7 :       u = gel(T,3);
     175           7 :       B = deg1pol_shallow(ginv(f),
     176             :                           gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
     177           7 :                           varn(T));
     178           7 :       return gerepileupto(av, idealhnf_two(nf, mkvec2(A,B)));
     179             :     }
     180     1156347 :     default: return idealhnf_principal(nf, x); /* PRINCIPAL */
     181             :   }
     182             : }
     183             : /* true nf */
     184             : GEN
     185         301 : idealhnf(GEN nf, GEN x)
     186             : {
     187         301 :   pari_sp av = avma;
     188         301 :   GEN y = idealhnf_shallow(nf, x);
     189         294 :   return (avma == av)? gcopy(y): gerepileupto(av, y);
     190             : }
     191             : 
     192             : /* GP functions */
     193             : 
     194             : GEN
     195          70 : idealtwoelt0(GEN nf, GEN x, GEN a)
     196             : {
     197          70 :   if (!a) return idealtwoelt(nf,x);
     198          42 :   return idealtwoelt2(nf,x,a);
     199             : }
     200             : 
     201             : GEN
     202          84 : idealpow0(GEN nf, GEN x, GEN n, long flag)
     203             : {
     204          84 :   if (flag) return idealpowred(nf,x,n);
     205          77 :   return idealpow(nf,x,n);
     206             : }
     207             : 
     208             : GEN
     209          70 : idealmul0(GEN nf, GEN x, GEN y, long flag)
     210             : {
     211          70 :   if (flag) return idealmulred(nf,x,y);
     212          63 :   return idealmul(nf,x,y);
     213             : }
     214             : 
     215             : GEN
     216          56 : idealdiv0(GEN nf, GEN x, GEN y, long flag)
     217             : {
     218          56 :   switch(flag)
     219             :   {
     220          28 :     case 0: return idealdiv(nf,x,y);
     221          28 :     case 1: return idealdivexact(nf,x,y);
     222           0 :     default: pari_err_FLAG("idealdiv");
     223             :   }
     224             :   return NULL; /* LCOV_EXCL_LINE */
     225             : }
     226             : 
     227             : GEN
     228          70 : idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
     229             : {
     230          70 :   if (!arg2) return idealaddmultoone(nf,arg1);
     231          35 :   return idealaddtoone(nf,arg1,arg2);
     232             : }
     233             : 
     234             : /* b not a scalar */
     235             : static GEN
     236          77 : hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
     237             : /* b not a scalar */
     238             : static GEN
     239          70 : hnf_Z_QC(GEN nf, GEN a, GEN b)
     240             : {
     241             :   GEN db;
     242          70 :   b = Q_remove_denom(b, &db);
     243          70 :   if (db) a = mulii(a, db);
     244          70 :   b = hnf_Z_ZC(nf,a,b);
     245          70 :   return db? RgM_Rg_div(b, db): b;
     246             : }
     247             : /* b not a scalar (not point in trying to optimize for this case) */
     248             : static GEN
     249          77 : hnf_Q_QC(GEN nf, GEN a, GEN b)
     250             : {
     251             :   GEN da, db;
     252          77 :   if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
     253           7 :   da = gel(a,2);
     254           7 :   a = gel(a,1);
     255           7 :   b = Q_remove_denom(b, &db);
     256             :   /* write da = d*A, db = d*B, gcd(A,B) = 1
     257             :    * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
     258           7 :   if (db)
     259             :   {
     260           7 :     GEN d = gcdii(da,db);
     261           7 :     if (!is_pm1(d)) db = diviiexact(db,d); /* B */
     262           7 :     if (!is_pm1(db))
     263             :     {
     264           7 :       a = mulii(a, db); /* a B */
     265           7 :       da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
     266             :     }
     267             :   }
     268           7 :   return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
     269             : }
     270             : static GEN
     271           7 : hnf_QC_QC(GEN nf, GEN a, GEN b)
     272             : {
     273             :   GEN da, db, d, x;
     274           7 :   a = Q_remove_denom(a, &da);
     275           7 :   b = Q_remove_denom(b, &db);
     276           7 :   if (da) b = ZC_Z_mul(b, da);
     277           7 :   if (db) a = ZC_Z_mul(a, db);
     278           7 :   d = mul_denom(da, db);
     279           7 :   a = zk_multable(nf,a); da = zkmultable_capZ(a);
     280           7 :   b = zk_multable(nf,b); db = zkmultable_capZ(b);
     281           7 :   x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
     282           7 :   return d? RgM_Rg_div(x, d): x;
     283             : }
     284             : static GEN
     285          21 : hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
     286             : GEN
     287         210 : idealhnf0(GEN nf, GEN a, GEN b)
     288             : {
     289             :   long ta, tb;
     290             :   pari_sp av;
     291             :   GEN x;
     292         210 :   nf = checknf(nf);
     293         210 :   if (!b) return idealhnf(nf,a);
     294             : 
     295             :   /* HNF of aZ_K+bZ_K */
     296         112 :   av = avma;
     297         112 :   a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
     298         112 :   b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
     299         105 :   if (ta == t_COL)
     300          14 :     x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
     301             :   else
     302          91 :     x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
     303         105 :   return gerepileupto(av, x);
     304             : }
     305             : 
     306             : /*******************************************************************/
     307             : /*                                                                 */
     308             : /*                       TWO-ELEMENT FORM                          */
     309             : /*                                                                 */
     310             : /*******************************************************************/
     311             : static GEN idealapprfact_i(GEN nf, GEN x, int nored);
     312             : 
     313             : static int
     314      226603 : ok_elt(GEN x, GEN xZ, GEN y)
     315             : {
     316      226603 :   pari_sp av = avma;
     317      226603 :   return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
     318             : }
     319             : 
     320             : /* a + s * b, a and b ZM, s integer */
     321             : static GEN
     322       65510 : addmul_mat(GEN a, GEN s, GEN b)
     323             : {
     324       65510 :   if (!signe(s)) return a;
     325       56824 :   if (!equali1(s)) b = ZM_Z_mul(b, s);
     326       56824 :   return a? ZM_add(a, b): b;
     327             : }
     328             : 
     329             : static GEN
     330      120160 : get_random_a(GEN nf, GEN x, GEN xZ)
     331             : {
     332             :   pari_sp av;
     333      120160 :   long i, lm, l = lg(x);
     334             :   GEN z, beta, mul;
     335             : 
     336      120160 :   beta= cgetg(l, t_MAT);
     337      120160 :   mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
     338             :   /* look for a in x such that a O/xZ = x O/xZ */
     339      251120 :   for (i = 2; i < l; i++)
     340             :   {
     341      241234 :     GEN xi = gel(x,i);
     342      241234 :     GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
     343      241229 :     if (gequal0(t)) continue;
     344      198605 :     if (ok_elt(x,xZ, t)) return xi;
     345       88332 :     gel(beta,lm) = xi;
     346             :     /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
     347       88332 :     gel(mul,lm) = t; lm++;
     348             :   }
     349        9886 :   setlg(mul, lm);
     350        9886 :   setlg(beta,lm); z = cgetg(lm, t_VEC);
     351       29862 :   for(av = avma;; set_avma(av))
     352       19976 :   {
     353       29862 :     GEN a = NULL;
     354       95372 :     for (i = 1; i < lm; i++)
     355             :     {
     356       65510 :       gel(z,i) = randomi(xZ);
     357       65510 :       a = addmul_mat(a, gel(z,i), gel(mul,i));
     358             :     }
     359             :     /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
     360       29862 :     if (a && ok_elt(x,xZ, a)) break;
     361             :   }
     362        9886 :   return ZM_ZC_mul(beta, z);
     363             : }
     364             : 
     365             : /* x square matrix, assume it is HNF */
     366             : static GEN
     367      254378 : mat_ideal_two_elt(GEN nf, GEN x)
     368             : {
     369             :   GEN y, a, cx, xZ;
     370      254378 :   long N = nf_get_degree(nf);
     371             :   pari_sp av, tetpil;
     372             : 
     373      254378 :   if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
     374      254364 :   if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
     375             : 
     376      136175 :   y = cgetg(3,t_VEC); av = avma;
     377      136176 :   cx = Q_content(x);
     378      136176 :   xZ = gcoeff(x,1,1);
     379      136176 :   if (gequal(xZ, cx)) /* x = (cx) */
     380             :   {
     381        5215 :     gel(y,1) = cx;
     382        5215 :     gel(y,2) = gen_0; return y;
     383             :   }
     384      130961 :   if (equali1(cx)) cx = NULL;
     385             :   else
     386             :   {
     387        3233 :     x = Q_div_to_int(x, cx);
     388        3233 :     xZ = gcoeff(x,1,1);
     389             :   }
     390      130961 :   if (N < 6)
     391      111770 :     a = get_random_a(nf, x, xZ);
     392             :   else
     393             :   {
     394       19191 :     const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
     395             :       2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
     396             :     };
     397       19191 :     GEN P, E, a1 = Z_lsmoothen(xZ, (GEN)FB, &P, &E);
     398       19191 :     if (!a1) /* factors completely */
     399       10801 :       a = idealapprfact_i(nf, idealfactor(nf,x), 1);
     400        8390 :     else if (lg(P) == 1) /* no small factors */
     401        2547 :       a = get_random_a(nf, x, xZ);
     402             :     else /* general case */
     403             :     {
     404             :       GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
     405        5843 :       a0 = diviiexact(xZ, a1);
     406        5843 :       A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
     407        5843 :       A1 = ZM_hnfmodid(x, a1); /* cofactor */
     408        5843 :       pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
     409        5843 :       pi1 = get_random_a(nf, A1, a1);
     410        5843 :       (void)bezout(a0, a1, &v0,&v1);
     411        5843 :       u0 = mulii(a0, v0);
     412        5843 :       u1 = mulii(a1, v1);
     413        5843 :       if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
     414             :       else
     415        5843 :       { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
     416        5843 :       u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
     417        5843 :       a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
     418             :     }
     419             :   }
     420      130961 :   if (cx)
     421             :   {
     422        3233 :     a = centermod(a, xZ);
     423        3233 :     tetpil = avma;
     424        3233 :     if (typ(cx) == t_INT)
     425             :     {
     426          77 :       gel(y,1) = mulii(xZ, cx);
     427          77 :       gel(y,2) = ZC_Z_mul(a, cx);
     428             :     }
     429             :     else
     430             :     {
     431        3156 :       gel(y,1) = gmul(xZ, cx);
     432        3156 :       gel(y,2) = RgC_Rg_mul(a, cx);
     433             :     }
     434             :   }
     435             :   else
     436             :   {
     437      127728 :     tetpil = avma;
     438      127728 :     gel(y,1) = icopy(xZ);
     439      127728 :     gel(y,2) = centermod(a, xZ);
     440             :   }
     441      130959 :   gerepilecoeffssp(av,tetpil,y+1,2); return y;
     442             : }
     443             : 
     444             : /* Given an ideal x, returns [a,alpha] such that a is in Q,
     445             :  * x = a Z_K + alpha Z_K, alpha in K^*
     446             :  * a = 0 or alpha = 0 are possible, but do not try to determine whether
     447             :  * x is principal. */
     448             : GEN
     449      102111 : idealtwoelt(GEN nf, GEN x)
     450             : {
     451             :   pari_sp av;
     452      102111 :   long tx = idealtyp(&x, NULL);
     453      102103 :   nf = checknf(nf);
     454      102103 :   if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
     455        1015 :   if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
     456             :   /* id_PRINCIPAL */
     457        1008 :   av = avma; x = nf_to_scalar_or_basis(nf, x);
     458        1820 :   return gerepilecopy(av, typ(x)==t_COL? mkvec2(gen_0,x):
     459         903 :                                          mkvec2(Q_abs_shallow(x),gen_0));
     460             : }
     461             : 
     462             : /*******************************************************************/
     463             : /*                                                                 */
     464             : /*                         FACTORIZATION                           */
     465             : /*                                                                 */
     466             : /*******************************************************************/
     467             : /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
     468             : static long
     469      731793 : idealHNF_norm_pval(GEN x, GEN p, long Zval)
     470             : {
     471      731793 :   long i, v = Zval, l = lg(x);
     472     2236782 :   for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
     473      731794 :   return v;
     474             : }
     475             : 
     476             : /* x integral in HNF, f0 = partial factorization of a multiple of
     477             :  * x[1,1] = x\cap Z */
     478             : GEN
     479      284537 : idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
     480             : {
     481      284537 :   GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
     482             :   long i, l;
     483      284554 :   P = gel(f,1); l = lg(P);
     484      284554 :   E = gel(f,2);
     485      284554 :   *pvN = vN = cgetg(l, t_VECSMALL);
     486      284560 :   *pvZ = vZ = cgetg(l, t_VECSMALL);
     487      711584 :   for (i = 1; i < l; i++)
     488             :   {
     489      427024 :     GEN p = gel(P,i);
     490      427024 :     vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
     491      427023 :     vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
     492             :   }
     493      284560 :   return P;
     494             : }
     495             : /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
     496             :  * x integral in HNF */
     497             : GEN
     498           0 : idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
     499           0 : { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
     500             : 
     501             : /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
     502             :  * Return v_P(A) */
     503             : static long
     504      868007 : idealHNF_val(GEN A, GEN P, long Nval, long Zval)
     505             : {
     506      868007 :   long f = pr_get_f(P), vmax, v, e, i, j, k, l;
     507             :   GEN mul, B, a, y, r, p, pk, cx, vals;
     508             :   pari_sp av;
     509             : 
     510      868006 :   if (Nval < f) return 0;
     511      867831 :   p = pr_get_p(P);
     512      867830 :   e = pr_get_e(P);
     513             :   /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
     514      867833 :   vmax = minss(Zval * e, Nval / f);
     515      867832 :   mul = pr_get_tau(P);
     516      867832 :   l = lg(mul);
     517      867832 :   B = cgetg(l,t_MAT);
     518             :   /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
     519      868093 :   gel(B,1) = gen_0; /* dummy */
     520     2556937 :   for (j = 2; j < l; j++)
     521             :   {
     522     1896856 :     GEN x = gel(A,j);
     523     1896856 :     gel(B,j) = y = cgetg(l, t_COL);
     524    15074727 :     for (i = 1; i < l; i++)
     525             :     { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
     526    13385883 :       a = mulii(gel(x,1), gcoeff(mul,i,1));
     527   123782130 :       for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
     528             :       /* p | a ? */
     529    13385595 :       gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
     530             :     }
     531             :   }
     532      660081 :   vals = cgetg(l, t_VECSMALL);
     533             :   /* vals[1] not needed */
     534     2199097 :   for (j = 2; j < l; j++)
     535             :   {
     536     1538929 :     gel(B,j) = Q_primitive_part(gel(B,j), &cx);
     537     1538950 :     vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
     538             :   }
     539      660168 :   pk = powiu(p, ceildivuu(vmax, e));
     540      660156 :   av = avma; y = cgetg(l,t_COL);
     541             :   /* can compute mod p^ceil((vmax-v)/e) */
     542     1260284 :   for (v = 1; v < vmax; v++)
     543             :   { /* we know v_pr(Bj) >= v for all j */
     544      623644 :     if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
     545     3152403 :     for (j = 2; j < l; j++)
     546             :     {
     547     2552298 :       GEN x = gel(B,j); if (v < vals[j]) continue;
     548    14497484 :       for (i = 1; i < l; i++)
     549             :       {
     550    13013999 :         pari_sp av2 = avma;
     551    13013999 :         a = mulii(gel(x,1), gcoeff(mul,i,1));
     552   217574629 :         for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
     553             :         /* a = (x.t_0)_i; p | a ? */
     554    13013684 :         a = dvmdii(a,p,&r); if (signe(r)) return v;
     555    12990088 :         if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
     556    12990082 :         gel(y,i) = gerepileuptoint(av2, a);
     557             :       }
     558     1483485 :       gel(B,j) = y; y = x;
     559     1483485 :       if (gc_needed(av,3))
     560             :       {
     561           0 :         if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
     562           0 :         gerepileall(av,3, &y,&B,&pk);
     563             :       }
     564             :     }
     565             :   }
     566      636640 :   return v;
     567             : }
     568             : /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
     569             :  * FA integer factorization matrix or NULL. Return partial factorization of
     570             :  * cx * x above primes in FA (complete factorization if !FA)*/
     571             : static GEN
     572      284534 : idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
     573             : {
     574      284534 :   const long N = lg(x)-1;
     575             :   long i, j, k, l, v;
     576      284534 :   GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
     577             : 
     578      284560 :   l = lg(vp);
     579      284560 :   i = cx? expi(cx)+1: 1;
     580      284562 :   vP = cgetg((l+i-2)*N+1, t_COL);
     581      284560 :   vE = cgetg((l+i-2)*N+1, t_COL);
     582      711583 :   for (i = k = 1; i < l; i++)
     583             :   {
     584      427020 :     GEN L, p = gel(vp,i);
     585      427020 :     long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
     586      427016 :     if (vc)
     587             :     {
     588       46047 :       L = idealprimedec(nf,p);
     589       46047 :       if (is_pm1(cx)) cx = NULL;
     590             :     }
     591             :     else
     592      380969 :       L = idealprimedec_limit_f(nf,p,Nval);
     593      990257 :     for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
     594             :     {
     595      563232 :       GEN P = gel(L,j);
     596      563232 :       pari_sp av = avma;
     597      563232 :       v = idealHNF_val(x, P, Nval, Zval);
     598      563219 :       set_avma(av);
     599      563219 :       Nval -= v*pr_get_f(P);
     600      563220 :       v += vc * pr_get_e(P); if (!v) continue;
     601      474188 :       gel(vP,k) = P;
     602      474188 :       gel(vE,k) = utoipos(v); k++;
     603             :     }
     604      470954 :     if (vc) for (; j<lg(L); j++)
     605             :     {
     606       43935 :       GEN P = gel(L,j);
     607       43935 :       gel(vP,k) = P;
     608       43935 :       gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
     609             :     }
     610             :   }
     611      284563 :   if (cx && !FA)
     612             :   { /* complete factorization */
     613       73492 :     GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
     614       73492 :     long lc = lg(cP);
     615      159822 :     for (i=1; i<lc; i++)
     616             :     {
     617       86330 :       GEN p = gel(cP,i), L = idealprimedec(nf,p);
     618       86329 :       long vc = itos(gel(cE,i));
     619      189039 :       for (j=1; j<lg(L); j++)
     620             :       {
     621      102709 :         GEN P = gel(L,j);
     622      102709 :         gel(vP,k) = P;
     623      102709 :         gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
     624             :       }
     625             :     }
     626             :   }
     627      284563 :   setlg(vP, k);
     628      284562 :   setlg(vE, k); return mkmat2(vP, vE);
     629             : }
     630             : /* true nf, x integral ideal */
     631             : static GEN
     632      239034 : idealHNF_factor(GEN nf, GEN x, ulong lim)
     633             : {
     634      239034 :   GEN cx, F = NULL;
     635      239034 :   if (lim)
     636             :   {
     637             :     GEN P, E;
     638             :     long i;
     639             :     /* strict useless because of prime table */
     640          70 :     F = absZ_factor_limit(gcoeff(x,1,1), lim);
     641          70 :     P = gel(F,1);
     642          70 :     E = gel(F,2);
     643             :     /* filter out entries > lim */
     644         119 :     for (i = lg(P)-1; i; i--)
     645         119 :       if (cmpiu(gel(P,i), lim) <= 0) break;
     646          70 :     setlg(P, i+1);
     647          70 :     setlg(E, i+1);
     648             :   }
     649      239034 :   x = Q_primitive_part(x, &cx);
     650      239009 :   return idealHNF_factor_i(nf, x, cx, F);
     651             : }
     652             : /* c * vector(#L,i,L[i].e), assume results fit in ulong */
     653             : static GEN
     654       27478 : prV_e_muls(GEN L, long c)
     655             : {
     656       27478 :   long j, l = lg(L);
     657       27478 :   GEN z = cgetg(l, t_COL);
     658       56549 :   for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
     659       27465 :   return z;
     660             : }
     661             : /* true nf, y in Q */
     662             : static GEN
     663       26851 : Q_nffactor(GEN nf, GEN y, ulong lim)
     664             : {
     665             :   GEN f, P, E;
     666             :   long l, i;
     667       26851 :   if (typ(y) == t_INT)
     668             :   {
     669       26826 :     if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
     670       26812 :     if (is_pm1(y)) return trivial_fact();
     671             :   }
     672       17118 :   y = Q_abs_shallow(y);
     673       17123 :   if (!lim) f = Q_factor(y);
     674             :   else
     675             :   {
     676         147 :     f = Q_factor_limit(y, lim);
     677         147 :     P = gel(f,1);
     678         147 :     E = gel(f,2);
     679         217 :     for (i = lg(P)-1; i > 0; i--)
     680         196 :       if (abscmpiu(gel(P,i), lim) < 0) break;
     681         147 :     setlg(P,i+1); setlg(E,i+1);
     682             :   }
     683       17126 :   P = gel(f,1); l = lg(P); if (l == 1) return f;
     684       17105 :   E = gel(f,2);
     685       44594 :   for (i = 1; i < l; i++)
     686             :   {
     687       27482 :     gel(P,i) = idealprimedec(nf, gel(P,i));
     688       27472 :     gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
     689             :   }
     690       17112 :   P = shallowconcat1(P); gel(f,1) = P; settyp(P, t_COL);
     691       17131 :   E = shallowconcat1(E); gel(f,2) = E; return f;
     692             : }
     693             : 
     694             : GEN
     695       25417 : idealfactor_partial(GEN nf, GEN x, GEN L)
     696             : {
     697       25417 :   pari_sp av = avma;
     698             :   long i, j, l;
     699             :   GEN P, E;
     700       25417 :   if (!L) return idealfactor(nf, x);
     701       24577 :   if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
     702       24556 :   l = lg(L); if (l == 1) return trivial_fact();
     703       23772 :   P = cgetg(l, t_VEC);
     704       89467 :   for (i = 1; i < l; i++)
     705             :   {
     706       65695 :     GEN p = gel(L,i);
     707       65695 :     gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
     708             :   }
     709       23772 :   P = shallowconcat1(P); settyp(P, t_COL);
     710       23772 :   P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
     711       23772 :   E = cgetg_copy(P, &l);
     712      113974 :   for (i = j = 1; i < l; i++)
     713             :   {
     714       90202 :     long v = idealval(nf, x, gel(P,i));
     715       90202 :     if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
     716             :   }
     717       23772 :   setlg(P,j);
     718       23772 :   setlg(E,j); return gerepilecopy(av, mkmat2(P, E));
     719             : }
     720             : GEN
     721      265975 : idealfactor_limit(GEN nf, GEN x, ulong lim)
     722             : {
     723      265975 :   pari_sp av = avma;
     724             :   GEN fa, y;
     725      265975 :   long tx = idealtyp(&x, NULL);
     726             : 
     727      265959 :   if (tx == id_PRIME)
     728             :   {
     729         105 :     if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
     730          98 :     retmkmat2(mkcolcopy(x), mkcol(gen_1));
     731             :   }
     732      265854 :   nf = checknf(nf);
     733      265852 :   if (tx == id_PRINCIPAL)
     734             :   {
     735       28067 :     y = nf_to_scalar_or_basis(nf, x);
     736       28071 :     if (typ(y) != t_COL) return gerepilecopy(av, Q_nffactor(nf, y, lim));
     737             :   }
     738      239003 :   y = idealnumden(nf, x);
     739      239021 :   fa = idealHNF_factor(nf, gel(y,1), lim);
     740      239022 :   if (!isint1(gel(y,2)))
     741          14 :     fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
     742      239023 :   fa = gerepilecopy(av, fa);
     743      239026 :   return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
     744             : }
     745             : GEN
     746      265603 : idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
     747             : GEN
     748         182 : gpidealfactor(GEN nf, GEN x, GEN lim)
     749             : {
     750         182 :   ulong L = 0;
     751         182 :   if (lim)
     752             :   {
     753          70 :     if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
     754          70 :     L = itou(lim);
     755             :   }
     756         182 :   return idealfactor_limit(nf, x, L);
     757             : }
     758             : 
     759             : static GEN
     760        7271 : ramified_root(GEN nf, GEN R, GEN A, long n)
     761             : {
     762        7271 :   GEN v, P = gel(idealfactor(nf, R), 1);
     763        7271 :   long i, l = lg(P);
     764        7271 :   v = cgetg(l, t_VECSMALL);
     765        7922 :   for (i = 1; i < l; i++)
     766             :   {
     767         658 :     long w = idealval(nf, A, gel(P,i));
     768         658 :     if (w % n) return NULL;
     769         651 :     v[i] = w / n;
     770             :   }
     771        7264 :   return idealfactorback(nf, P, v, 0);
     772             : }
     773             : static int
     774           7 : ramified_root_simple(GEN nf, long n, GEN P, GEN v)
     775             : {
     776           7 :   long i, l = lg(v);
     777          21 :   for (i = 1; i < l; i++)
     778             :   {
     779          14 :     long w = v[i] % n;
     780          14 :     if (w)
     781             :     {
     782           7 :       GEN vpr = idealprimedec(nf, gel(P,i));
     783           7 :       long lpr = lg(vpr), j;
     784          14 :       for (j = 1; j < lpr; j++)
     785             :       {
     786           7 :         long e = pr_get_e(gel(vpr,j));
     787           7 :         if ((e * w) % n) return 0;
     788             :       }
     789             :     }
     790             :   }
     791           7 :   return 1;
     792             : }
     793             : /* true nf, n > 1, A a non-zero integral ideal; check whether A is the n-th
     794             :  * power of an ideal and set *pB to its n-th root if so */
     795             : static long
     796        7278 : idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
     797             : {
     798             :   GEN C, root;
     799             :   long i, l;
     800             : 
     801        7278 :   if (typ(A) == t_MAT && ZM_isscalar(A, NULL)) A = gcoeff(A,1,1);
     802        7278 :   if (typ(A) == t_INT) /* > 0 */
     803             :   {
     804        5060 :     GEN P = nf_get_ramified_primes(nf), v, q;
     805        5060 :     l = lg(P); v = cgetg(l, t_VECSMALL);
     806       23980 :     for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
     807        5060 :     C = gen_1;
     808        5060 :     if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
     809        5060 :     if (!pB) return ramified_root_simple(nf, n, P, v);
     810        5053 :     q = factorback2(P, v);
     811        5053 :     root = ramified_root(nf, q, q, n);
     812        5053 :     if (!root) return 0;
     813        5053 :     if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
     814        5053 :     *pB = root; return 1;
     815             :   }
     816             :   /* compute valuations at ramified primes */
     817        2218 :   root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
     818        2218 :   if (!root) return 0;
     819             :   /* remove ramified primes */
     820        2211 :   if (isint1(root))
     821        1868 :     root = matid(nf_get_degree(nf));
     822             :   else
     823         343 :     A = idealdivexact(nf, A, idealpows(nf,root,n));
     824        2211 :   A = Q_primitive_part(A, &C);
     825        2211 :   if (C)
     826             :   {
     827          28 :     if (!Z_ispowerall(C,n,&C)) return 0;
     828          21 :     if (pB) root = ZM_Z_mul(root, C);
     829             :   }
     830             : 
     831             :   /* compute final n-th root, at most degree(nf)-1 iterations */
     832        2204 :   for (i = 0;; i++)
     833        2071 :   {
     834        4275 :     GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
     835        4275 :     if (is_pm1(a)) break;
     836        2099 :     if (!Z_ispowerall(a,n,&b)) return 0;
     837        2071 :     J = idealadd(nf, b, A);
     838        2071 :     A = idealdivexact(nf, idealpows(nf,J,n), A);
     839             :     /* div and not divexact here */
     840        2071 :     if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
     841             :   }
     842        2176 :   if (pB) *pB = root;
     843        2176 :   return 1;
     844             : }
     845             : 
     846             : /* A is assumed to be the n-th power of an ideal in nf
     847             :  returns its n-th root. */
     848             : long
     849        3667 : idealispower(GEN nf, GEN A, long n, GEN *pB)
     850             : {
     851        3667 :   pari_sp av = avma;
     852             :   GEN v, N, D;
     853        3667 :   nf = checknf(nf);
     854        3667 :   if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
     855        3667 :   if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
     856        3660 :   v = idealnumden(nf,A);
     857        3660 :   if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
     858        3660 :   if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
     859        3618 :   if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
     860        3618 :   if (pB) *pB = gerepileupto(av, idealdiv(nf,N,D)); else set_avma(av);
     861        3618 :   return 1;
     862             : }
     863             : 
     864             : /* x t_INT or integral nonzero ideal in HNF */
     865             : static GEN
     866       97159 : idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
     867             : {
     868             :   GEN cx, y, U, N, F, Q;
     869       97159 :   if (typ(x) == t_INT)
     870             :   {
     871       51142 :     if (!signe(x) || is_pm1(x)) return gen_1;
     872        1911 :     F = Z_factor_limit(x, B);
     873        1911 :     gel(F,2) = gdiventgs(gel(F,2), k);
     874        1911 :     return ginv(factorback(F));
     875             :   }
     876       46017 :   N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
     877       45527 :   F = absZ_factor_limit_strict(N, B, &U);
     878       45527 :   if (U)
     879             :   {
     880         153 :     GEN M = powii(gel(U,1), gel(U,2));
     881         153 :     y = hnfmodid(x, M); /* coprime part to B! */
     882         153 :     if (!idealispower(nf, y, k, &U)) U = NULL;
     883         153 :     x = hnfmodid(x, diviiexact(N, M));
     884             :   }
     885             :   /* x = B-smooth part of initial x */
     886       45527 :   x = Q_primitive_part(x, &cx);
     887       45524 :   F = idealHNF_factor_i(nf, x, cx, F);
     888       45527 :   gel(F,2) = gdiventgs(gel(F,2), k);
     889       45527 :   Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
     890       45526 :   if (U) Q = idealmul(nf,Q,U);
     891       45527 :   if (typ(Q) == t_INT) return Q;
     892       12563 :   y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
     893       12563 :   return gdiv(y, gcoeff(Q,1,1));
     894             : }
     895             : GEN
     896       48587 : idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
     897             : {
     898       48587 :   pari_sp av = avma;
     899             :   GEN a, b;
     900       48587 :   nf = checknf(nf);
     901       48587 :   if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
     902       48587 :   x = idealnumden(nf, x);
     903       48587 :   a = gel(x,1);
     904       48587 :   if (isintzero(a)) { set_avma(av); return gen_1; }
     905       48580 :   a = idealredmodpower_i(nf, gel(x,1), n, B);
     906       48580 :   b = idealredmodpower_i(nf, gel(x,2), n, B);
     907       48580 :   if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
     908       48580 :   return gerepilecopy(av, a);
     909             : }
     910             : 
     911             : /* P prime ideal in idealprimedec format. Return valuation(A) at P */
     912             : long
     913     2156522 : idealval(GEN nf, GEN A, GEN P)
     914             : {
     915     2156522 :   pari_sp av = avma;
     916             :   GEN p, cA;
     917     2156522 :   long vcA, v, Zval, tx = idealtyp(&A, NULL);
     918             : 
     919     2156518 :   if (tx == id_PRINCIPAL) return nfval(nf,A,P);
     920     2057485 :   checkprid(P);
     921     2057485 :   if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
     922             :   /* id_MAT */
     923     2057457 :   nf = checknf(nf);
     924     2057457 :   A = Q_primitive_part(A, &cA);
     925     2057457 :   p = pr_get_p(P);
     926     2057457 :   vcA = cA? Q_pval(cA,p): 0;
     927     2057457 :   if (pr_is_inert(P)) return gc_long(av,vcA);
     928     1659157 :   Zval = Z_pval(gcoeff(A,1,1), p);
     929     1659158 :   if (!Zval) v = 0;
     930             :   else
     931             :   {
     932      304774 :     long Nval = idealHNF_norm_pval(A, p, Zval);
     933      304777 :     v = idealHNF_val(A, P, Nval, Zval);
     934             :   }
     935     1659159 :   return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
     936             : }
     937             : GEN
     938        7105 : gpidealval(GEN nf, GEN ix, GEN P)
     939             : {
     940        7105 :   long v = idealval(nf,ix,P);
     941        7105 :   return v == LONG_MAX? mkoo(): stoi(v);
     942             : }
     943             : 
     944             : /* gcd and generalized Bezout */
     945             : 
     946             : GEN
     947      106417 : idealadd(GEN nf, GEN x, GEN y)
     948             : {
     949      106417 :   pari_sp av = avma;
     950             :   long tx, ty;
     951             :   GEN z, a, dx, dy, dz;
     952             : 
     953      106417 :   tx = idealtyp(&x, NULL);
     954      106417 :   ty = idealtyp(&y, NULL); nf = checknf(nf);
     955      106417 :   if (tx != id_MAT) x = idealhnf_shallow(nf,x);
     956      106417 :   if (ty != id_MAT) y = idealhnf_shallow(nf,y);
     957      106417 :   if (lg(x) == 1) return gerepilecopy(av,y);
     958      105423 :   if (lg(y) == 1) return gerepilecopy(av,x); /* check for 0 ideal */
     959      104968 :   dx = Q_denom(x);
     960      104968 :   dy = Q_denom(y); dz = lcmii(dx,dy);
     961      104968 :   if (is_pm1(dz)) dz = NULL; else {
     962       15246 :     x = Q_muli_to_int(x, dz);
     963       15246 :     y = Q_muli_to_int(y, dz);
     964             :   }
     965      104968 :   a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
     966      104968 :   if (is_pm1(a))
     967             :   {
     968       37274 :     long N = lg(x)-1;
     969       37274 :     if (!dz) { set_avma(av); return matid(N); }
     970        3871 :     return gerepileupto(av, scalarmat(ginv(dz), N));
     971             :   }
     972       67694 :   z = ZM_hnfmodid(shallowconcat(x,y), a);
     973       67694 :   if (dz) z = RgM_Rg_div(z,dz);
     974       67694 :   return gerepileupto(av,z);
     975             : }
     976             : 
     977             : static GEN
     978          28 : trivial_merge(GEN x)
     979          28 : { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
     980             : /* true nf */
     981             : static GEN
     982      731275 : _idealaddtoone(GEN nf, GEN x, GEN y, long red)
     983             : {
     984             :   GEN a;
     985      731275 :   long tx = idealtyp(&x, NULL);
     986      731260 :   long ty = idealtyp(&y, NULL);
     987             :   long ea;
     988      731258 :   if (tx != id_MAT) x = idealhnf_shallow(nf, x);
     989      731280 :   if (ty != id_MAT) y = idealhnf_shallow(nf, y);
     990      731280 :   if (lg(x) == 1)
     991          14 :     a = trivial_merge(y);
     992      731266 :   else if (lg(y) == 1)
     993          14 :     a = trivial_merge(x);
     994             :   else
     995      731252 :     a = hnfmerge_get_1(x, y);
     996      731243 :   if (!a) pari_err_COPRIME("idealaddtoone",x,y);
     997      731226 :   if (red && (ea = gexpo(a)) > 10)
     998             :   {
     999        5124 :     GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
    1000        5124 :     b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
    1001        5124 :     if (gexpo(b) < ea) a = b;
    1002             :   }
    1003      731226 :   return a;
    1004             : }
    1005             : /* true nf */
    1006             : GEN
    1007       19446 : idealaddtoone_i(GEN nf, GEN x, GEN y)
    1008       19446 : { return _idealaddtoone(nf, x, y, 1); }
    1009             : /* true nf */
    1010             : GEN
    1011      711830 : idealaddtoone_raw(GEN nf, GEN x, GEN y)
    1012      711830 : { return _idealaddtoone(nf, x, y, 0); }
    1013             : 
    1014             : GEN
    1015          98 : idealaddtoone(GEN nf, GEN x, GEN y)
    1016             : {
    1017          98 :   GEN z = cgetg(3,t_VEC), a;
    1018          98 :   pari_sp av = avma;
    1019          98 :   nf = checknf(nf);
    1020          98 :   a = gerepileupto(av, idealaddtoone_i(nf,x,y));
    1021          84 :   gel(z,1) = a;
    1022          84 :   gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
    1023          84 :   return z;
    1024             : }
    1025             : 
    1026             : /* assume elements of list are integral ideals */
    1027             : GEN
    1028          35 : idealaddmultoone(GEN nf, GEN list)
    1029             : {
    1030          35 :   pari_sp av = avma;
    1031          35 :   long N, i, l, nz, tx = typ(list);
    1032             :   GEN H, U, perm, L;
    1033             : 
    1034          35 :   nf = checknf(nf); N = nf_get_degree(nf);
    1035          35 :   if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
    1036          35 :   l = lg(list);
    1037          35 :   L = cgetg(l, t_VEC);
    1038          35 :   if (l == 1)
    1039           0 :     pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
    1040          35 :   nz = 0; /* number of nonzero ideals in L */
    1041          98 :   for (i=1; i<l; i++)
    1042             :   {
    1043          70 :     GEN I = gel(list,i);
    1044          70 :     if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
    1045          70 :     if (lg(I) != 1)
    1046             :     {
    1047          42 :       nz++; RgM_check_ZM(I,"idealaddmultoone");
    1048          35 :       if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
    1049             :     }
    1050          63 :     gel(L,i) = I;
    1051             :   }
    1052          28 :   H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
    1053          28 :   if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
    1054           7 :     pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
    1055          49 :   for (i=1; i<=N; i++)
    1056          49 :     if (perm[i] == 1) break;
    1057          21 :   U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
    1058          21 :   nz = 0;
    1059          63 :   for (i=1; i<l; i++)
    1060             :   {
    1061          42 :     GEN c = gel(L,i);
    1062          42 :     if (lg(c) == 1)
    1063          14 :       c = gen_0;
    1064             :     else {
    1065          28 :       c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
    1066          28 :       nz++;
    1067             :     }
    1068          42 :     gel(L,i) = c;
    1069             :   }
    1070          21 :   return gerepilecopy(av, L);
    1071             : }
    1072             : 
    1073             : /* multiplication */
    1074             : 
    1075             : /* x integral ideal (without archimedean component) in HNF form
    1076             :  * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
    1077             :  * alpha a ZV or a ZM (multiplication table). Multiply them */
    1078             : static GEN
    1079      830173 : idealHNF_mul_two(GEN nf, GEN x, GEN y)
    1080             : {
    1081      830173 :   GEN m, a = gel(y,1), alpha = gel(y,2);
    1082             :   long i, N;
    1083             : 
    1084      830173 :   if (typ(alpha) != t_MAT)
    1085             :   {
    1086      494169 :     alpha = zk_scalar_or_multable(nf, alpha);
    1087      494174 :     if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
    1088       18292 :       return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
    1089             :   }
    1090      811886 :   N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
    1091     3404447 :   for (i=1; i<=N; i++) gel(m,i)   = ZM_ZC_mul(alpha,gel(x,i));
    1092     3403158 :   for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
    1093      810867 :   return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
    1094             : }
    1095             : 
    1096             : /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
    1097             :  * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
    1098             : GEN
    1099      394897 : idealHNF_mul(GEN nf, GEN x, GEN y)
    1100             : {
    1101             :   GEN z;
    1102      394897 :   if (typ(y) == t_VEC)
    1103      217308 :     z = idealHNF_mul_two(nf,x,y);
    1104             :   else
    1105             :   { /* reduce one ideal to two-elt form. The smallest */
    1106      177589 :     GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
    1107      177589 :     if (cmpii(xZ, yZ) < 0)
    1108             :     {
    1109       37893 :       if (is_pm1(xZ)) return gcopy(y);
    1110       21978 :       z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
    1111             :     }
    1112             :     else
    1113             :     {
    1114      139696 :       if (is_pm1(yZ)) return gcopy(x);
    1115       35448 :       z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
    1116             :     }
    1117             :   }
    1118      274737 :   return z;
    1119             : }
    1120             : 
    1121             : /* operations on elements in factored form */
    1122             : 
    1123             : GEN
    1124      200014 : famat_mul_shallow(GEN f, GEN g)
    1125             : {
    1126      200014 :   if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
    1127      200014 :   if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
    1128      200014 :   if (lgcols(f) == 1) return g;
    1129      148811 :   if (lgcols(g) == 1) return f;
    1130      146959 :   return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
    1131      146959 :                 shallowconcat(gel(f,2), gel(g,2)));
    1132             : }
    1133             : GEN
    1134       88438 : famat_mulpow_shallow(GEN f, GEN g, GEN e)
    1135             : {
    1136       88438 :   if (!signe(e)) return f;
    1137       55170 :   return famat_mul_shallow(f, famat_pow_shallow(g, e));
    1138             : }
    1139             : 
    1140             : GEN
    1141      117983 : famat_mulpows_shallow(GEN f, GEN g, long e)
    1142             : {
    1143      117983 :   if (e==0) return f;
    1144       94569 :   return famat_mul_shallow(f, famat_pows_shallow(g, e));
    1145             : }
    1146             : 
    1147             : GEN
    1148       10311 : famat_div_shallow(GEN f, GEN g)
    1149       10311 : { return famat_mul_shallow(f, famat_inv_shallow(g)); }
    1150             : 
    1151             : GEN
    1152           0 : to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
    1153             : GEN
    1154     2461490 : to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
    1155             : 
    1156             : /* concat the single elt x; not gconcat since x may be a t_COL */
    1157             : static GEN
    1158      179288 : append(GEN v, GEN x)
    1159             : {
    1160      179288 :   long i, l = lg(v);
    1161      179288 :   GEN w = cgetg(l+1, typ(v));
    1162     1032679 :   for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
    1163      179288 :   gel(w,i) = gcopy(x); return w;
    1164             : }
    1165             : /* add x^1 to famat f */
    1166             : static GEN
    1167      168143 : famat_add(GEN f, GEN x)
    1168             : {
    1169      168143 :   GEN h = cgetg(3,t_MAT);
    1170      168143 :   if (lgcols(f) == 1)
    1171             :   {
    1172       10151 :     gel(h,1) = mkcolcopy(x);
    1173       10151 :     gel(h,2) = mkcol(gen_1);
    1174             :   }
    1175             :   else
    1176             :   {
    1177      157992 :     gel(h,1) = append(gel(f,1), x);
    1178      157992 :     gel(h,2) = gconcat(gel(f,2), gen_1);
    1179             :   }
    1180      168143 :   return h;
    1181             : }
    1182             : /* add x^-1 to famat f */
    1183             : static GEN
    1184       37996 : famat_sub(GEN f, GEN x)
    1185             : {
    1186       37996 :   GEN h = cgetg(3,t_MAT);
    1187       37996 :   if (lgcols(f) == 1)
    1188             :   {
    1189       16700 :     gel(h,1) = mkcolcopy(x);
    1190       16700 :     gel(h,2) = mkcol(gen_m1);
    1191             :   }
    1192             :   else
    1193             :   {
    1194       21296 :     gel(h,1) = append(gel(f,1), x);
    1195       21296 :     gel(h,2) = gconcat(gel(f,2), gen_m1);
    1196             :   }
    1197       37996 :   return h;
    1198             : }
    1199             : 
    1200             : GEN
    1201      262793 : famat_mul(GEN f, GEN g)
    1202             : {
    1203             :   GEN h;
    1204      262793 :   if (typ(g) != t_MAT) {
    1205       40043 :     if (typ(f) == t_MAT) return famat_add(f, g);
    1206           0 :     h = cgetg(3, t_MAT);
    1207           0 :     gel(h,1) = mkcol2(gcopy(f), gcopy(g));
    1208           0 :     gel(h,2) = mkcol2(gen_1, gen_1);
    1209             :   }
    1210      222750 :   if (typ(f) != t_MAT) return famat_add(g, f);
    1211       94650 :   if (lgcols(f) == 1) return gcopy(g);
    1212       72445 :   if (lgcols(g) == 1) return gcopy(f);
    1213       68650 :   h = cgetg(3,t_MAT);
    1214       68650 :   gel(h,1) = gconcat(gel(f,1), gel(g,1));
    1215       68651 :   gel(h,2) = gconcat(gel(f,2), gel(g,2));
    1216       68651 :   return h;
    1217             : }
    1218             : 
    1219             : GEN
    1220       38003 : famat_div(GEN f, GEN g)
    1221             : {
    1222             :   GEN h;
    1223       38003 :   if (typ(g) != t_MAT) {
    1224       37954 :     if (typ(f) == t_MAT) return famat_sub(f, g);
    1225           0 :     h = cgetg(3, t_MAT);
    1226           0 :     gel(h,1) = mkcol2(gcopy(f), gcopy(g));
    1227           0 :     gel(h,2) = mkcol2(gen_1, gen_m1);
    1228             :   }
    1229          49 :   if (typ(f) != t_MAT) return famat_sub(g, f);
    1230           7 :   if (lgcols(f) == 1) return famat_inv(g);
    1231           7 :   if (lgcols(g) == 1) return gcopy(f);
    1232           7 :   h = cgetg(3,t_MAT);
    1233           7 :   gel(h,1) = gconcat(gel(f,1), gel(g,1));
    1234           7 :   gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
    1235           7 :   return h;
    1236             : }
    1237             : 
    1238             : GEN
    1239       21609 : famat_sqr(GEN f)
    1240             : {
    1241             :   GEN h;
    1242       21609 :   if (typ(f) != t_MAT) return to_famat(f,gen_2);
    1243       21609 :   if (lgcols(f) == 1) return gcopy(f);
    1244       12938 :   h = cgetg(3,t_MAT);
    1245       12938 :   gel(h,1) = gcopy(gel(f,1));
    1246       12938 :   gel(h,2) = gmul2n(gel(f,2),1);
    1247       12938 :   return h;
    1248             : }
    1249             : 
    1250             : GEN
    1251       27087 : famat_inv_shallow(GEN f)
    1252             : {
    1253       27087 :   if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
    1254       10444 :   if (lgcols(f) == 1) return f;
    1255       10444 :   return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
    1256             : }
    1257             : GEN
    1258       20439 : famat_inv(GEN f)
    1259             : {
    1260       20439 :   if (typ(f) != t_MAT) return to_famat(f,gen_m1);
    1261       20439 :   if (lgcols(f) == 1) return gcopy(f);
    1262        1803 :   retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
    1263             : }
    1264             : GEN
    1265       60642 : famat_pow(GEN f, GEN n)
    1266             : {
    1267       60642 :   if (typ(f) != t_MAT) return to_famat(f,n);
    1268       60642 :   if (lgcols(f) == 1) return gcopy(f);
    1269       60642 :   retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
    1270             : }
    1271             : GEN
    1272       62603 : famat_pow_shallow(GEN f, GEN n)
    1273             : {
    1274       62603 :   if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
    1275       36162 :   if (typ(f) != t_MAT) return to_famat_shallow(f,n);
    1276        7925 :   if (lgcols(f) == 1) return f;
    1277        6507 :   return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
    1278             : }
    1279             : 
    1280             : GEN
    1281      123045 : famat_pows_shallow(GEN f, long n)
    1282             : {
    1283      123045 :   if (n==1) return f;
    1284       34741 :   if (n==-1) return famat_inv_shallow(f);
    1285       34741 :   if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
    1286       26483 :   if (lgcols(f) == 1) return f;
    1287       26483 :   return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
    1288             : }
    1289             : 
    1290             : GEN
    1291           0 : famat_Z_gcd(GEN M, GEN n)
    1292             : {
    1293           0 :   pari_sp av=avma;
    1294           0 :   long i, j, l=lgcols(M);
    1295           0 :   GEN F=cgetg(3,t_MAT);
    1296           0 :   gel(F,1)=cgetg(l,t_COL);
    1297           0 :   gel(F,2)=cgetg(l,t_COL);
    1298           0 :   for (i=1, j=1; i<l; i++)
    1299             :   {
    1300           0 :     GEN p = gcoeff(M,i,1);
    1301           0 :     GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
    1302           0 :     if (signe(e))
    1303             :     {
    1304           0 :       gcoeff(F,j,1)=p;
    1305           0 :       gcoeff(F,j,2)=e;
    1306           0 :       j++;
    1307             :     }
    1308             :   }
    1309           0 :   setlg(gel(F,1),j); setlg(gel(F,2),j);
    1310           0 :   return gerepilecopy(av,F);
    1311             : }
    1312             : 
    1313             : /* x assumed to be a t_MATs (factorization matrix), or compatible with
    1314             :  * the element_* functions. */
    1315             : static GEN
    1316       32557 : ext_sqr(GEN nf, GEN x)
    1317       32557 : { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
    1318             : static GEN
    1319       87621 : ext_mul(GEN nf, GEN x, GEN y)
    1320       87621 : { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
    1321             : static GEN
    1322       20439 : ext_inv(GEN nf, GEN x)
    1323       20439 : { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
    1324             : static GEN
    1325           0 : ext_pow(GEN nf, GEN x, GEN n)
    1326           0 : { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
    1327             : 
    1328             : GEN
    1329           0 : famat_to_nf(GEN nf, GEN f)
    1330             : {
    1331             :   GEN t, x, e;
    1332             :   long i;
    1333           0 :   if (lgcols(f) == 1) return gen_1;
    1334           0 :   x = gel(f,1);
    1335           0 :   e = gel(f,2);
    1336           0 :   t = nfpow(nf, gel(x,1), gel(e,1));
    1337           0 :   for (i=lg(x)-1; i>1; i--)
    1338           0 :     t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
    1339           0 :   return t;
    1340             : }
    1341             : 
    1342             : GEN
    1343           0 : famat_idealfactor(GEN nf, GEN x)
    1344             : {
    1345             :   long i, l;
    1346           0 :   GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
    1347           0 :   for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
    1348           0 :   h = famat_reduce(famatV_factorback(h,e));
    1349           0 :   return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
    1350             : }
    1351             : 
    1352             : GEN
    1353      295005 : famat_reduce(GEN fa)
    1354             : {
    1355             :   GEN E, G, L, g, e;
    1356             :   long i, k, l;
    1357             : 
    1358      295005 :   if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
    1359      284810 :   g = gel(fa,1); l = lg(g);
    1360      284810 :   e = gel(fa,2);
    1361      284810 :   L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
    1362      284809 :   G = cgetg(l, t_COL);
    1363      284810 :   E = cgetg(l, t_COL);
    1364             :   /* merge */
    1365     2275016 :   for (k=i=1; i<l; i++,k++)
    1366             :   {
    1367     1990206 :     gel(G,k) = gel(g,L[i]);
    1368     1990206 :     gel(E,k) = gel(e,L[i]);
    1369     1990206 :     if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
    1370             :     {
    1371      813565 :       gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
    1372      813564 :       k--;
    1373             :     }
    1374             :   }
    1375             :   /* kill 0 exponents */
    1376      284810 :   l = k;
    1377     1461452 :   for (k=i=1; i<l; i++)
    1378     1176642 :     if (!gequal0(gel(E,i)))
    1379             :     {
    1380     1159807 :       gel(G,k) = gel(G,i);
    1381     1159807 :       gel(E,k) = gel(E,i); k++;
    1382             :     }
    1383      284810 :   setlg(G, k);
    1384      284810 :   setlg(E, k); return mkmat2(G,E);
    1385             : }
    1386             : GEN
    1387          63 : matreduce(GEN f)
    1388          63 : { pari_sp av = avma;
    1389          63 :   switch(typ(f))
    1390             :   {
    1391          21 :     case t_VEC: case t_COL:
    1392             :     {
    1393          21 :       GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
    1394          21 :       return gerepilecopy(av, mkmat2(f, zc_to_ZC(e)));
    1395             :     }
    1396          35 :     case t_MAT:
    1397          35 :       if (lg(f) == 3) break;
    1398             :     default:
    1399          14 :       pari_err_TYPE("matreduce", f);
    1400             :   }
    1401          28 :   if (typ(gel(f,1)) == t_VECSMALL)
    1402           0 :     f = famatsmall_reduce(f);
    1403             :   else
    1404             :   {
    1405          28 :     if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
    1406          21 :     f = famat_reduce(f);
    1407             :   }
    1408          21 :   return gerepilecopy(av, f);
    1409             : }
    1410             : 
    1411             : GEN
    1412      173790 : famatsmall_reduce(GEN fa)
    1413             : {
    1414             :   GEN E, G, L, g, e;
    1415             :   long i, k, l;
    1416      173790 :   if (lgcols(fa) == 1) return fa;
    1417      173790 :   g = gel(fa,1); l = lg(g);
    1418      173790 :   e = gel(fa,2);
    1419      173790 :   L = vecsmall_indexsort(g);
    1420      173791 :   G = cgetg(l, t_VECSMALL);
    1421      173790 :   E = cgetg(l, t_VECSMALL);
    1422             :   /* merge */
    1423      480228 :   for (k=i=1; i<l; i++,k++)
    1424             :   {
    1425      306437 :     G[k] = g[L[i]];
    1426      306437 :     E[k] = e[L[i]];
    1427      306437 :     if (k > 1 && G[k] == G[k-1])
    1428             :     {
    1429        8099 :       E[k-1] += E[k];
    1430        8099 :       k--;
    1431             :     }
    1432             :   }
    1433             :   /* kill 0 exponents */
    1434      173791 :   l = k;
    1435      472129 :   for (k=i=1; i<l; i++)
    1436      298338 :     if (E[i])
    1437             :     {
    1438      294583 :       G[k] = G[i];
    1439      294583 :       E[k] = E[i]; k++;
    1440             :     }
    1441      173791 :   setlg(G, k);
    1442      173791 :   setlg(E, k); return mkmat2(G,E);
    1443             : }
    1444             : 
    1445             : GEN
    1446       53372 : famat_remove_trivial(GEN fa)
    1447             : {
    1448       53372 :   GEN P, E, p = gel(fa,1), e = gel(fa,2);
    1449       53372 :   long j, k, l = lg(p);
    1450       53372 :   P = cgetg(l, t_COL);
    1451       53372 :   E = cgetg(l, t_COL);
    1452     1722675 :   for (j = k = 1; j < l; j++)
    1453     1669303 :     if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
    1454       53372 :   setlg(P, k); setlg(E, k); return mkmat2(P,E);
    1455             : }
    1456             : 
    1457             : GEN
    1458       13556 : famatV_factorback(GEN v, GEN e)
    1459             : {
    1460       13556 :   long i, l = lg(e);
    1461             :   GEN V;
    1462       13556 :   if (l == 1) return trivial_fact();
    1463       13171 :   V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
    1464       55910 :   for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
    1465       13171 :   return V;
    1466             : }
    1467             : 
    1468             : GEN
    1469       46529 : famatV_zv_factorback(GEN v, GEN e)
    1470             : {
    1471       46529 :   long i, l = lg(e);
    1472             :   GEN V;
    1473       46529 :   if (l == 1) return trivial_fact();
    1474       44142 :   V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
    1475      138677 :   for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
    1476       44142 :   return V;
    1477             : }
    1478             : 
    1479             : GEN
    1480      567880 : ZM_famat_limit(GEN fa, GEN limit)
    1481             : {
    1482             :   pari_sp av;
    1483             :   GEN E, G, g, e, r;
    1484             :   long i, k, l, n, lG;
    1485             : 
    1486      567880 :   if (lgcols(fa) == 1) return fa;
    1487      567873 :   g = gel(fa,1); l = lg(g);
    1488      567873 :   e = gel(fa,2);
    1489     1136894 :   for(n=0, i=1; i<l; i++)
    1490      569021 :     if (cmpii(gel(g,i),limit)<=0) n++;
    1491      567873 :   lG = n<l-1 ? n+2 : n+1;
    1492      567873 :   G = cgetg(lG, t_COL);
    1493      567872 :   E = cgetg(lG, t_COL);
    1494      567872 :   av = avma;
    1495     1136892 :   for (i=1, k=1, r = gen_1; i<l; i++)
    1496             :   {
    1497      569020 :     if (cmpii(gel(g,i),limit)<=0)
    1498             :     {
    1499      568887 :       gel(G,k) = gel(g,i);
    1500      568887 :       gel(E,k) = gel(e,i);
    1501      568887 :       k++;
    1502         133 :     } else r = mulii(r, powii(gel(g,i), gel(e,i)));
    1503             :   }
    1504      567872 :   if (k<i)
    1505             :   {
    1506         133 :     gel(G, k) = gerepileuptoint(av, r);
    1507         133 :     gel(E, k) = gen_1;
    1508             :   }
    1509      567872 :   return mkmat2(G,E);
    1510             : }
    1511             : 
    1512             : /* assume pr has degree 1 and coprime to Q_denom(x) */
    1513             : static GEN
    1514      122514 : to_Fp_coprime(GEN nf, GEN x, GEN modpr)
    1515             : {
    1516      122514 :   GEN d, r, p = modpr_get_p(modpr);
    1517      122514 :   x = nf_to_scalar_or_basis(nf,x);
    1518      122514 :   if (typ(x) != t_COL) return Rg_to_Fp(x,p);
    1519      121030 :   x = Q_remove_denom(x, &d);
    1520      121030 :   r = zk_to_Fq(x, modpr);
    1521      121029 :   if (d) r = Fp_div(r, d, p);
    1522      121029 :   return r;
    1523             : }
    1524             : 
    1525             : /* pr coprime to all denominators occurring in x */
    1526             : static GEN
    1527         763 : famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
    1528             : {
    1529         763 :   GEN p = modpr_get_p(modpr);
    1530         763 :   GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
    1531         763 :   long i, l = lg(g);
    1532        4711 :   for (i = 1; i < l; i++)
    1533             :   {
    1534        3948 :     GEN n = modii(gel(e,i), q);
    1535        3948 :     if (signe(n))
    1536             :     {
    1537        3941 :       GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
    1538        3941 :       h = Fp_pow(h, n, p);
    1539        3941 :       t = t? Fp_mul(t, h, p): h;
    1540             :     }
    1541             :   }
    1542         763 :   return t? modii(t, p): gen_1;
    1543             : }
    1544             : 
    1545             : /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
    1546             : GEN
    1547      119336 : nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
    1548             : {
    1549         763 :   return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
    1550      120099 :                       : to_Fp_coprime(nf, x, modpr);
    1551             : }
    1552             : 
    1553             : static long
    1554     3922610 : zk_pvalrem(GEN x, GEN p, GEN *py)
    1555     3922610 : { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
    1556             : /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
    1557             :  * such that x = p^v (newx / dx); dx = NULL if 1 */
    1558             : static GEN
    1559     4021201 : nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
    1560             : {
    1561             :   long vcx;
    1562             :   GEN dx;
    1563     4021201 :   x = nf_to_scalar_or_basis(nf, x);
    1564     4021206 :   x = Q_remove_denom(x, &dx);
    1565     4021204 :   if (dx)
    1566             :   {
    1567      347847 :     vcx = - Z_pvalrem(dx, p, &dx);
    1568      347848 :     if (!vcx) vcx = zk_pvalrem(x, p, &x);
    1569      347848 :     if (isint1(dx)) dx = NULL;
    1570             :   }
    1571             :   else
    1572             :   {
    1573     3673357 :     vcx = zk_pvalrem(x, p, &x);
    1574     3673359 :     dx = NULL;
    1575             :   }
    1576     4021207 :   *pv = vcx;
    1577     4021207 :   *pdx = dx; return x;
    1578             : }
    1579             : /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
    1580             :  * if p inert (instead of 1) */
    1581             : static GEN
    1582       91498 : p_makecoprime(GEN pr)
    1583             : {
    1584       91498 :   GEN B = pr_get_tau(pr), b;
    1585             :   long i, e;
    1586             : 
    1587       91498 :   if (typ(B) == t_INT) return NULL;
    1588       68741 :   b = gel(B,1); /* B = multiplication table by b */
    1589       68741 :   e = pr_get_e(pr);
    1590       68741 :   if (e == 1) return b;
    1591             :   /* one could also divide (exactly) by p in each iteration */
    1592       45519 :   for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
    1593       22215 :   return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
    1594             : }
    1595             : 
    1596             : /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
    1597             :  * Method: modify each g[i] so that it becomes coprime to pr,
    1598             :  * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
    1599             :  * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
    1600             :  * Optimizations:
    1601             :  * 1) remove all powers of p from contents, and consider extra generator p^vp;
    1602             :  * modified as p * (b/p)^e = b^e / p^(e-1)
    1603             :  * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
    1604             :  *
    1605             :  * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
    1606             :  * case the e[i] are large */
    1607             : GEN
    1608     2000714 : famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
    1609             : {
    1610     2000714 :   GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
    1611     2000713 :   long i, l = lg(g);
    1612             : 
    1613     2000713 :   G = cgetg(l+1, t_VEC);
    1614     2000724 :   E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
    1615     6021905 :   for (i=1; i < l; i++)
    1616             :   {
    1617             :     long vcx;
    1618     4021197 :     GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
    1619     4021204 :     if (vcx) /* = v_p(content(g[i])) */
    1620             :     {
    1621      148503 :       GEN a = mulsi(vcx, gel(e,i));
    1622      148505 :       vp = vp? addii(vp, a): a;
    1623             :     }
    1624             :     /* x integral, content coprime to p; dx coprime to p */
    1625     4021206 :     if (typ(x) == t_INT)
    1626             :     { /* x coprime to p, hence to pr */
    1627     1014639 :       x = modii(x, prkZ);
    1628     1014637 :       if (dx) x = Fp_div(x, dx, prkZ);
    1629             :     }
    1630             :     else
    1631             :     {
    1632     3006567 :       (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
    1633     3006537 :       x = ZC_hnfrem(FpC_red(x,prkZ), prk);
    1634     3006545 :       if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
    1635             :     }
    1636     4021176 :     gel(G,i) = x;
    1637     4021176 :     gel(E,i) = gel(e,i);
    1638             :   }
    1639             : 
    1640     2000708 :   t = vp? p_makecoprime(pr): NULL;
    1641     2000712 :   if (!t)
    1642             :   { /* no need for extra generator */
    1643     1932048 :     setlg(G,l);
    1644     1932052 :     setlg(E,l);
    1645             :   }
    1646             :   else
    1647             :   {
    1648       68664 :     gel(G,i) = FpC_red(t, prkZ);
    1649       68663 :     gel(E,i) = vp;
    1650             :   }
    1651     2000719 :   return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
    1652             : }
    1653             : 
    1654             : /* simplified version of famat_makecoprime for X = SUnits[1] */
    1655             : GEN
    1656          98 : sunits_makecoprime(GEN X, GEN pr, GEN prk)
    1657             : {
    1658          98 :   GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
    1659          98 :   long i, l = lg(X);
    1660             : 
    1661          98 :   G = cgetg(l, t_VEC);
    1662        9086 :   for (i = 1; i < l; i++)
    1663             :   {
    1664        8988 :     GEN x = gel(X,i);
    1665        8988 :     if (typ(x) == t_INT) /* a prime */
    1666        1414 :       x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
    1667             :     else
    1668             :     {
    1669        7574 :       (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
    1670        7574 :       x = ZC_hnfrem(FpC_red(x,prkZ), prk);
    1671             :     }
    1672        8988 :     gel(G,i) = x;
    1673             :   }
    1674          98 :   return G;
    1675             : }
    1676             : 
    1677             : /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
    1678             : GEN
    1679       20069 : famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
    1680             : {
    1681       20069 :   GEN t, cyc = bid_get_cyc(bid);
    1682       20069 :   if (lg(cyc) == 1)
    1683           0 :     t = gen_1;
    1684             :   else
    1685       20069 :     t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
    1686             :                                      cyc_get_expo(cyc));
    1687       20069 :   return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
    1688             : }
    1689             : 
    1690             : GEN
    1691    10870434 : vecmul(GEN x, GEN y)
    1692             : {
    1693    10870434 :   if (is_scalar_t(typ(x))) return gmul(x,y);
    1694     2440440 :   pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
    1695             : }
    1696             : 
    1697             : GEN
    1698      136899 : vecsqr(GEN x)
    1699             : {
    1700      136899 :   if (is_scalar_t(typ(x))) return gsqr(x);
    1701       41478 :   pari_APPLY_same(vecsqr(gel(x,i)))
    1702             : }
    1703             : 
    1704             : GEN
    1705         770 : vecinv(GEN x)
    1706             : {
    1707         770 :   if (is_scalar_t(typ(x))) return ginv(x);
    1708           0 :   pari_APPLY_same(vecinv(gel(x,i)))
    1709             : }
    1710             : 
    1711             : GEN
    1712           0 : vecpow(GEN x, GEN n)
    1713             : {
    1714           0 :   if (is_scalar_t(typ(x))) return powgi(x,n);
    1715           0 :   pari_APPLY_same(vecpow(gel(x,i), n))
    1716             : }
    1717             : 
    1718             : GEN
    1719         903 : vecdiv(GEN x, GEN y)
    1720             : {
    1721         903 :   if (is_scalar_t(typ(x))) return gdiv(x,y);
    1722         903 :   pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
    1723             : }
    1724             : 
    1725             : /* A ideal as a square t_MAT */
    1726             : static GEN
    1727      301896 : idealmulelt(GEN nf, GEN x, GEN A)
    1728             : {
    1729             :   long i, lx;
    1730             :   GEN dx, dA, D;
    1731      301896 :   if (lg(A) == 1) return cgetg(1, t_MAT);
    1732      301896 :   x = nf_to_scalar_or_basis(nf,x);
    1733      301896 :   if (typ(x) != t_COL)
    1734       97580 :     return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
    1735      204316 :   x = Q_remove_denom(x, &dx);
    1736      204316 :   A = Q_remove_denom(A, &dA);
    1737      204316 :   x = zk_multable(nf, x);
    1738      204316 :   D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
    1739      204316 :   x = zkC_multable_mul(A, x);
    1740      204316 :   settyp(x, t_MAT); lx = lg(x);
    1741             :   /* x may contain scalars (at most 1 since the ideal is nonzero)*/
    1742      778267 :   for (i=1; i<lx; i++)
    1743      588098 :     if (typ(gel(x,i)) == t_INT)
    1744             :     {
    1745       14147 :       if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
    1746       14147 :       gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
    1747       14147 :       break;
    1748             :     }
    1749      204316 :   x = ZM_hnfmodid(x, D);
    1750      204316 :   dx = mul_denom(dx,dA);
    1751      204316 :   return dx? gdiv(x,dx): x;
    1752             : }
    1753             : 
    1754             : /* nf a true nf, tx <= ty */
    1755             : static GEN
    1756      569029 : idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
    1757             : {
    1758             :   GEN z, cx, cy;
    1759      569029 :   switch(tx)
    1760             :   {
    1761      357961 :     case id_PRINCIPAL:
    1762      357961 :       switch(ty)
    1763             :       {
    1764       55673 :         case id_PRINCIPAL:
    1765       55673 :           return idealhnf_principal(nf, nfmul(nf,x,y));
    1766         392 :         case id_PRIME:
    1767             :         {
    1768         392 :           GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
    1769         392 :           if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
    1770             : 
    1771         217 :           x = nf_to_scalar_or_basis(nf, x);
    1772         217 :           switch(typ(x))
    1773             :           {
    1774         203 :             case t_INT:
    1775         203 :               if (!signe(x)) return cgetg(1,t_MAT);
    1776         203 :               return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
    1777           7 :             case t_FRAC:
    1778           7 :               return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
    1779             :           }
    1780             :           /* t_COL */
    1781           7 :           x = Q_primitive_part(x, &cx);
    1782           7 :           x = zk_multable(nf, x);
    1783           7 :           z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
    1784           7 :           z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
    1785           7 :           return cx? ZM_Q_mul(z, cx): z;
    1786             :         }
    1787      301896 :         default: /* id_MAT */
    1788      301896 :           return idealmulelt(nf, x,y);
    1789             :       }
    1790       42432 :     case id_PRIME:
    1791       42432 :       if (ty==id_PRIME)
    1792        4340 :       { y = pr_hnf(nf,y); cy = NULL; }
    1793             :       else
    1794       38092 :         y = Q_primitive_part(y, &cy);
    1795       42432 :       y = idealHNF_mul_two(nf,y,x);
    1796       42432 :       return cy? ZM_Q_mul(y,cy): y;
    1797             : 
    1798      168636 :     default: /* id_MAT */
    1799             :     {
    1800      168636 :       long N = nf_get_degree(nf);
    1801      168636 :       if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
    1802      168622 :       x = Q_primitive_part(x, &cx);
    1803      168622 :       y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
    1804      168622 :       y = idealHNF_mul(nf,x,y);
    1805      168622 :       return cx? ZM_Q_mul(y,cx): y;
    1806             :     }
    1807             :   }
    1808             : }
    1809             : 
    1810             : /* output the ideal product x.y */
    1811             : GEN
    1812      569026 : idealmul(GEN nf, GEN x, GEN y)
    1813             : {
    1814             :   pari_sp av;
    1815             :   GEN res, ax, ay, z;
    1816      569026 :   long tx = idealtyp(&x,&ax);
    1817      569028 :   long ty = idealtyp(&y,&ay), f;
    1818      569028 :   if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
    1819      569028 :   f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
    1820      569028 :   av = avma;
    1821      569028 :   z = gerepileupto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
    1822      569015 :   if (!f) return z;
    1823       26841 :   if (ax && ay)
    1824       25364 :     ax = ext_mul(nf, ax, ay);
    1825             :   else
    1826        1477 :     ax = gcopy(ax? ax: ay);
    1827       26841 :   gel(res,1) = z; gel(res,2) = ax; return res;
    1828             : }
    1829             : 
    1830             : /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
    1831             :  * nf = true nf */
    1832             : static GEN
    1833      285212 : idealsqrprime(GEN nf, GEN pr, GEN *pc)
    1834             : {
    1835      285212 :   GEN p = pr_get_p(pr), q, gen;
    1836      285212 :   long e = pr_get_e(pr), f = pr_get_f(pr);
    1837             : 
    1838      285215 :   q = (e == 1)? sqri(p): p;
    1839      285207 :   if (e <= 2 && e * f == nf_get_degree(nf))
    1840             :   { /* pr^e = (p) */
    1841       45718 :     *pc = q;
    1842       45718 :     return mkvec2(gen_1,gen_0);
    1843             :   }
    1844      239492 :   gen = nfsqr(nf, pr_get_gen(pr));
    1845      239490 :   gen = FpC_red(gen, q);
    1846      239486 :   *pc = NULL;
    1847      239486 :   return mkvec2(q, gen);
    1848             : }
    1849             : /* cf idealpow_aux */
    1850             : static GEN
    1851       37226 : idealsqr_aux(GEN nf, GEN x, long tx)
    1852             : {
    1853       37226 :   GEN T = nf_get_pol(nf), m, cx, a, alpha;
    1854       37226 :   long N = degpol(T);
    1855       37226 :   switch(tx)
    1856             :   {
    1857          84 :     case id_PRINCIPAL:
    1858          84 :       return idealhnf_principal(nf, nfsqr(nf,x));
    1859       10423 :     case id_PRIME:
    1860       10423 :       if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
    1861       10255 :       x = idealsqrprime(nf, x, &cx);
    1862       10255 :       x = idealhnf_two(nf,x);
    1863       10255 :       return cx? ZM_Z_mul(x, cx): x;
    1864       26719 :     default:
    1865       26719 :       x = Q_primitive_part(x, &cx);
    1866       26719 :       a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
    1867       26719 :       alpha = nfsqr(nf,alpha);
    1868       26719 :       m = zk_scalar_or_multable(nf, alpha);
    1869       26719 :       if (typ(m) == t_INT) {
    1870        1635 :         x = gcdii(sqri(a), m);
    1871        1635 :         if (cx) x = gmul(x, gsqr(cx));
    1872        1635 :         x = scalarmat(x, N);
    1873             :       }
    1874             :       else
    1875             :       { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
    1876       25084 :         x = ZM_hnfmodid(m, sqri(a));
    1877       25084 :         if (cx) cx = gsqr(cx);
    1878       25084 :         if (cx) x = ZM_Q_mul(x, cx);
    1879             :       }
    1880       26719 :       return x;
    1881             :   }
    1882             : }
    1883             : GEN
    1884       37226 : idealsqr(GEN nf, GEN x)
    1885             : {
    1886             :   pari_sp av;
    1887             :   GEN res, ax, z;
    1888       37226 :   long tx = idealtyp(&x,&ax);
    1889       37226 :   res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
    1890       37226 :   av = avma;
    1891       37226 :   z = gerepileupto(av, idealsqr_aux(checknf(nf), x, tx));
    1892       37226 :   if (!ax) return z;
    1893       32557 :   gel(res,1) = z;
    1894       32557 :   gel(res,2) = ext_sqr(nf, ax); return res;
    1895             : }
    1896             : 
    1897             : /* norm of an ideal */
    1898             : GEN
    1899       99801 : idealnorm(GEN nf, GEN x)
    1900             : {
    1901             :   pari_sp av;
    1902             :   long tx;
    1903             : 
    1904       99801 :   switch(idealtyp(&x, NULL))
    1905             :   {
    1906         952 :     case id_PRIME: return pr_norm(x);
    1907        9697 :     case id_MAT: return RgM_det_triangular(x);
    1908             :   }
    1909             :   /* id_PRINCIPAL */
    1910       89152 :   nf = checknf(nf); av = avma;
    1911       89152 :   x = nfnorm(nf, x);
    1912       89152 :   tx = typ(x);
    1913       89152 :   if (tx == t_INT) return gerepileuptoint(av, absi(x));
    1914         406 :   if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
    1915         406 :   return gerepileupto(av, Q_abs(x));
    1916             : }
    1917             : 
    1918             : /* x \cap Z */
    1919             : GEN
    1920        2982 : idealdown(GEN nf, GEN x)
    1921             : {
    1922        2982 :   pari_sp av = avma;
    1923             :   GEN y, c;
    1924        2982 :   switch(idealtyp(&x, NULL))
    1925             :   {
    1926           7 :     case id_PRIME: return icopy(pr_get_p(x));
    1927        2107 :     case id_MAT: return gcopy(gcoeff(x,1,1));
    1928             :   }
    1929             :   /* id_PRINCIPAL */
    1930         868 :   nf = checknf(nf); av = avma;
    1931         868 :   x = nf_to_scalar_or_basis(nf, x);
    1932         868 :   if (is_rational_t(typ(x))) return Q_abs(x);
    1933          14 :   x = Q_primitive_part(x, &c);
    1934          14 :   y = zkmultable_capZ(zk_multable(nf, x));
    1935          14 :   return gerepilecopy(av, mul_content(c, y));
    1936             : }
    1937             : 
    1938             : /* true nf */
    1939             : static GEN
    1940          35 : idealismaximal_int(GEN nf, GEN p)
    1941             : {
    1942             :   GEN L;
    1943          35 :   if (!BPSW_psp(p)) return NULL;
    1944          70 :   if (!dvdii(nf_get_index(nf), p) &&
    1945          49 :       !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
    1946          21 :   L = idealprimedec(nf, p);
    1947          21 :   return lg(L) == 2? gel(L,1): NULL;
    1948             : }
    1949             : /* true nf */
    1950             : static GEN
    1951          21 : idealismaximal_mat(GEN nf, GEN x)
    1952             : {
    1953             :   GEN p, c, L;
    1954             :   long i, l, f;
    1955          21 :   x = Q_primitive_part(x, &c);
    1956          21 :   p = gcoeff(x,1,1);
    1957          21 :   if (c)
    1958             :   {
    1959           7 :     if (typ(c) == t_FRAC || !equali1(p)) return NULL;
    1960           7 :     return idealismaximal_int(nf, c);
    1961             :   }
    1962          14 :   if (!BPSW_psp(p)) return NULL;
    1963          14 :   l = lg(x); f = 1;
    1964          35 :   for (i = 2; i < l; i++)
    1965             :   {
    1966          21 :     c = gcoeff(x,i,i);
    1967          21 :     if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
    1968             :   }
    1969          14 :   L = idealprimedec_limit_f(nf, p, f);
    1970          28 :   for (i = lg(L)-1; i; i--)
    1971             :   {
    1972          28 :     GEN pr = gel(L,i);
    1973          28 :     if (pr_get_f(pr) != f) break;
    1974          28 :     if (idealval(nf, x, pr) == 1) return pr;
    1975             :   }
    1976           0 :   return NULL;
    1977             : }
    1978             : /* true nf */
    1979             : static GEN
    1980          63 : idealismaximal_i(GEN nf, GEN x)
    1981             : {
    1982             :   GEN L, p, pr, c;
    1983             :   long i, l;
    1984          63 :   switch(idealtyp(&x, NULL))
    1985             :   {
    1986           7 :     case id_PRIME: return x;
    1987          21 :     case id_MAT: return idealismaximal_mat(nf, x);
    1988             :   }
    1989             :   /* id_PRINCIPAL */
    1990          35 :   x = nf_to_scalar_or_basis(nf, x);
    1991          35 :   switch(typ(x))
    1992             :   {
    1993          28 :     case t_INT: return idealismaximal_int(nf, absi_shallow(x));
    1994           0 :     case t_FRAC: return NULL;
    1995             :   }
    1996           7 :   x = Q_primitive_part(x, &c);
    1997           7 :   if (c) return NULL;
    1998           7 :   p = zkmultable_capZ(zk_multable(nf, x));
    1999           7 :   L = idealprimedec(nf, p); l = lg(L); pr = NULL;
    2000          21 :   for (i = 1; i < l; i++)
    2001             :   {
    2002          14 :     long v = ZC_nfval(x, gel(L,i));
    2003          14 :     if (v > 1 || (v && pr)) return NULL;
    2004          14 :     pr = gel(L,i);
    2005             :   }
    2006           7 :   return pr;
    2007             : }
    2008             : GEN
    2009          63 : idealismaximal(GEN nf, GEN x)
    2010             : {
    2011          63 :   pari_sp av = avma;
    2012          63 :   x = idealismaximal_i(checknf(nf), x);
    2013          63 :   if (!x) { set_avma(av); return gen_0; }
    2014          49 :   return gerepilecopy(av, x);
    2015             : }
    2016             : 
    2017             : /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
    2018             :  *
    2019             :  * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
    2020             :  * nf[5][7] = same in 2-elt form.
    2021             :  * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
    2022             : GEN
    2023      215660 : idealHNF_inv_Z(GEN nf, GEN I)
    2024             : {
    2025      215660 :   GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
    2026      215660 :   if (isint1(IZ)) return matid(lg(I)-1);
    2027      195921 :   J = idealHNF_mul(nf,I, gmael(nf,5,7));
    2028             :  /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
    2029             :   * missing content cancels while solving the linear equation */
    2030      195923 :   dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
    2031      195924 :   return ZM_hnfmodid(dual, IZ);
    2032             : }
    2033             : /* I HNF with rational coefficients (denominator d). */
    2034             : GEN
    2035       77627 : idealHNF_inv(GEN nf, GEN I)
    2036             : {
    2037       77627 :   GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
    2038       77627 :   J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
    2039       77627 :   return equali1(IQ)? J: RgM_Rg_div(J, IQ);
    2040             : }
    2041             : 
    2042             : /* return p * P^(-1)  [integral] */
    2043             : GEN
    2044       38855 : pr_inv_p(GEN pr)
    2045             : {
    2046       38855 :   if (pr_is_inert(pr)) return matid(pr_get_f(pr));
    2047       38071 :   return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
    2048             : }
    2049             : GEN
    2050       18259 : pr_inv(GEN pr)
    2051             : {
    2052       18259 :   GEN p = pr_get_p(pr);
    2053       18259 :   if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
    2054       17867 :   return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
    2055             : }
    2056             : 
    2057             : GEN
    2058      147137 : idealinv(GEN nf, GEN x)
    2059             : {
    2060             :   GEN res, ax;
    2061             :   pari_sp av;
    2062      147137 :   long tx = idealtyp(&x,&ax), N;
    2063             : 
    2064      147137 :   res = ax? cgetg(3,t_VEC): NULL;
    2065      147137 :   nf = checknf(nf); av = avma;
    2066      147137 :   N = nf_get_degree(nf);
    2067      147137 :   switch (tx)
    2068             :   {
    2069       70888 :     case id_MAT:
    2070       70888 :       if (lg(x)-1 != N) pari_err_DIM("idealinv");
    2071       70888 :       x = idealHNF_inv(nf,x); break;
    2072       59228 :     case id_PRINCIPAL:
    2073       59228 :       x = nf_to_scalar_or_basis(nf, x);
    2074       59228 :       if (typ(x) != t_COL)
    2075       59179 :         x = idealhnf_principal(nf,ginv(x));
    2076             :       else
    2077             :       { /* nfinv + idealhnf where we already know (x) \cap Z */
    2078             :         GEN c, d;
    2079          49 :         x = Q_remove_denom(x, &c);
    2080          49 :         x = zk_inv(nf, x);
    2081          49 :         x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
    2082          49 :         if (!d) /* x and x^(-1) integral => x a unit */
    2083          14 :           x = c? scalarmat(c, N): matid(N);
    2084             :         else
    2085             :         {
    2086          35 :           c = c? gdiv(c,d): ginv(d);
    2087          35 :           x = zk_multable(nf, x);
    2088          35 :           x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
    2089             :         }
    2090             :       }
    2091       59228 :       break;
    2092       17021 :     case id_PRIME:
    2093       17021 :       x = pr_inv(x); break;
    2094             :   }
    2095      147136 :   x = gerepileupto(av,x); if (!ax) return x;
    2096       20439 :   gel(res,1) = x;
    2097       20439 :   gel(res,2) = ext_inv(nf, ax); return res;
    2098             : }
    2099             : 
    2100             : /* write x = A/B, A,B coprime integral ideals */
    2101             : GEN
    2102      377819 : idealnumden(GEN nf, GEN x)
    2103             : {
    2104      377819 :   pari_sp av = avma;
    2105             :   GEN x0, c, d, A, B, J;
    2106      377819 :   long tx = idealtyp(&x, NULL);
    2107      377816 :   nf = checknf(nf);
    2108      377820 :   switch (tx)
    2109             :   {
    2110           7 :     case id_PRIME:
    2111           7 :       retmkvec2(idealhnf(nf, x), gen_1);
    2112      137452 :     case id_PRINCIPAL:
    2113             :     {
    2114             :       GEN xZ, mx;
    2115      137452 :       x = nf_to_scalar_or_basis(nf, x);
    2116      137452 :       switch(typ(x))
    2117             :       {
    2118       86191 :         case t_INT: return gerepilecopy(av, mkvec2(absi_shallow(x),gen_1));
    2119        2639 :         case t_FRAC:return gerepilecopy(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
    2120             :       }
    2121             :       /* t_COL */
    2122       48622 :       x = Q_remove_denom(x, &d);
    2123       48622 :       if (!d) return gerepilecopy(av, mkvec2(idealhnf_shallow(nf, x), gen_1));
    2124         133 :       mx = zk_multable(nf, x);
    2125         133 :       xZ = zkmultable_capZ(mx);
    2126         133 :       x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
    2127         133 :       x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
    2128         133 :       break;
    2129             :     }
    2130      240361 :     default: /* id_MAT */
    2131             :     {
    2132      240361 :       long n = lg(x)-1;
    2133      240361 :       if (n == 0) return mkvec2(gen_0, gen_1);
    2134      240361 :       if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
    2135      240361 :       x0 = x = Q_remove_denom(x, &d);
    2136      240350 :       if (!d) return gerepilecopy(av, mkvec2(x, gen_1));
    2137          21 :       break;
    2138             :     }
    2139             :   }
    2140         154 :   J = hnfmodid(x, d); /* = d/B */
    2141         154 :   c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
    2142         154 :   B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
    2143         154 :   if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
    2144         154 :   A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
    2145         154 :   A = ZM_Z_divexact(A, d); /* = A ! */
    2146         154 :   return gerepilecopy(av, mkvec2(A, B));
    2147             : }
    2148             : 
    2149             : /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
    2150             :  * nf = true nf */
    2151             : static GEN
    2152     1141853 : idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
    2153             : {
    2154     1141853 :   GEN p = pr_get_p(pr), q, gen;
    2155             : 
    2156     1141835 :   *pc = NULL;
    2157     1141835 :   if (is_pm1(n)) /* n = 1 special cased for efficiency */
    2158             :   {
    2159      601002 :     q = p;
    2160      601002 :     if (typ(pr_get_tau(pr)) == t_INT) /* inert */
    2161             :     {
    2162           0 :       *pc = (signe(n) >= 0)? p: ginv(p);
    2163           0 :       return mkvec2(gen_1,gen_0);
    2164             :     }
    2165      600995 :     if (signe(n) >= 0) gen = pr_get_gen(pr);
    2166             :     else
    2167             :     {
    2168      155487 :       gen = pr_get_tau(pr); /* possibly t_MAT */
    2169      155502 :       *pc = ginv(p);
    2170             :     }
    2171             :   }
    2172      540901 :   else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
    2173             :   else
    2174             :   {
    2175      265944 :     long e = pr_get_e(pr), f = pr_get_f(pr);
    2176      265951 :     GEN r, m = truedvmdis(n, e, &r);
    2177      265938 :     if (e * f == nf_get_degree(nf))
    2178             :     { /* pr^e = (p) */
    2179       86489 :       if (signe(m)) *pc = powii(p,m);
    2180       86489 :       if (!signe(r)) return mkvec2(gen_1,gen_0);
    2181       42564 :       q = p;
    2182       42564 :       gen = nfpow(nf, pr_get_gen(pr), r);
    2183             :     }
    2184             :     else
    2185             :     {
    2186      179461 :       m = absi_shallow(m);
    2187      179462 :       if (signe(r)) m = addiu(m,1);
    2188      179462 :       q = powii(p,m); /* m = ceil(|n|/e) */
    2189      179462 :       if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
    2190             :       else
    2191             :       {
    2192       24257 :         gen = pr_get_tau(pr);
    2193       24257 :         if (typ(gen) == t_MAT) gen = gel(gen,1);
    2194       24257 :         n = negi(n);
    2195       24257 :         gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
    2196       24256 :         *pc = ginv(q);
    2197             :       }
    2198             :     }
    2199      222030 :     gen = FpC_red(gen, q);
    2200             :   }
    2201      823025 :   return mkvec2(q, gen);
    2202             : }
    2203             : 
    2204             : /* True nf. x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
    2205             : GEN
    2206      753641 : idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
    2207             : {
    2208             :   GEN c, cx, y;
    2209      753641 :   long N = nf_get_degree(nf);
    2210             : 
    2211      753641 :   if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
    2212             : 
    2213             :   /* inert, special cased for efficiency */
    2214      753634 :   if (pr_is_inert(pr))
    2215             :   {
    2216       74974 :     GEN q = powii(pr_get_p(pr), n);
    2217       72910 :     return typ(x) == t_MAT? RgM_Rg_mul(x,q)
    2218      147859 :                           : scalarmat_shallow(gmul(Q_abs(x),q), N);
    2219             :   }
    2220             : 
    2221      678656 :   y = idealpowprime(nf, pr, n, &c);
    2222      678656 :   if (typ(x) == t_MAT)
    2223      675648 :   { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
    2224             :   else
    2225        3008 :   { cx = x; x = NULL; }
    2226      678580 :   cx = mul_content(c,cx);
    2227      678578 :   if (x)
    2228      512979 :     x = idealHNF_mul_two(nf,x,y);
    2229             :   else
    2230      165599 :     x = idealhnf_two(nf,y);
    2231      678771 :   if (cx) x = ZM_Q_mul(x,cx);
    2232      678421 :   return x;
    2233             : }
    2234             : GEN
    2235        9983 : idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
    2236             : {
    2237        9983 :   return idealmulpowprime(nf,x,pr, negi(n));
    2238             : }
    2239             : 
    2240             : /* nf = true nf */
    2241             : static GEN
    2242      838649 : idealpow_aux(GEN nf, GEN x, long tx, GEN n)
    2243             : {
    2244      838649 :   GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
    2245      838648 :   long N = degpol(T), s = signe(n);
    2246      838650 :   if (!s) return matid(N);
    2247      823378 :   switch(tx)
    2248             :   {
    2249       75234 :     case id_PRINCIPAL:
    2250       75234 :       return idealhnf_principal(nf, nfpow(nf,x,n));
    2251      554726 :     case id_PRIME:
    2252      554726 :       if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
    2253      463190 :       x = idealpowprime(nf, x, n, &cx);
    2254      463173 :       x = idealhnf_two(nf,x);
    2255      463191 :       return cx? ZM_Q_mul(x, cx): x;
    2256      193418 :     default:
    2257      193418 :       if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
    2258       69145 :       n1 = (s < 0)? negi(n): n;
    2259             : 
    2260       69145 :       x = Q_primitive_part(x, &cx);
    2261       69145 :       a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
    2262       69145 :       alpha = nfpow(nf,alpha,n1);
    2263       69145 :       m = zk_scalar_or_multable(nf, alpha);
    2264       69145 :       if (typ(m) == t_INT) {
    2265         553 :         x = gcdii(powii(a,n1), m);
    2266         553 :         if (s<0) x = ginv(x);
    2267         553 :         if (cx) x = gmul(x, powgi(cx,n));
    2268         553 :         x = scalarmat(x, N);
    2269             :       }
    2270             :       else
    2271             :       { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
    2272       68592 :         x = ZM_hnfmodid(m, powii(a,n1));
    2273       68592 :         if (cx) cx = powgi(cx,n);
    2274       68592 :         if (s<0) {
    2275           7 :           GEN xZ = gcoeff(x,1,1);
    2276           7 :           cx = cx ? gdiv(cx, xZ): ginv(xZ);
    2277           7 :           x = idealHNF_inv_Z(nf,x);
    2278             :         }
    2279       68592 :         if (cx) x = ZM_Q_mul(x, cx);
    2280             :       }
    2281       69145 :       return x;
    2282             :   }
    2283             : }
    2284             : 
    2285             : /* raise the ideal x to the power n (in Z) */
    2286             : GEN
    2287      838649 : idealpow(GEN nf, GEN x, GEN n)
    2288             : {
    2289             :   pari_sp av;
    2290             :   long tx;
    2291             :   GEN res, ax;
    2292             : 
    2293      838649 :   if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
    2294      838649 :   tx = idealtyp(&x,&ax);
    2295      838653 :   res = ax? cgetg(3,t_VEC): NULL;
    2296      838653 :   av = avma;
    2297      838653 :   x = gerepileupto(av, idealpow_aux(checknf(nf), x, tx, n));
    2298      838652 :   if (!ax) return x;
    2299           0 :   gel(res,1) = x;
    2300           0 :   gel(res,2) = ext_pow(nf, ax, n);
    2301           0 :   return res;
    2302             : }
    2303             : 
    2304             : /* Return ideal^e in number field nf. e is a C integer. */
    2305             : GEN
    2306      248012 : idealpows(GEN nf, GEN ideal, long e)
    2307             : {
    2308      248012 :   long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
    2309      248012 :   affsi(e,court); return idealpow(nf,ideal,court);
    2310             : }
    2311             : 
    2312             : static GEN
    2313       27415 : _idealmulred(GEN nf, GEN x, GEN y)
    2314       27415 : { return idealred(nf,idealmul(nf,x,y)); }
    2315             : static GEN
    2316       34342 : _idealsqrred(GEN nf, GEN x)
    2317       34342 : { return idealred(nf,idealsqr(nf,x)); }
    2318             : static GEN
    2319       10698 : _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
    2320             : static GEN
    2321       34342 : _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
    2322             : 
    2323             : /* compute x^n (x ideal, n integer), reducing along the way */
    2324             : GEN
    2325       79453 : idealpowred(GEN nf, GEN x, GEN n)
    2326             : {
    2327       79453 :   pari_sp av = avma, av2;
    2328             :   long s;
    2329             :   GEN y;
    2330             : 
    2331       79453 :   if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
    2332       79453 :   s = signe(n); if (s == 0) return idealpow(nf,x,n);
    2333       79453 :   y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
    2334       79453 :   av2 = avma;
    2335       79453 :   if (s < 0) y = idealinv(nf,y);
    2336       79452 :   if (s < 0 || is_pm1(n)) y = idealred(nf,y);
    2337       79454 :   return avma == av2? gerepilecopy(av,y): gerepileupto(av,y);
    2338             : }
    2339             : 
    2340             : GEN
    2341       16717 : idealmulred(GEN nf, GEN x, GEN y)
    2342             : {
    2343       16717 :   pari_sp av = avma;
    2344       16717 :   return gerepileupto(av, _idealmulred(nf,x,y));
    2345             : }
    2346             : 
    2347             : long
    2348          91 : isideal(GEN nf,GEN x)
    2349             : {
    2350          91 :   long N, i, j, lx, tx = typ(x);
    2351             :   pari_sp av;
    2352             :   GEN T, xZ;
    2353             : 
    2354          91 :   nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
    2355          91 :   if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
    2356          91 :   switch(tx)
    2357             :   {
    2358          14 :     case t_INT: case t_FRAC: return 1;
    2359           7 :     case t_POL: return varn(x) == varn(T);
    2360           7 :     case t_POLMOD: return RgX_equal_var(T, gel(x,1));
    2361          14 :     case t_VEC: return get_prid(x)? 1 : 0;
    2362          42 :     case t_MAT: break;
    2363           7 :     default: return 0;
    2364             :   }
    2365          42 :   N = degpol(T);
    2366          42 :   if (lx-1 != N) return (lx == 1);
    2367          28 :   if (nbrows(x) != N) return 0;
    2368             : 
    2369          28 :   av = avma; x = Q_primpart(x);
    2370          28 :   if (!ZM_ishnf(x)) return 0;
    2371          14 :   xZ = gcoeff(x,1,1);
    2372          21 :   for (j=2; j<=N; j++)
    2373          14 :     if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
    2374          14 :   for (i=2; i<=N; i++)
    2375          14 :     for (j=2; j<=N; j++)
    2376           7 :        if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
    2377           7 :   return gc_long(av,1);
    2378             : }
    2379             : 
    2380             : GEN
    2381       39470 : idealdiv(GEN nf, GEN x, GEN y)
    2382             : {
    2383       39470 :   pari_sp av = avma, tetpil;
    2384       39470 :   GEN z = idealinv(nf,y);
    2385       39470 :   tetpil = avma; return gerepile(av,tetpil, idealmul(nf,x,z));
    2386             : }
    2387             : 
    2388             : /* This routine computes the quotient x/y of two ideals in the number field nf.
    2389             :  * It assumes that the quotient is an integral ideal.  The idea is to find an
    2390             :  * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1.  Then
    2391             :  *
    2392             :  *   x + (Nx/Nz)    x
    2393             :  *   ----------- = ---
    2394             :  *   y + (Ny/Nz)    y
    2395             :  *
    2396             :  * Proof: we can assume x and y are integral. Let p be any prime ideal
    2397             :  *
    2398             :  * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
    2399             :  * product of the integers N(x/y) and N(y/z)).  Both the numerator and the
    2400             :  * denominator on the left will be coprime to p.  So will x/y, since x/y is
    2401             :  * assumed integral and its norm N(x/y) is coprime to p.
    2402             :  *
    2403             :  * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
    2404             :  * Hence v_p (x + Nx/Nz) = v_p(x).  Likewise for the denominators.  QED.
    2405             :  *
    2406             :  *                Peter Montgomery.  July, 1994. */
    2407             : static void
    2408           7 : err_divexact(GEN x, GEN y)
    2409           7 : { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
    2410           0 :                   gen_1,mkvec2(x,y)); }
    2411             : GEN
    2412        4411 : idealdivexact(GEN nf, GEN x0, GEN y0)
    2413             : {
    2414        4411 :   pari_sp av = avma;
    2415             :   GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
    2416             : 
    2417        4411 :   nf = checknf(nf);
    2418        4411 :   x = idealhnf_shallow(nf, x0);
    2419        4411 :   y = idealhnf_shallow(nf, y0);
    2420        4411 :   if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
    2421        4404 :   if (lg(x) == 1) { set_avma(av); return cgetg(1, t_MAT); } /* numerator is zero */
    2422        4404 :   y = Q_primitive_part(y, &cy);
    2423        4404 :   if (cy) x = RgM_Rg_div(x,cy);
    2424        4404 :   xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
    2425        4397 :   yZ = gcoeff(y,1,1); if (isint1(yZ)) return gerepilecopy(av, x);
    2426        2668 :   Nx = idealnorm(nf,x);
    2427        2668 :   Ny = idealnorm(nf,y);
    2428        2668 :   if (typ(Nx) != t_INT) err_divexact(x,y);
    2429        2668 :   q = dvmdii(Nx,Ny, &r);
    2430        2668 :   if (signe(r)) err_divexact(x,y);
    2431        2668 :   if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
    2432             :   /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
    2433         478 :   for (Nz = Ny;;) /* q = Nx/Nz */
    2434         458 :   {
    2435         936 :     GEN p1 = gcdii(Nz, q);
    2436         936 :     if (is_pm1(p1)) break;
    2437         458 :     Nz = diviiexact(Nz,p1);
    2438         458 :     q = mulii(q,p1);
    2439             :   }
    2440         478 :   xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
    2441         478 :   if (!equalii(xZ,q))
    2442             :   { /* Replace x/y  by  x+(Nx/Nz) / y+(Ny/Nz) */
    2443         328 :     x = ZM_hnfmodid(x, q);
    2444             :     /* y reduced to unit ideal ? */
    2445         328 :     if (Nz == Ny) return gerepileupto(av, x);
    2446             : 
    2447         111 :     yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
    2448         111 :     y = ZM_hnfmodid(y, q);
    2449             :   }
    2450         261 :   yZ = gcoeff(y,1,1);
    2451         261 :   y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
    2452         261 :   return gerepileupto(av, ZM_Z_divexact(y, yZ));
    2453             : }
    2454             : 
    2455             : GEN
    2456          21 : idealintersect(GEN nf, GEN x, GEN y)
    2457             : {
    2458          21 :   pari_sp av = avma;
    2459             :   long lz, lx, i;
    2460             :   GEN z, dx, dy, xZ, yZ;;
    2461             : 
    2462          21 :   nf = checknf(nf);
    2463          21 :   x = idealhnf_shallow(nf,x);
    2464          21 :   y = idealhnf_shallow(nf,y);
    2465          21 :   if (lg(x) == 1 || lg(y) == 1) { set_avma(av); return cgetg(1,t_MAT); }
    2466          14 :   x = Q_remove_denom(x, &dx);
    2467          14 :   y = Q_remove_denom(y, &dy);
    2468          14 :   if (dx) y = ZM_Z_mul(y, dx);
    2469          14 :   if (dy) x = ZM_Z_mul(x, dy);
    2470          14 :   xZ = gcoeff(x,1,1);
    2471          14 :   yZ = gcoeff(y,1,1);
    2472          14 :   dx = mul_denom(dx,dy);
    2473          14 :   z = ZM_lll(shallowconcat(x,y), 0.99, LLL_KER); lz = lg(z);
    2474          14 :   lx = lg(x);
    2475          63 :   for (i=1; i<lz; i++) setlg(z[i], lx);
    2476          14 :   z = ZM_hnfmodid(ZM_mul(x,z), lcmii(xZ, yZ));
    2477          14 :   if (dx) z = RgM_Rg_div(z,dx);
    2478          14 :   return gerepileupto(av,z);
    2479             : }
    2480             : 
    2481             : /*******************************************************************/
    2482             : /*                                                                 */
    2483             : /*                      T2-IDEAL REDUCTION                         */
    2484             : /*                                                                 */
    2485             : /*******************************************************************/
    2486             : 
    2487             : static GEN
    2488          21 : chk_vdir(GEN nf, GEN vdir)
    2489             : {
    2490          21 :   long i, l = lg(vdir);
    2491             :   GEN v;
    2492          21 :   if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
    2493          14 :   switch(typ(vdir))
    2494             :   {
    2495           0 :     case t_VECSMALL: return vdir;
    2496          14 :     case t_VEC: break;
    2497           0 :     default: pari_err_TYPE("idealred",vdir);
    2498             :   }
    2499          14 :   v = cgetg(l, t_VECSMALL);
    2500          56 :   for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
    2501          14 :   return v;
    2502             : }
    2503             : 
    2504             : static void
    2505       12613 : twistG(GEN G, long r1, long i, long v)
    2506             : {
    2507       12613 :   long j, lG = lg(G);
    2508       12613 :   if (i <= r1) {
    2509       37209 :     for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
    2510             :   } else {
    2511         565 :     long k = (i<<1) - r1;
    2512        4222 :     for (j=1; j<lG; j++)
    2513             :     {
    2514        3657 :       gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
    2515        3657 :       gcoeff(G,k  ,j) = gmul2n(gcoeff(G,k  ,j), v);
    2516             :     }
    2517             :   }
    2518       12613 : }
    2519             : 
    2520             : GEN
    2521      135823 : nf_get_Gtwist(GEN nf, GEN vdir)
    2522             : {
    2523             :   long i, l, v, r1;
    2524             :   GEN G;
    2525             : 
    2526      135823 :   if (!vdir) return nf_get_roundG(nf);
    2527          21 :   if (typ(vdir) == t_MAT)
    2528             :   {
    2529           0 :     long N = nf_get_degree(nf);
    2530           0 :     if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
    2531           0 :     return vdir;
    2532             :   }
    2533          21 :   vdir = chk_vdir(nf, vdir);
    2534          14 :   G = RgM_shallowcopy(nf_get_G(nf));
    2535          14 :   r1 = nf_get_r1(nf);
    2536          14 :   l = lg(vdir);
    2537          56 :   for (i=1; i<l; i++)
    2538             :   {
    2539          42 :     v = vdir[i]; if (!v) continue;
    2540          42 :     twistG(G, r1, i, v);
    2541             :   }
    2542          14 :   return RM_round_maxrank(G);
    2543             : }
    2544             : GEN
    2545       12571 : nf_get_Gtwist1(GEN nf, long i)
    2546             : {
    2547       12571 :   GEN G = RgM_shallowcopy( nf_get_G(nf) );
    2548       12571 :   long r1 = nf_get_r1(nf);
    2549       12571 :   twistG(G, r1, i, 10);
    2550       12571 :   return RM_round_maxrank(G);
    2551             : }
    2552             : 
    2553             : GEN
    2554       96208 : RM_round_maxrank(GEN G0)
    2555             : {
    2556       96208 :   long e, r = lg(G0)-1;
    2557       96208 :   pari_sp av = avma;
    2558       96208 :   for (e = 4; ; e <<= 1, set_avma(av))
    2559           0 :   {
    2560       96208 :     GEN G = gmul2n(G0, e), H = ground(G);
    2561       96210 :     if (ZM_rank(H) == r) return H; /* maximal rank ? */
    2562             :   }
    2563             : }
    2564             : 
    2565             : GEN
    2566      135816 : idealred0(GEN nf, GEN I, GEN vdir)
    2567             : {
    2568      135816 :   pari_sp av = avma;
    2569      135816 :   GEN G, aI, IZ, J, y, my, yi, c1 = NULL;
    2570             :   long N;
    2571             : 
    2572      135816 :   nf = checknf(nf);
    2573      135816 :   N = nf_get_degree(nf);
    2574             :   /* put first for sanity checks, unused when I obviously principal */
    2575      135816 :   G = nf_get_Gtwist(nf, vdir);
    2576      135809 :   switch (idealtyp(&I,&aI))
    2577             :   {
    2578       36774 :     case id_PRIME:
    2579       36774 :       if (pr_is_inert(I)) {
    2580         655 :         if (!aI) { set_avma(av); return matid(N); }
    2581         655 :         c1 = gel(I,1); I = matid(N);
    2582         655 :         goto END;
    2583             :       }
    2584       36119 :       IZ = pr_get_p(I);
    2585       36119 :       J = pr_inv_p(I);
    2586       36120 :       I = idealhnf_two(nf,I);
    2587       36120 :       break;
    2588       99008 :     case id_MAT:
    2589       99008 :       if (lg(I)-1 != N) pari_err_DIM("idealred");
    2590       99001 :       I = Q_primitive_part(I, &c1);
    2591       99000 :       IZ = gcoeff(I,1,1);
    2592       99000 :       if (is_pm1(IZ))
    2593             :       {
    2594        8810 :         if (!aI) { set_avma(av); return matid(N); }
    2595        8726 :         goto END;
    2596             :       }
    2597       90190 :       J = idealHNF_inv_Z(nf, I);
    2598       90191 :       break;
    2599          21 :     default: /* id_PRINCIPAL, silly case */
    2600          21 :       if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
    2601          21 :       if (!aI) return I;
    2602          14 :       goto END;
    2603             :   }
    2604             :   /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
    2605      126311 :   y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
    2606      126311 :   if (equalii(ZV_content(y), IZ))
    2607             :   { /* already reduced */
    2608       70000 :     if (!aI) return gerepilecopy(av, I);
    2609       66512 :     goto END;
    2610             :   }
    2611             : 
    2612       56313 :   my = zk_multable(nf, y);
    2613       56313 :   I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
    2614       56312 :   c1 = mul_content(c1, IZ);
    2615       56312 :   if (equali1(c1)) c1 = NULL; /* can be simplified with IZ */
    2616       56312 :   yi = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
    2617       56313 :   I = hnfmodid(I, Q_denom(yi)); /* denom(yi) generates (y) \cap Z */
    2618       56312 :   if (!aI) return gerepileupto(av, I);
    2619       54648 :   if (typ(aI) == t_MAT) /* yi is not integral and usually larger than y */
    2620       37912 :     aI = famat_div(aI, y);
    2621             :   else
    2622       16736 :     c1 = c1? RgC_Rg_mul(yi, c1): yi;
    2623      130555 : END:
    2624      130555 :   if (c1) aI = ext_mul(nf, aI,c1);
    2625      130555 :   return gerepilecopy(av, mkvec2(I, aI));
    2626             : }
    2627             : 
    2628             : /* I integral ZM (not HNF), G ZM, rounded Cholesky form of a weighted
    2629             :  * T2 matrix. Reduce I wrt G */
    2630             : GEN
    2631     1283455 : idealpseudored(GEN I, GEN G)
    2632     1283455 : { return ZM_mul(I, ZM_lll(ZM_mul(G, I), 0.99, LLL_IM)); }
    2633             : 
    2634             : /* Same I, G; m in I with T2(m) small */
    2635             : GEN
    2636      138901 : idealpseudomin(GEN I, GEN G)
    2637             : {
    2638      138901 :   GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
    2639      138904 :   return ZM_ZC_mul(I, gel(u,1));
    2640             : }
    2641             : /* Same I,G; irrational m in I with T2(m) small */
    2642             : GEN
    2643           0 : idealpseudomin_nonscalar(GEN I, GEN G)
    2644             : {
    2645           0 :   GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
    2646           0 :   GEN m = ZM_ZC_mul(I, gel(u,1));
    2647           0 :   if (ZV_isscalar(m) && lg(u) > 2) m = ZM_ZC_mul(I, gel(u,2));
    2648           0 :   return m;
    2649             : }
    2650             : /* Same I,G; t_VEC of irrational m in I with T2(m) small */
    2651             : GEN
    2652     1203772 : idealpseudominvec(GEN I, GEN G)
    2653             : {
    2654     1203772 :   long i, j, k, n = lg(I)-1;
    2655     1203772 :   GEN x, L, b = idealpseudored(I, G);
    2656     1203772 :   L = cgetg(1 + (n*(n+1))/2, t_VEC);
    2657     4254431 :   for (i = k = 1; i <= n; i++)
    2658             :   {
    2659     3050658 :     x = gel(b,i);
    2660     3050658 :     if (!ZV_isscalar(x)) gel(L,k++) = x;
    2661             :   }
    2662     3050659 :   for (i = 2; i <= n; i++)
    2663             :   {
    2664     1846887 :     long J = minss(i, 4);
    2665     4586879 :     for (j = 1; j < J; j++)
    2666             :     {
    2667     2739993 :       x = ZC_add(gel(b,i),gel(b,j));
    2668     2739992 :       if (!ZV_isscalar(x)) gel(L,k++) = x;
    2669             :     }
    2670             :   }
    2671     1203772 :   setlg(L,k); return L;
    2672             : }
    2673             : 
    2674             : GEN
    2675       12584 : idealred_elt(GEN nf, GEN I)
    2676             : {
    2677       12584 :   pari_sp av = avma;
    2678       12584 :   GEN u = idealpseudomin(I, nf_get_roundG(nf));
    2679       12584 :   return gerepileupto(av, u);
    2680             : }
    2681             : 
    2682             : GEN
    2683           7 : idealmin(GEN nf, GEN x, GEN vdir)
    2684             : {
    2685           7 :   pari_sp av = avma;
    2686             :   GEN y, dx;
    2687           7 :   nf = checknf(nf);
    2688           7 :   switch( idealtyp(&x, NULL) )
    2689             :   {
    2690           0 :     case id_PRINCIPAL: return gcopy(x);
    2691           0 :     case id_PRIME: x = pr_hnf(nf,x); break;
    2692           7 :     case id_MAT: if (lg(x) == 1) return gen_0;
    2693             :   }
    2694           7 :   x = Q_remove_denom(x, &dx);
    2695           7 :   y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
    2696           7 :   if (dx) y = RgC_Rg_div(y, dx);
    2697           7 :   return gerepileupto(av, y);
    2698             : }
    2699             : 
    2700             : /*******************************************************************/
    2701             : /*                                                                 */
    2702             : /*                   APPROXIMATION THEOREM                         */
    2703             : /*                                                                 */
    2704             : /*******************************************************************/
    2705             : /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
    2706             :  * and ppo(a,b) = Z_ppo(a,b) */
    2707             : /* return gcd(a,b),ppi(a,b),ppo(a,b) */
    2708             : GEN
    2709      512533 : Z_ppio(GEN a, GEN b)
    2710             : {
    2711      512533 :   GEN x, y, d = gcdii(a,b);
    2712      512533 :   if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
    2713      395122 :   x = d; y = diviiexact(a,d);
    2714             :   for(;;)
    2715       67837 :   {
    2716      462959 :     GEN g = gcdii(x,y);
    2717      462959 :     if (is_pm1(g)) return mkvec3(d, x, y);
    2718       67837 :     x = mulii(x,g); y = diviiexact(y,g);
    2719             :   }
    2720             : }
    2721             : /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
    2722             :  * and pple all others */
    2723             : /* return gcd(a,b),ppg(a,b),pple(a,b) */
    2724             : GEN
    2725           0 : Z_ppgle(GEN a, GEN b)
    2726             : {
    2727           0 :   GEN x, y, g, d = gcdii(a,b);
    2728           0 :   if (equalii(a, d)) return mkvec3(a, gen_1, a);
    2729           0 :   x = diviiexact(a,d); y = d;
    2730             :   for(;;)
    2731             :   {
    2732           0 :     g = gcdii(x,y);
    2733           0 :     if (is_pm1(g)) return mkvec3(d, x, y);
    2734           0 :     x = mulii(x,g); y = diviiexact(y,g);
    2735             :   }
    2736             : }
    2737             : static void
    2738           0 : Z_dcba_rec(GEN L, GEN a, GEN b)
    2739             : {
    2740             :   GEN x, r, v, g, h, c, c0;
    2741             :   long n;
    2742           0 :   if (is_pm1(b)) {
    2743           0 :     if (!is_pm1(a)) vectrunc_append(L, a);
    2744           0 :     return;
    2745             :   }
    2746           0 :   v = Z_ppio(a,b);
    2747           0 :   a = gel(v,2);
    2748           0 :   r = gel(v,3);
    2749           0 :   if (!is_pm1(r)) vectrunc_append(L, r);
    2750           0 :   v = Z_ppgle(a,b);
    2751           0 :   g = gel(v,1);
    2752           0 :   h = gel(v,2);
    2753           0 :   x = c0 = gel(v,3);
    2754           0 :   for (n = 1; !is_pm1(h); n++)
    2755             :   {
    2756             :     GEN d, y;
    2757             :     long i;
    2758           0 :     v = Z_ppgle(h,sqri(g));
    2759           0 :     g = gel(v,1);
    2760           0 :     h = gel(v,2);
    2761           0 :     c = gel(v,3); if (is_pm1(c)) continue;
    2762           0 :     d = gcdii(c,b);
    2763           0 :     x = mulii(x,d);
    2764           0 :     y = d; for (i=1; i < n; i++) y = sqri(y);
    2765           0 :     Z_dcba_rec(L, diviiexact(c,y), d);
    2766             :   }
    2767           0 :   Z_dcba_rec(L,diviiexact(b,x), c0);
    2768             : }
    2769             : static GEN
    2770     3608668 : Z_cba_rec(GEN L, GEN a, GEN b)
    2771             : {
    2772             :   GEN g;
    2773             :   /* a few naive steps before switching to dcba */
    2774     3608668 :   if (lg(L) > 10) { Z_dcba_rec(L, a, b); return veclast(L); }
    2775     3608668 :   if (is_pm1(a)) return b;
    2776     2149973 :   g = gcdii(a,b);
    2777     2149973 :   if (is_pm1(g)) { vectrunc_append(L, a); return b; }
    2778     1606871 :   a = diviiexact(a,g);
    2779     1606871 :   b = diviiexact(b,g);
    2780     1606871 :   return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
    2781             : }
    2782             : GEN
    2783      394926 : Z_cba(GEN a, GEN b)
    2784             : {
    2785      394926 :   GEN L = vectrunc_init(expi(a) + expi(b) + 2);
    2786      394926 :   GEN t = Z_cba_rec(L, a, b);
    2787      394926 :   if (!is_pm1(t)) vectrunc_append(L, t);
    2788      394926 :   return L;
    2789             : }
    2790             : /* P = coprime base, extend it by b; TODO: quadratic for now */
    2791             : GEN
    2792          35 : ZV_cba_extend(GEN P, GEN b)
    2793             : {
    2794          35 :   long i, l = lg(P);
    2795          35 :   GEN w = cgetg(l+1, t_VEC);
    2796         133 :   for (i = 1; i < l; i++)
    2797             :   {
    2798          98 :     GEN v = Z_cba(gel(P,i), b);
    2799          98 :     long nv = lg(v)-1;
    2800          98 :     gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
    2801          98 :     b = gel(v,nv);
    2802             :   }
    2803          35 :   gel(w,l) = b; return shallowconcat1(w);
    2804             : }
    2805             : GEN
    2806          28 : ZV_cba(GEN v)
    2807             : {
    2808          28 :   long i, l = lg(v);
    2809             :   GEN P;
    2810          28 :   if (l <= 2) return v;
    2811          14 :   P = Z_cba(gel(v,1), gel(v,2));
    2812          42 :   for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
    2813          14 :   return P;
    2814             : }
    2815             : 
    2816             : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
    2817             : GEN
    2818     3348635 : Z_ppo(GEN x, GEN f)
    2819             : {
    2820             :   for (;;)
    2821             :   {
    2822     3348635 :     f = gcdii(x, f); if (is_pm1(f)) break;
    2823     2274002 :     x = diviiexact(x, f);
    2824             :   }
    2825     1074632 :   return x;
    2826             : }
    2827             : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
    2828             : ulong
    2829    69652412 : u_ppo(ulong x, ulong f)
    2830             : {
    2831             :   for (;;)
    2832             :   {
    2833    69652412 :     f = ugcd(x, f); if (f == 1) break;
    2834    15785520 :     x /= f;
    2835             :   }
    2836    53866844 :   return x;
    2837             : }
    2838             : 
    2839             : /* result known to be representable as an ulong */
    2840             : static ulong
    2841     1519723 : lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
    2842             : 
    2843             : /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
    2844             :  * set *pd = gcd(x,N) */
    2845             : ulong
    2846     5511069 : Fl_invgen(ulong x, ulong N, ulong *pd)
    2847             : {
    2848             :   ulong d, d0, e, v, v1;
    2849             :   long s;
    2850     5511069 :   *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
    2851     5511334 :   if (s > 0) v = N - v;
    2852     5511334 :   if (d == 1) return v;
    2853             :   /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
    2854     2599599 :   e = N / d;
    2855     2599599 :   d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
    2856     2599720 :   if (d0 == 1) return v;
    2857     1519682 :   e = lcmuu(e, d / d0);
    2858     1519712 :   return u_chinese_coprime(v, 1, e, d0, e*d0);
    2859             : }
    2860             : 
    2861             : /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
    2862             : static GEN
    2863         126 : nf_coprime_part(GEN nf, GEN x, GEN listpr)
    2864             : {
    2865         126 :   long v, j, lp = lg(listpr), N = nf_get_degree(nf);
    2866             :   GEN x1, x2, ex;
    2867             : 
    2868             : #if 0 /*1) via many gcds. Expensive ! */
    2869             :   GEN f = idealprodprime(nf, listpr);
    2870             :   f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
    2871             :   x = scalarmat(x, N);
    2872             :   for (;;)
    2873             :   {
    2874             :     if (gequal1(gcoeff(f,1,1))) break;
    2875             :     x = idealdivexact(nf, x, f);
    2876             :     f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
    2877             :   }
    2878             :   x2 = x;
    2879             : #else /*2) from prime decomposition */
    2880         126 :   x1 = NULL;
    2881         350 :   for (j=1; j<lp; j++)
    2882             :   {
    2883         224 :     GEN pr = gel(listpr,j);
    2884         224 :     v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
    2885             : 
    2886         126 :     ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
    2887         126 :     x1 = x1? idealmulpowprime(nf, x1, pr, ex)
    2888         126 :            : idealpow(nf, pr, ex);
    2889             :   }
    2890         126 :   x = scalarmat(x, N);
    2891         126 :   x2 = x1? idealdivexact(nf, x, x1): x;
    2892             : #endif
    2893         126 :   return x2;
    2894             : }
    2895             : 
    2896             : /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f  */
    2897             : GEN
    2898       10976 : make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
    2899             : {
    2900             :   GEN fZ, t, L, D2, d1, d2, d;
    2901             : 
    2902       10976 :   L = Q_remove_denom(L0, &d);
    2903       10976 :   if (!d) return L0;
    2904             : 
    2905             :   /* L0 = L / d, L integral */
    2906         518 :   fZ = gcoeff(f,1,1);
    2907         518 :   if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
    2908             :   /* Kill denom part coprime to fZ */
    2909         126 :   d2 = Z_ppo(d, fZ);
    2910         126 :   t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
    2911         126 :   if (equalii(d, d2)) return L;
    2912             : 
    2913         126 :   d1 = diviiexact(d, d2);
    2914             :   /* L0 = (L / d1) mod f. d1 not coprime to f
    2915             :    * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
    2916         126 :   D2 = nf_coprime_part(nf, d1, listpr);
    2917         126 :   t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
    2918         126 :   L = nfmuli(nf,t,L);
    2919             : 
    2920             :   /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
    2921         126 :   return Q_div_to_int(L, d1); /* exact division */
    2922             : }
    2923             : 
    2924             : /* assume L is a list of prime ideals. Return the product */
    2925             : GEN
    2926         336 : idealprodprime(GEN nf, GEN L)
    2927             : {
    2928         336 :   long l = lg(L), i;
    2929             :   GEN z;
    2930         336 :   if (l == 1) return matid(nf_get_degree(nf));
    2931         336 :   z = pr_hnf(nf, gel(L,1));
    2932         364 :   for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
    2933         336 :   return z;
    2934             : }
    2935             : 
    2936             : /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
    2937             : GEN
    2938         462 : idealprod(GEN nf, GEN I)
    2939             : {
    2940         462 :   long i, l = lg(I);
    2941             :   GEN z;
    2942        1134 :   for (i = 1; i < l; i++)
    2943        1127 :     if (!equali1(gel(I,i))) break;
    2944         462 :   if (i == l) return gen_1;
    2945         455 :   z = gel(I,i);
    2946         763 :   for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
    2947         455 :   return z;
    2948             : }
    2949             : 
    2950             : /* v_pr(idealprod(nf,I)) */
    2951             : long
    2952        2722 : idealprodval(GEN nf, GEN I, GEN pr)
    2953             : {
    2954        2722 :   long i, l = lg(I), v = 0;
    2955       15586 :   for (i = 1; i < l; i++)
    2956       12864 :     if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
    2957        2722 :   return v;
    2958             : }
    2959             : 
    2960             : /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
    2961             : GEN
    2962       57538 : factorbackprime(GEN nf, GEN L, GEN e)
    2963             : {
    2964       57538 :   long l = lg(L), i;
    2965             :   GEN z;
    2966             : 
    2967       57538 :   if (l == 1) return matid(nf_get_degree(nf));
    2968       44126 :   z = idealpow(nf, gel(L,1), gel(e,1));
    2969       73714 :   for (i=2; i<l; i++)
    2970       29588 :     if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
    2971       44126 :   return z;
    2972             : }
    2973             : 
    2974             : /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
    2975             :  * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
    2976             : GEN
    2977       58056 : pr_uniformizer(GEN pr, GEN F)
    2978             : {
    2979       58056 :   GEN p = pr_get_p(pr), t = pr_get_gen(pr);
    2980       58056 :   if (!equalii(F, p))
    2981             :   {
    2982       36779 :     long e = pr_get_e(pr);
    2983       36779 :     GEN u, v, q = (e == 1)? sqri(p): p;
    2984       36779 :     u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
    2985       36779 :     v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
    2986       36779 :     if (pr_is_inert(pr))
    2987          28 :       t = addii(mulii(p, v), u);
    2988             :     else
    2989             :     {
    2990       36751 :       t = ZC_Z_mul(t, v);
    2991       36751 :       gel(t,1) = addii(gel(t,1), u); /* return u + vt */
    2992             :     }
    2993             :   }
    2994       58056 :   return t;
    2995             : }
    2996             : /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
    2997             : GEN
    2998       81835 : prV_lcm_capZ(GEN L)
    2999             : {
    3000       81835 :   long i, r = lg(L);
    3001             :   GEN F;
    3002       81835 :   if (r == 1) return gen_1;
    3003       69284 :   F = pr_get_p(gel(L,1));
    3004      122814 :   for (i = 2; i < r; i++)
    3005             :   {
    3006       53529 :     GEN pr = gel(L,i), p = pr_get_p(pr);
    3007       53529 :     if (!dvdii(F, p)) F = mulii(F,p);
    3008             :   }
    3009       69285 :   return F;
    3010             : }
    3011             : /* v vector of prid. Return underlying list of rational primes */
    3012             : GEN
    3013       60466 : prV_primes(GEN v)
    3014             : {
    3015       60466 :   long i, l = lg(v);
    3016       60466 :   GEN w = cgetg(l,t_VEC);
    3017      195965 :   for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
    3018       60466 :   return ZV_sort_uniq(w);
    3019             : }
    3020             : 
    3021             : /* Given a prime ideal factorization with possibly zero or negative
    3022             :  * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
    3023             :  * and v_pr(b) >= 0 for all other pr.
    3024             :  * For optimal performance, all [anti-]uniformizers should be precomputed,
    3025             :  * but no support for this yet. If nored, do not reduce result. */
    3026             : static GEN
    3027       53884 : idealapprfact_i(GEN nf, GEN x, int nored)
    3028             : {
    3029       53884 :   GEN d = NULL, z, L, e, e2, F;
    3030             :   long i, r;
    3031       53884 :   int hasden = 0;
    3032             : 
    3033       53884 :   nf = checknf(nf);
    3034       53884 :   L = gel(x,1);
    3035       53884 :   e = gel(x,2);
    3036       53884 :   F = prV_lcm_capZ(L);
    3037       53884 :   z = NULL; r = lg(e);
    3038      136307 :   for (i = 1; i < r; i++)
    3039             :   {
    3040       82423 :     long s = signe(gel(e,i));
    3041             :     GEN pi, q;
    3042       82423 :     if (!s) continue;
    3043       53674 :     if (s < 0) hasden = 1;
    3044       53674 :     pi = pr_uniformizer(gel(L,i), F);
    3045       53674 :     q = nfpow(nf, pi, gel(e,i));
    3046       53674 :     z = z? nfmul(nf, z, q): q;
    3047             :   }
    3048       53884 :   if (!z) return gen_1;
    3049       26786 :   if (hasden) /* denominator */
    3050             :   {
    3051       10114 :     z = Q_remove_denom(z, &d);
    3052       10114 :     d = diviiexact(d, Z_ppo(d, F));
    3053             :   }
    3054       26786 :   if (nored || typ(z) != t_COL) return d? gdiv(z, d): z;
    3055       10114 :   e2 = cgetg(r, t_VEC);
    3056       28694 :   for (i = 1; i < r; i++) gel(e2,i) = addiu(gel(e,i), 1);
    3057       10114 :   x = factorbackprime(nf, L, e2);
    3058       10114 :   if (d) x = RgM_Rg_mul(x, d);
    3059       10112 :   z = ZC_reducemodlll(z, x);
    3060       10114 :   return d? RgC_Rg_div(z,d): z;
    3061             : }
    3062             : 
    3063             : GEN
    3064           0 : idealapprfact(GEN nf, GEN x) {
    3065           0 :   pari_sp av = avma;
    3066           0 :   return gerepileupto(av, idealapprfact_i(nf, x, 0));
    3067             : }
    3068             : GEN
    3069          14 : idealappr(GEN nf, GEN x) {
    3070          14 :   pari_sp av = avma;
    3071          14 :   if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
    3072          14 :   return gerepileupto(av, idealapprfact_i(nf, x, 0));
    3073             : }
    3074             : 
    3075             : /* OBSOLETE */
    3076             : GEN
    3077          14 : idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
    3078             : 
    3079             : static GEN
    3080          21 : mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
    3081             : {
    3082          21 :   GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
    3083          21 :   long i, r = lg(E);
    3084          84 :   for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
    3085          21 :   return idealapprfact_i(nf,F,1);
    3086             : }
    3087             : 
    3088             : static void
    3089          14 : not_in_ideal(GEN a) {
    3090          14 :   pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
    3091           0 : }
    3092             : /* x integral in HNF, a an 'nf' */
    3093             : static int
    3094          28 : in_ideal(GEN x, GEN a)
    3095             : {
    3096          28 :   switch(typ(a))
    3097             :   {
    3098          14 :     case t_INT: return dvdii(a, gcoeff(x,1,1));
    3099           7 :     case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
    3100           7 :     default: return 0;
    3101             :   }
    3102             : }
    3103             : 
    3104             : /* Given an integral ideal x and a in x, gives a b such that
    3105             :  * x = aZ_K + bZ_K using the approximation theorem */
    3106             : GEN
    3107          42 : idealtwoelt2(GEN nf, GEN x, GEN a)
    3108             : {
    3109          42 :   pari_sp av = avma;
    3110             :   GEN cx, b;
    3111             : 
    3112          42 :   nf = checknf(nf);
    3113          42 :   a = nf_to_scalar_or_basis(nf, a);
    3114          42 :   x = idealhnf_shallow(nf,x);
    3115          42 :   if (lg(x) == 1)
    3116             :   {
    3117          14 :     if (!isintzero(a)) not_in_ideal(a);
    3118           7 :     set_avma(av); return gen_0;
    3119             :   }
    3120          28 :   x = Q_primitive_part(x, &cx);
    3121          28 :   if (cx) a = gdiv(a, cx);
    3122          28 :   if (!in_ideal(x, a)) not_in_ideal(a);
    3123          21 :   b = mat_ideal_two_elt2(nf, x, a);
    3124          21 :   if (typ(b) == t_COL)
    3125             :   {
    3126          14 :     GEN mod = idealhnf_principal(nf,a);
    3127          14 :     b = ZC_hnfrem(b,mod);
    3128          14 :     if (ZV_isscalar(b)) b = gel(b,1);
    3129             :   }
    3130             :   else
    3131             :   {
    3132           7 :     GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
    3133           7 :     b = centermodii(b, aZ, shifti(aZ,-1));
    3134             :   }
    3135          21 :   b = cx? gmul(b,cx): gcopy(b);
    3136          21 :   return gerepileupto(av, b);
    3137             : }
    3138             : 
    3139             : /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
    3140             :  * beta * x is an integral ideal coprime to y */
    3141             : GEN
    3142       37203 : idealcoprimefact(GEN nf, GEN x, GEN fy)
    3143             : {
    3144       37203 :   GEN L = gel(fy,1), e;
    3145       37203 :   long i, r = lg(L);
    3146             : 
    3147       37203 :   e = cgetg(r, t_COL);
    3148       76068 :   for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
    3149       37205 :   return idealapprfact_i(nf, mkmat2(L,e), 0);
    3150             : }
    3151             : GEN
    3152          84 : idealcoprime(GEN nf, GEN x, GEN y)
    3153             : {
    3154          84 :   pari_sp av = avma;
    3155          84 :   return gerepileupto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
    3156             : }
    3157             : 
    3158             : GEN
    3159           7 : nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
    3160             : {
    3161           7 :   pari_sp av = avma;
    3162           7 :   GEN z, p, pr = modpr, T;
    3163             : 
    3164           7 :   nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
    3165           0 :   x = nf_to_Fq(nf,x,modpr);
    3166           0 :   y = nf_to_Fq(nf,y,modpr);
    3167           0 :   z = Fq_mul(x,y,T,p);
    3168           0 :   return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
    3169             : }
    3170             : 
    3171             : GEN
    3172           0 : nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
    3173             : {
    3174           0 :   pari_sp av = avma;
    3175           0 :   nf = checknf(nf);
    3176           0 :   return gerepileupto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
    3177             : }
    3178             : 
    3179             : GEN
    3180           0 : nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
    3181             : {
    3182           0 :   pari_sp av=avma;
    3183           0 :   GEN z, T, p, pr = modpr;
    3184             : 
    3185           0 :   nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
    3186           0 :   z = nf_to_Fq(nf,x,modpr);
    3187           0 :   z = Fq_pow(z,k,T,p);
    3188           0 :   return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
    3189             : }
    3190             : 
    3191             : GEN
    3192           0 : nfkermodpr(GEN nf, GEN x, GEN modpr)
    3193             : {
    3194           0 :   pari_sp av = avma;
    3195           0 :   GEN T, p, pr = modpr;
    3196             : 
    3197           0 :   nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
    3198           0 :   if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
    3199           0 :   x = nfM_to_FqM(x, nf, modpr);
    3200           0 :   return gerepilecopy(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
    3201             : }
    3202             : 
    3203             : GEN
    3204           0 : nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
    3205             : {
    3206           0 :   const char *f = "nfsolvemodpr";
    3207           0 :   pari_sp av = avma;
    3208             :   GEN T, p, modpr;
    3209             : 
    3210           0 :   nf = checknf(nf);
    3211           0 :   modpr = nf_to_Fq_init(nf, &pr,&T,&p);
    3212           0 :   if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
    3213           0 :   a = nfM_to_FqM(a, nf, modpr);
    3214           0 :   switch(typ(b))
    3215             :   {
    3216           0 :     case t_MAT:
    3217           0 :       b = nfM_to_FqM(b, nf, modpr);
    3218           0 :       b = FqM_gauss(a,b,T,p);
    3219           0 :       if (!b) pari_err_INV(f,a);
    3220           0 :       a = FqM_to_nfM(b, modpr);
    3221           0 :       break;
    3222           0 :     case t_COL:
    3223           0 :       b = nfV_to_FqV(b, nf, modpr);
    3224           0 :       b = FqM_FqC_gauss(a,b,T,p);
    3225           0 :       if (!b) pari_err_INV(f,a);
    3226           0 :       a = FqV_to_nfV(b, modpr);
    3227           0 :       break;
    3228           0 :     default: pari_err_TYPE(f,b);
    3229             :   }
    3230           0 :   return gerepilecopy(av, a);
    3231             : }

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