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 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 - kummer.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.10.0 lcov report (development 21947-4fc3047) Lines: 736 852 86.4 %
Date: 2018-02-24 06:16:21 Functions: 54 59 91.5 %
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. It is distributed in the hope that it will be useful, but WITHOUT
       8             : ANY WARRANTY WHATSOEVER.
       9             : 
      10             : Check the License for details. You should have received a copy of it, along
      11             : with the package; see the file 'COPYING'. If not, write to the Free Software
      12             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      13             : 
      14             : /*******************************************************************/
      15             : /*                                                                 */
      16             : /*                      KUMMER EXTENSIONS                          */
      17             : /*                                                                 */
      18             : /*******************************************************************/
      19             : #include "pari.h"
      20             : #include "paripriv.h"
      21             : 
      22             : typedef struct {
      23             :   GEN R; /* nf.pol */
      24             :   GEN x; /* tau ( Mod(x, R) ) */
      25             :   GEN zk;/* action of tau on nf.zk (as t_MAT) */
      26             : } tau_s;
      27             : 
      28             : typedef struct {
      29             :   GEN polnf, invexpoteta1, powg;
      30             :   tau_s *tau;
      31             :   long m;
      32             : } toK_s;
      33             : 
      34             : typedef struct {
      35             :   GEN R; /* ZX, compositum(P,Q) */
      36             :   GEN p; /* QX, Mod(p,R) root of P */
      37             :   GEN q; /* QX, Mod(q,R) root of Q */
      38             :   long k; /* Q[X]/R generated by q + k p */
      39             :   GEN rev;
      40             : } compo_s;
      41             : 
      42             : static long
      43        1085 : prank(GEN cyc, long ell)
      44             : {
      45             :   long i;
      46        3101 :   for (i=1; i<lg(cyc); i++)
      47        2338 :     if (smodis(gel(cyc,i),ell)) break;
      48        1085 :   return i-1;
      49             : }
      50             : 
      51             : /* increment y, which runs through [0,d-1]^(k-1). Return 0 when done. */
      52             : static int
      53         140 : increment(GEN y, long k, long d)
      54             : {
      55         140 :   long i = k, j;
      56             :   do
      57             :   {
      58         161 :     if (--i == 0) return 0;
      59         133 :     y[i]++;
      60         133 :   } while (y[i] >= d);
      61         112 :   for (j = i+1; j < k; j++) y[j] = 0;
      62         112 :   return 1;
      63             : }
      64             : 
      65             : static int
      66         539 : ok_congruence(GEN X, ulong ell, long lW, GEN vecMsup)
      67             : {
      68             :   long i, l;
      69         539 :   if (zv_equal0(X)) return 0;
      70         539 :   l = lg(X);
      71         924 :   for (i=lW; i<l; i++)
      72         413 :     if (X[i] == 0) return 0;
      73         511 :   l = lg(vecMsup);
      74         798 :   for (i=1; i<l; i++)
      75         287 :     if (zv_equal0(Flm_Flc_mul(gel(vecMsup,i),X, ell))) return 0;
      76         511 :   return 1;
      77             : }
      78             : 
      79             : static int
      80         259 : ok_sign(GEN X, GEN msign, GEN arch)
      81             : {
      82         259 :   return zv_equal(Flm_Flc_mul(msign, X, 2), arch);
      83             : }
      84             : 
      85             : /* REDUCTION MOD ell-TH POWERS */
      86             : 
      87             : #if 0
      88             : static GEN
      89             : logarch2arch(GEN x, long r1, long prec)
      90             : {
      91             :   long i, lx;
      92             :   GEN y = cgetg_copy(x, &lx);
      93             :   if (typ(x) == t_MAT)
      94             :   {
      95             :     for (i=1; i<lx; i++) gel(y,i) = logarch2arch(gel(x,i), r1, prec);
      96             :   }
      97             :   else
      98             :   {
      99             :     for (i=1; i<=r1;i++) gel(y,i) = gexp(gel(x,i),prec);
     100             :     for (   ; i<lx; i++) gel(y,i) = gexp(gmul2n(gel(x,i),-1),prec);
     101             :   }
     102             :   return y;
     103             : }
     104             : #endif
     105             : 
     106             : /* make be integral by multiplying by t in (Q^*)^ell */
     107             : static GEN
     108         434 : reduce_mod_Qell(GEN bnfz, GEN be, GEN gell)
     109             : {
     110             :   GEN c;
     111         434 :   be = nf_to_scalar_or_basis(bnfz, be);
     112         434 :   be = Q_primitive_part(be, &c);
     113         434 :   if (c)
     114             :   {
     115         238 :     GEN d, fa = factor(c);
     116         238 :     gel(fa,2) = FpC_red(gel(fa,2), gell);
     117         238 :     d = factorback(fa);
     118         238 :     be = typ(be) == t_INT? mulii(be,d): ZC_Z_mul(be, d);
     119             :   }
     120         434 :   return be;
     121             : }
     122             : 
     123             : /* return q, q^n r = x, v_pr(r) < n for all pr. Insist q is a genuine n-th
     124             :  * root (i.e r = 1) if strict != 0. */
     125             : static GEN
     126        1260 : idealsqrtn(GEN nf, GEN x, GEN gn, int strict)
     127             : {
     128        1260 :   long i, l, n = itos(gn);
     129             :   GEN fa, q, Ex, Pr;
     130             : 
     131        1260 :   fa = idealfactor(nf, x);
     132        1260 :   Pr = gel(fa,1); l = lg(Pr);
     133        1260 :   Ex = gel(fa,2); q = NULL;
     134        3395 :   for (i=1; i<l; i++)
     135             :   {
     136        2135 :     long ex = itos(gel(Ex,i));
     137        2135 :     GEN e = stoi(ex / n);
     138        2135 :     if (strict && ex % n) pari_err_SQRTN("idealsqrtn", fa);
     139        2135 :     if (q) q = idealmulpowprime(nf, q, gel(Pr,i), e);
     140         469 :     else   q = idealpow(nf, gel(Pr,i), e);
     141             :   }
     142        1260 :   return q? q: gen_1;
     143             : }
     144             : 
     145             : static GEN
     146         434 : reducebeta(GEN bnfz, GEN b, GEN ell)
     147             : {
     148         434 :   long prec = nf_get_prec(bnfz);
     149         434 :   GEN y, elllogfu, nf = bnf_get_nf(bnfz), fu = bnf_get_fu_nocheck(bnfz);
     150             : 
     151         434 :   if (DEBUGLEVEL>1) err_printf("reducing beta = %Ps\n",b);
     152         434 :   b = reduce_mod_Qell(nf, b, ell);
     153             :   /* reduce l-th root */
     154         434 :   y = idealsqrtn(nf, b, ell, 0); /* (b) = y^ell I, I integral */
     155         434 :   if (typ(y) == t_MAT && !is_pm1(gcoeff(y,1,1)))
     156             :   {
     157         168 :     GEN T = idealred(nf, mkvec2(y, gen_1)), t = gel(T,2);
     158             :     /* (t)*T[1] = y, T[1] integral and small */
     159         168 :     if (gcmp(idealnorm(nf,t), gen_1) > 0)
     160         154 :       b = nfmul(nf, b, nfpow(nf, t, negi(ell)));
     161             :   }
     162         434 :   if (DEBUGLEVEL>1) err_printf("beta reduced via ell-th root = %Ps\n",b);
     163             :   /* log. embeddings of fu^ell */
     164         434 :   elllogfu = RgM_Rg_mul(real_i(bnf_get_logfu(bnfz)), ell);
     165             :   for (;;)
     166             :   {
     167         455 :     GEN emb, z = get_arch_real(nf, b, &emb, prec);
     168         455 :     if (z)
     169             :     {
     170         434 :       GEN ex = RgM_Babai(elllogfu, z);
     171         434 :       if (ex)
     172             :       {
     173         434 :         b = nfdiv(nf, b, nffactorback(nf, fu, RgC_Rg_mul(ex,ell)));
     174         434 :         break;
     175             :       }
     176             :     }
     177          21 :     prec = precdbl(prec);
     178          21 :     if (DEBUGLEVEL) pari_warn(warnprec,"reducebeta",prec);
     179          21 :     nf = nfnewprec_shallow(nf,prec);
     180          21 :   }
     181         434 :   if (DEBUGLEVEL>1) err_printf("beta LLL-reduced mod U^l = %Ps\n",b);
     182         434 :   return b;
     183             : }
     184             : 
     185             : /* FIXME: remove */
     186             : static GEN
     187         455 : tauofalg(GEN x, tau_s *tau) {
     188         455 :   long tx = typ(x);
     189         455 :   if (tx == t_POLMOD) { x = gel(x,2); tx = typ(x); }
     190         455 :   if (tx == t_POL) x = RgX_RgXQ_eval(x, tau->x, tau->R);
     191         455 :   return mkpolmod(x, tau->R);
     192             : }
     193             : 
     194             : /* compute Gal(K(\zeta_l)/K) */
     195             : static void
     196         231 : get_tau(tau_s *tau, GEN nf, compo_s *C, ulong g)
     197             : {
     198             :   GEN U;
     199             : 
     200             :   /* compute action of tau: q^g + kp */
     201         231 :   U = RgX_add(RgXQ_powu(C->q, g, C->R), RgX_muls(C->p, C->k));
     202         231 :   U = RgX_RgXQ_eval(C->rev, U, C->R);
     203             : 
     204         231 :   tau->x  = U;
     205         231 :   tau->R  = C->R;
     206         231 :   tau->zk = nfgaloismatrix(nf, U);
     207         231 : }
     208             : 
     209             : static GEN tauoffamat(GEN x, tau_s *tau);
     210             : 
     211             : static GEN
     212        9898 : tauofelt(GEN x, tau_s *tau)
     213             : {
     214        9898 :   switch(typ(x))
     215             :   {
     216        8281 :     case t_COL: return RgM_RgC_mul(tau->zk, x);
     217        1162 :     case t_MAT: return tauoffamat(x, tau);
     218         455 :     default: return tauofalg(x, tau);
     219             :   }
     220             : }
     221             : static GEN
     222        1330 : tauofvec(GEN x, tau_s *tau)
     223             : {
     224             :   long i, l;
     225        1330 :   GEN y = cgetg_copy(x, &l);
     226        1330 :   for (i=1; i<l; i++) gel(y,i) = tauofelt(gel(x,i), tau);
     227        1330 :   return y;
     228             : }
     229             : /* [x, tau(x), ..., tau^(m-1)(x)] */
     230             : static GEN
     231         630 : powtau(GEN x, long m, tau_s *tau)
     232             : {
     233         630 :   GEN y = cgetg(m+1, t_VEC);
     234             :   long i;
     235         630 :   gel(y,1) = x;
     236         630 :   for (i=2; i<=m; i++) gel(y,i) = tauofelt(gel(y,i-1), tau);
     237         630 :   return y;
     238             : }
     239             : /* x^lambda */
     240             : static GEN
     241         539 : lambdaofelt(GEN x, toK_s *T)
     242             : {
     243         539 :   tau_s *tau = T->tau;
     244         539 :   long i, m = T->m;
     245         539 :   GEN y = cgetg(1, t_MAT), powg = T->powg; /* powg[i] = g^i */
     246        1372 :   for (i=1; i<m; i++)
     247             :   {
     248         833 :     y = famat_mulpows_shallow(y, x, uel(powg,m-i+1));
     249         833 :     x = tauofelt(x, tau);
     250             :   }
     251         539 :   return famat_mul_shallow(y, x);
     252             : }
     253             : static GEN
     254         448 : lambdaofvec(GEN x, toK_s *T)
     255             : {
     256             :   long i, l;
     257         448 :   GEN y = cgetg_copy(x, &l);
     258         448 :   for (i=1; i<l; i++) gel(y,i) = lambdaofelt(gel(x,i), T);
     259         448 :   return y;
     260             : }
     261             : 
     262             : static GEN
     263        1162 : tauoffamat(GEN x, tau_s *tau)
     264             : {
     265        1162 :   return mkmat2(tauofvec(gel(x,1), tau), gel(x,2));
     266             : }
     267             : 
     268             : static GEN
     269         140 : tauofideal(GEN id, tau_s *tau)
     270             : {
     271         140 :   return ZM_hnfmodid(RgM_mul(tau->zk, id), gcoeff(id, 1,1));
     272             : }
     273             : 
     274             : static int
     275         644 : isprimeidealconj(GEN P, GEN Q, tau_s *tau)
     276             : {
     277         644 :   GEN p = pr_get_p(P);
     278         644 :   GEN x = pr_get_gen(P);
     279         644 :   if (!equalii(p, pr_get_p(Q))
     280         490 :    || pr_get_e(P) != pr_get_e(Q)
     281         490 :    || pr_get_f(P) != pr_get_f(Q)) return 0;
     282         483 :   if (ZV_equal(x, pr_get_gen(Q))) return 1;
     283             :   for(;;)
     284             :   {
     285        1295 :     if (ZC_prdvd(x,Q)) return 1;
     286         980 :     x = FpC_red(tauofelt(x, tau), p);
     287         980 :     if (ZC_prdvd(x,P)) return 0;
     288         812 :   }
     289             : }
     290             : 
     291             : static int
     292        1092 : isconjinprimelist(GEN S, GEN pr, tau_s *tau)
     293             : {
     294             :   long i, l;
     295             : 
     296        1092 :   if (!tau) return 0;
     297         756 :   l = lg(S);
     298        1085 :   for (i=1; i<l; i++)
     299         644 :     if (isprimeidealconj(gel(S,i),pr,tau)) return 1;
     300         441 :   return 0;
     301             : }
     302             : 
     303             : /* assume x in basistoalg form */
     304             : static GEN
     305        1036 : downtoK(toK_s *T, GEN x)
     306             : {
     307        1036 :   long degKz = lg(T->invexpoteta1) - 1;
     308        1036 :   GEN y = gmul(T->invexpoteta1, Rg_to_RgC(lift_shallow(x), degKz));
     309        1036 :   return gmodulo(gtopolyrev(y,varn(T->polnf)), T->polnf);
     310             : }
     311             : 
     312             : static GEN
     313           0 : no_sol(long all, long i)
     314             : {
     315           0 :   if (!all) pari_err_BUG(stack_sprintf("kummer [bug%ld]", i));
     316           0 :   return cgetg(1,t_VEC);
     317             : }
     318             : 
     319             : static GEN
     320         420 : get_gell(GEN bnr, GEN subgp, long all)
     321             : {
     322             :   GEN gell;
     323         420 :   if (all && all != -1) return utoipos(labs(all));
     324         399 :   if (!subgp) return ZV_prod(bnr_get_cyc(bnr));
     325         399 :   gell = det(subgp);
     326         399 :   if (typ(gell) != t_INT) pari_err_TYPE("rnfkummer",gell);
     327         399 :   return gell;
     328             : }
     329             : 
     330             : typedef struct {
     331             :   GEN Sm, Sml1, Sml2, Sl, ESml2;
     332             : } primlist;
     333             : 
     334             : static int
     335         413 : build_list_Hecke(primlist *L, GEN nfz, GEN fa, GEN gothf, GEN gell, tau_s *tau)
     336             : {
     337             :   GEN listpr, listex, pr, factell;
     338         413 :   long vp, i, l, ell = itos(gell), degKz = nf_get_degree(nfz);
     339             : 
     340         413 :   if (!fa) fa = idealfactor(nfz, gothf);
     341         413 :   listpr = gel(fa,1);
     342         413 :   listex = gel(fa,2); l = lg(listpr);
     343         413 :   L->Sm  = vectrunc_init(l);
     344         413 :   L->Sml1= vectrunc_init(l);
     345         413 :   L->Sml2= vectrunc_init(l);
     346         413 :   L->Sl  = vectrunc_init(l+degKz);
     347         413 :   L->ESml2=vecsmalltrunc_init(l);
     348        1274 :   for (i=1; i<l; i++)
     349             :   {
     350         861 :     pr = gel(listpr,i);
     351         861 :     vp = itos(gel(listex,i));
     352         861 :     if (!equalii(pr_get_p(pr), gell))
     353             :     {
     354         574 :       if (vp != 1) return 1;
     355         574 :       if (!isconjinprimelist(L->Sm,pr,tau)) vectrunc_append(L->Sm,pr);
     356             :     }
     357             :     else
     358             :     {
     359         287 :       long e = pr_get_e(pr), vd = (vp-1)*(ell-1)-ell*e;
     360         287 :       if (vd > 0) return 4;
     361         287 :       if (vd==0)
     362             :       {
     363          63 :         if (!isconjinprimelist(L->Sml1,pr,tau)) vectrunc_append(L->Sml1, pr);
     364             :       }
     365             :       else
     366             :       {
     367         224 :         if (vp==1) return 2;
     368         224 :         if (!isconjinprimelist(L->Sml2,pr,tau))
     369             :         {
     370         224 :           vectrunc_append(L->Sml2, pr);
     371         224 :           vecsmalltrunc_append(L->ESml2, vp);
     372             :         }
     373             :       }
     374             :     }
     375             :   }
     376         413 :   factell = idealprimedec(nfz,gell); l = lg(factell);
     377         931 :   for (i=1; i<l; i++)
     378             :   {
     379         518 :     pr = gel(factell,i);
     380         518 :     if (!idealval(nfz,gothf,pr) && !isconjinprimelist(L->Sl,pr,tau))
     381         224 :       vectrunc_append(L->Sl, pr);
     382             :   }
     383         413 :   return 0; /* OK */
     384             : }
     385             : 
     386             : /* Return a Flm */
     387             : static GEN
     388         672 : logall(GEN nf, GEN vec, long lW, long mginv, long ell, GEN pr, long ex)
     389             : {
     390         672 :   GEN m, M, sprk = zlog_pr_init(nf, pr, ex);
     391         672 :   long ellrank, i, l = lg(vec);
     392             : 
     393         672 :   ellrank = prank(gel(sprk,1), ell);
     394         672 :   M = cgetg(l,t_MAT);
     395        2835 :   for (i=1; i<l; i++)
     396             :   {
     397        2163 :     m = zlog_pr(nf, gel(vec,i), sprk);
     398        2163 :     setlg(m, ellrank+1);
     399        2163 :     if (i < lW) m = gmulsg(mginv, m);
     400        2163 :     gel(M,i) = ZV_to_Flv(m, ell);
     401             :   }
     402         672 :   return M;
     403             : }
     404             : 
     405             : /* compute the u_j (see remark 5.2.15.) */
     406             : static GEN
     407         413 : get_u(GEN cyc, long rc, ulong ell)
     408             : {
     409         413 :   long i, l = lg(cyc);
     410         413 :   GEN u = cgetg(l,t_VECSMALL);
     411         413 :   for (i=1; i<=rc; i++) uel(u,i) = 0;
     412         413 :   for (   ; i<  l; i++) uel(u,i) = Fl_inv(uel(cyc,i), ell);
     413         413 :   return u;
     414             : }
     415             : 
     416             : /* alg. 5.2.15. with remark */
     417             : static GEN
     418         469 : isprincipalell(GEN bnfz, GEN id, GEN cycgen, GEN u, ulong ell, long rc)
     419             : {
     420         469 :   long i, l = lg(cycgen);
     421         469 :   GEN v, b, db, y = bnfisprincipal0(bnfz, id, nf_FORCE|nf_GENMAT);
     422             : 
     423         469 :   v = ZV_to_Flv(gel(y,1), ell);
     424         469 :   b = gel(y,2);
     425         469 :   if (typ(b) == t_COL)
     426             :   {
     427         406 :     b = Q_remove_denom(gel(y,2), &db);
     428         406 :     if (db) b = famat_mulpows_shallow(b, db, -1);
     429             :   }
     430         672 :   for (i=rc+1; i<l; i++)
     431             :   {
     432         203 :     ulong e = Fl_mul( uel(v,i), uel(u,i), ell);
     433         203 :     b = famat_mulpows_shallow(b, gel(cycgen,i), e);
     434             :   }
     435         469 :   setlg(v,rc+1); return mkvec2(v, b);
     436             : }
     437             : 
     438             : static GEN
     439         140 : famat_factorback(GEN v, GEN e)
     440             : {
     441         140 :   long i, l = lg(e);
     442         140 :   GEN V = cgetg(1, t_MAT);
     443         140 :   for (i=1; i<l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
     444         140 :   return V;
     445             : }
     446             : 
     447             : static GEN
     448        1190 : famat_factorbacks(GEN v, GEN e)
     449             : {
     450        1190 :   long i, l = lg(e);
     451        1190 :   GEN V = cgetg(1, t_MAT);
     452        1190 :   for (i=1; i<l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
     453        1190 :   return V;
     454             : }
     455             : 
     456             : static GEN
     457         434 : compute_beta(GEN X, GEN vecWB, GEN ell, GEN bnfz)
     458             : {
     459             :   GEN BE, be;
     460         434 :   BE = famat_reduce(famat_factorbacks(vecWB, X));
     461         434 :   gel(BE,2) = centermod(gel(BE,2), ell);
     462         434 :   be = nffactorback(bnfz, BE, NULL);
     463         434 :   be = reducebeta(bnfz, be, ell);
     464         434 :   if (DEBUGLEVEL>1) err_printf("beta reduced = %Ps\n",be);
     465         434 :   return be;
     466             : }
     467             : 
     468             : static GEN
     469         413 : get_Selmer(GEN bnf, GEN cycgen, long rc)
     470             : {
     471         413 :   GEN U = bnf_build_units(bnf), tu = gel(U,1), fu = vecslice(U, 2, lg(U)-1);
     472         413 :   return shallowconcat(shallowconcat(fu,mkvec(tu)), vecslice(cycgen,1,rc));
     473             : }
     474             : 
     475             : GEN
     476       35105 : lift_if_rational(GEN x)
     477             : {
     478             :   long lx, i;
     479             :   GEN y;
     480             : 
     481       35105 :   switch(typ(x))
     482             :   {
     483        4816 :     default: break;
     484             : 
     485             :     case t_POLMOD:
     486       20657 :       y = gel(x,2);
     487       20657 :       if (typ(y) == t_POL)
     488             :       {
     489        6664 :         long d = degpol(y);
     490        6664 :         if (d > 0) return x;
     491        1148 :         return (d < 0)? gen_0: gel(y,2);
     492             :       }
     493       13993 :       return y;
     494             : 
     495        4011 :     case t_POL: lx = lg(x);
     496        4011 :       for (i=2; i<lx; i++) gel(x,i) = lift_if_rational(gel(x,i));
     497        4011 :       break;
     498        5621 :     case t_VEC: case t_COL: case t_MAT: lx = lg(x);
     499        5621 :       for (i=1; i<lx; i++) gel(x,i) = lift_if_rational(gel(x,i));
     500             :   }
     501       14448 :   return x;
     502             : }
     503             : 
     504             : /* A column vector representing a subgroup of prime index */
     505             : static GEN
     506           0 : grptocol(GEN H)
     507             : {
     508           0 :   long i, j, l = lg(H);
     509           0 :   GEN col = cgetg(l, t_VECSMALL);
     510           0 :   for (i = 1; i < l; i++)
     511             :   {
     512           0 :     ulong ell = itou( gcoeff(H,i,i) );
     513           0 :     if (ell == 1) col[i] = 0; else { col[i] = ell-1; break; }
     514             :   }
     515           0 :   for (j=i; ++j < l; ) col[j] = itou( gcoeff(H,i,j) );
     516           0 :   return col;
     517             : }
     518             : 
     519             : /* Reorganize kernel basis so that the tests of ok_congruence can be ok
     520             :  * for y[ncyc]=1 and y[1..ncyc]=1 */
     521             : static GEN
     522           0 : fix_kernel(GEN K, GEN M, GEN vecMsup, long lW, long ell)
     523             : {
     524           0 :   pari_sp av = avma;
     525           0 :   long i, j, idx, ffree, dK = lg(K)-1;
     526           0 :   GEN Ki, Kidx = cgetg(dK+1, t_VECSMALL);
     527             : 
     528             :   /* First step: Gauss elimination on vectors lW...lg(M) */
     529           0 :   for (idx = lg(K), i=lg(M); --i >= lW; )
     530             :   {
     531           0 :     for (j=dK; j > 0; j--) if (coeff(K, i, j)) break;
     532           0 :     if (!j)
     533             :     { /* Do our best to ensure that K[dK,i] != 0 */
     534           0 :       if (coeff(K, i, dK)) continue;
     535           0 :       for (j = idx; j < dK; j++)
     536           0 :         if (coeff(K, i, j) && coeff(K, Kidx[j], dK) != ell - 1)
     537           0 :           Flv_add_inplace(gel(K,dK), gel(K,j), ell);
     538             :     }
     539           0 :     if (j != --idx) swap(gel(K, j), gel(K, idx));
     540           0 :     Kidx[idx] = i;
     541           0 :     if (coeff(K,i,idx) != 1)
     542           0 :       Flv_Fl_div_inplace(gel(K,idx), coeff(K,i,idx), ell);
     543           0 :     Ki = gel(K,idx);
     544           0 :     if (coeff(K,i,dK) != 1)
     545             :     {
     546           0 :       ulong t = Fl_sub(coeff(K,i,dK), 1, ell);
     547           0 :       Flv_sub_inplace(gel(K,dK), Flv_Fl_mul(Ki, t, ell), ell);
     548             :     }
     549           0 :     for (j = dK; --j > 0; )
     550             :     {
     551           0 :       if (j == idx) continue;
     552           0 :       if (coeff(K,i,j))
     553           0 :         Flv_sub_inplace(gel(K,j), Flv_Fl_mul(Ki, coeff(K,i,j), ell), ell);
     554             :     }
     555             :   }
     556             :   /* ffree = first vector that is not "free" for the scalar products */
     557           0 :   ffree = idx;
     558             :   /* Second step: for each hyperplane equation in vecMsup, do the same
     559             :    * thing as before. */
     560           0 :   for (i=1; i < lg(vecMsup); i++)
     561             :   {
     562           0 :     GEN Msup = gel(vecMsup,i);
     563             :     ulong dotprod;
     564           0 :     if (lgcols(Msup) != 2) continue;
     565           0 :     Msup = zm_row(Msup, 1);
     566           0 :     for (j=ffree; --j > 0; )
     567             :     {
     568           0 :       dotprod = Flv_dotproduct(Msup, gel(K,j), ell);
     569           0 :       if (dotprod)
     570             :       {
     571           0 :         if (j != --ffree) swap(gel(K, j), gel(K, ffree));
     572           0 :         if (dotprod != 1) Flv_Fl_div_inplace(gel(K, ffree), dotprod, ell);
     573           0 :         break;
     574             :       }
     575             :     }
     576           0 :     if (!j)
     577             :     { /* Do our best to ensure that vecMsup.K[dK] != 0 */
     578           0 :       if (Flv_dotproduct(Msup, gel(K,dK), ell) == 0)
     579             :       {
     580           0 :         for (j = ffree-1; j <= dK; j++)
     581           0 :           if (Flv_dotproduct(Msup, gel(K,j), ell)
     582           0 :               && coeff(K,Kidx[j],dK) != ell-1)
     583           0 :             Flv_add_inplace(gel(K,dK), gel(K,j), ell);
     584             :       }
     585           0 :       continue;
     586             :     }
     587           0 :     Ki = gel(K,ffree);
     588           0 :     dotprod = Flv_dotproduct(Msup, gel(K,dK), ell);
     589           0 :     if (dotprod != 1)
     590             :     {
     591           0 :       ulong t = Fl_sub(dotprod,1,ell);
     592           0 :       Flv_sub_inplace(gel(K,dK), Flv_Fl_mul(Ki,t,ell), ell);
     593             :     }
     594           0 :     for (j = dK; --j > 0; )
     595             :     {
     596           0 :       if (j == ffree) continue;
     597           0 :       dotprod = Flv_dotproduct(Msup, gel(K,j), ell);
     598           0 :       if (dotprod) Flv_sub_inplace(gel(K,j), Flv_Fl_mul(Ki,dotprod,ell), ell);
     599             :     }
     600             :   }
     601           0 :   if (ell == 2)
     602             :   {
     603           0 :     for (i = ffree, j = ffree-1; i <= dK && j; i++, j--)
     604           0 :     { swap(gel(K,i), gel(K,j)); }
     605             :   }
     606             :   /* Try to ensure that y = vec_ei(n, i) gives a good candidate */
     607           0 :   for (i = 1; i < dK; i++) Flv_add_inplace(gel(K,i), gel(K,dK), ell);
     608           0 :   return gerepilecopy(av, K);
     609             : }
     610             : 
     611             : static GEN
     612           0 : Flm_init(long m, long n)
     613             : {
     614           0 :   GEN M = cgetg(n+1, t_MAT);
     615           0 :   long i; for (i = 1; i <= n; i++) gel(M,i) = cgetg(m+1, t_VECSMALL);
     616           0 :   return M;
     617             : }
     618             : static void
     619           0 : Flv_fill(GEN v, GEN y)
     620             : {
     621           0 :   long i, l = lg(y);
     622           0 :   for (i = 1; i < l; i++) v[i] = y[i];
     623           0 : }
     624             : 
     625             : static GEN
     626         616 : get_badbnf(GEN bnf)
     627             : {
     628             :   long i, l;
     629         616 :   GEN bad = gen_1, gen = bnf_get_gen(bnf);
     630         616 :   l = lg(gen);
     631        1064 :   for (i = 1; i < l; i++)
     632             :   {
     633         448 :     GEN g = gel(gen,i);
     634         448 :     bad = lcmii(bad, gcoeff(g,1,1));
     635             :   }
     636         616 :   return bad;
     637             : }
     638             : /* Let K base field, L/K described by bnr (conductor f) + H. Return a list of
     639             :  * primes coprime to f*ell of degree 1 in K whose images in Cl_f(K) generate H:
     640             :  * thus they all split in Lz/Kz; t in Kz is such that
     641             :  * t^(1/p) generates Lz => t is an ell-th power in k(pr) for all such primes.
     642             :  * Restrict to primes not dividing
     643             :  * - the index fz of the polynomial defining Kz, or
     644             :  * - the modulus, or
     645             :  * - ell, or
     646             :  * - a generator in bnf.gen or bnfz.gen */
     647             : static GEN
     648         399 : get_prlist(GEN bnr, GEN H, ulong ell, GEN bnfz)
     649             : {
     650         399 :   pari_sp av0 = avma;
     651             :   forprime_t T;
     652             :   ulong p;
     653             :   GEN L, nf, cyc, bad, cond, condZ, Hsofar;
     654         399 :   L = cgetg(1, t_VEC);
     655         399 :   cyc = bnr_get_cyc(bnr);
     656         399 :   nf = bnr_get_nf(bnr);
     657             : 
     658         399 :   cond = gel(bnr_get_mod(bnr), 1);
     659         399 :   condZ = gcoeff(cond,1,1);
     660         399 :   bad = get_badbnf(bnr_get_bnf(bnr));
     661         399 :   if (bnfz)
     662             :   {
     663         217 :     GEN badz = lcmii(get_badbnf(bnfz), nf_get_index(bnf_get_nf(bnfz)));
     664         217 :     bad = mulii(bad,badz);
     665             :   }
     666         399 :   bad = lcmii(muliu(condZ, ell), bad);
     667             :   /* restrict to primes not dividing bad */
     668             : 
     669         399 :   u_forprime_init(&T, 2, ULONG_MAX);
     670         399 :   Hsofar = cgetg(1, t_MAT);
     671        7266 :   while ((p = u_forprime_next(&T)))
     672             :   {
     673             :     GEN LP;
     674             :     long i, l;
     675        6867 :     if (p == ell || !umodiu(bad, p)) continue;
     676        5775 :     LP = idealprimedec_limit_f(nf, utoipos(p), 1);
     677        5775 :     l = lg(LP);
     678        9387 :     for (i = 1; i < l; i++)
     679             :     {
     680        4011 :       pari_sp av = avma;
     681        4011 :       GEN M, P = gel(LP,i), v = bnrisprincipal(bnr, P, 0);
     682        4011 :       if (!hnf_invimage(H, v)) { avma = av; continue; }
     683        1218 :       M = shallowconcat(Hsofar, v);
     684        1218 :       M = ZM_hnfmodid(M, cyc);
     685        1218 :       if (ZM_equal(M, Hsofar)) continue;
     686         833 :       L = shallowconcat(L, mkvec(P));
     687         833 :       Hsofar = M;
     688             :       /* the primes in L generate H */
     689         833 :       if (ZM_equal(M, H)) return gerepilecopy(av0, L);
     690             :     }
     691             :   }
     692           0 :   pari_err_BUG("rnfkummer [get_prlist]");
     693           0 :   return NULL;
     694             : }
     695             : /*Lprz list of prime ideals in Kz that must split completely in Lz/Kz, vecWA
     696             :  * generators for the S-units used to build the Kummer generators. Return
     697             :  * matsmall M such that \prod WA[j]^x[j] ell-th power mod pr[i] iff
     698             :  * \sum M[i,j] x[j] = 0 (mod ell) */
     699             : static GEN
     700         399 : subgroup_info(GEN bnfz, GEN Lprz, long ell, GEN vecWA)
     701             : {
     702         399 :   GEN nfz = bnf_get_nf(bnfz), M, gell = utoipos(ell), Lell = mkvec(gell);
     703         399 :   long i, j, l = lg(vecWA), lz = lg(Lprz);
     704         399 :   M = cgetg(l, t_MAT);
     705         399 :   for (j=1; j<l; j++) gel(M,j) = cgetg(lz, t_VECSMALL);
     706        1232 :   for (i=1; i < lz; i++)
     707             :   {
     708         833 :     GEN pr = gel(Lprz,i), EX = subiu(pr_norm(pr), 1);
     709         833 :     GEN N, g,T,p, prM = idealhnf(nfz, pr);
     710         833 :     GEN modpr = zk_to_Fq_init(nfz, &pr,&T,&p);
     711         833 :     long v = Z_lvalrem(divis(EX,ell), ell, &N) + 1; /* Norm(pr)-1 = N * ell^v */
     712         833 :     GEN ellv = powuu(ell, v);
     713         833 :     g = gener_Fq_local(T,p, Lell);
     714         833 :     g = Fq_pow(g,N, T,p); /* order ell^v */
     715        4186 :     for (j=1; j < l; j++)
     716             :     {
     717        3353 :       GEN logc, c = gel(vecWA,j);
     718        3353 :       if (typ(c) == t_MAT) /* famat */
     719        1071 :         c = famat_makecoprime(nfz, gel(c,1), gel(c,2), pr, prM, EX);
     720        3353 :       c = nf_to_Fq(nfz, c, modpr);
     721        3353 :       c = Fq_pow(c, N, T,p);
     722        3353 :       logc = Fq_log(c, g, ellv, T,p);
     723        3353 :       ucoeff(M, i,j) = umodiu(logc, ell);
     724             :     }
     725             :   }
     726         399 :   return M;
     727             : }
     728             : 
     729             : /* if all!=0, give all equations of degree 'all'. Assume bnr modulus is the
     730             :  * conductor */
     731             : static GEN
     732         182 : rnfkummersimple(GEN bnr, GEN subgroup, GEN gell, long all)
     733             : {
     734             :   long ell, i, j, degK, dK;
     735             :   long lSml2, lSl2, lSp, rc, lW;
     736             :   long prec;
     737         182 :   long rk=0, ncyc=0;
     738         182 :   GEN mat=NULL, matgrp=NULL, xell, be1 = NULL;
     739         182 :   long firstpass = all<0;
     740             : 
     741             :   GEN bnf,nf,bid,ideal,arch,cycgen;
     742             :   GEN cyc;
     743             :   GEN Sp,listprSp,matP;
     744         182 :   GEN res=NULL,u,M,K,y,vecMsup,vecW,vecWB,vecBp,msign;
     745             :   primlist L;
     746             : 
     747         182 :   bnf = bnr_get_bnf(bnr); (void)bnf_build_units(bnf);
     748         182 :   nf  = bnf_get_nf(bnf);
     749         182 :   degK = nf_get_degree(nf);
     750             : 
     751         182 :   bid = bnr_get_bid(bnr);
     752         182 :   ideal= bid_get_ideal(bid);
     753         182 :   arch = bid_get_arch(bid); /* this is the conductor */
     754         182 :   ell = itos(gell);
     755         182 :   i = build_list_Hecke(&L, nf, bid_get_fact2(bid), ideal, gell, NULL);
     756         182 :   if (i) return no_sol(all,i);
     757             : 
     758         182 :   lSml2 = lg(L.Sml2);
     759         182 :   Sp = shallowconcat(L.Sm, L.Sml1); lSp = lg(Sp);
     760         182 :   listprSp = shallowconcat(L.Sml2, L.Sl); lSl2 = lg(listprSp);
     761             : 
     762         182 :   cycgen = bnf_build_cycgen(bnf);
     763         182 :   cyc = bnf_get_cyc(bnf); rc = prank(cyc, ell);
     764             : 
     765         182 :   vecW = get_Selmer(bnf, cycgen, rc);
     766         182 :   u = get_u(ZV_to_Flv(cyc,ell), rc, ell);
     767             : 
     768         182 :   vecBp = cgetg(lSp, t_VEC);
     769         182 :   matP  = cgetg(lSp, t_MAT);
     770         357 :   for (j = 1; j < lSp; j++)
     771             :   {
     772         175 :     GEN L = isprincipalell(bnf,gel(Sp,j), cycgen,u,ell,rc);
     773         175 :     gel( matP,j) = gel(L,1);
     774         175 :     gel(vecBp,j) = gel(L,2);
     775             :   }
     776         182 :   vecWB = shallowconcat(vecW, vecBp);
     777             : 
     778         182 :   prec = DEFAULTPREC +
     779         182 :       nbits2extraprec(((degK-1) * (gexpo(vecWB) + gexpo(nf_get_M(nf)))));
     780         182 :   if (nf_get_prec(nf) < prec) nf = nfnewprec_shallow(nf, prec);
     781         182 :   msign = nfsign(nf, vecWB);
     782         182 :   arch = ZV_to_zv(arch);
     783             : 
     784         182 :   vecMsup = cgetg(lSml2,t_VEC);
     785         182 :   M = NULL;
     786         343 :   for (i = 1; i < lSl2; i++)
     787             :   {
     788         161 :     GEN pr = gel(listprSp,i);
     789         161 :     long e = pr_get_e(pr), z = ell * (e / (ell-1));
     790             : 
     791         161 :     if (i < lSml2)
     792             :     {
     793          91 :       z += 1 - L.ESml2[i];
     794          91 :       gel(vecMsup,i) = logall(nf, vecWB, 0,0, ell, pr,z+1);
     795             :     }
     796         161 :     M = vconcat(M, logall(nf, vecWB, 0,0, ell, pr,z));
     797             :   }
     798         182 :   lW = lg(vecW);
     799         182 :   M = vconcat(M, shallowconcat(zero_Flm(rc,lW-1), matP));
     800         182 :   if (!all)
     801             :   { /* primes landing in subgroup must be totally split */
     802         182 :     GEN Lpr = get_prlist(bnr, subgroup, ell, NULL);
     803         182 :     GEN M2 = subgroup_info(bnf, Lpr, ell, vecWB);
     804         182 :     M = vconcat(M, M2);
     805             :   }
     806         182 :   K = Flm_ker(M, ell);
     807         182 :   dK = lg(K)-1;
     808             : 
     809         182 :   if (all < 0)
     810           0 :     K = fix_kernel(K, M, vecMsup, lW, ell);
     811             : 
     812         182 :   y = cgetg(dK+1,t_VECSMALL);
     813         182 :   if (all) res = cgetg(1,t_VEC); /* in case all = 1 */
     814         182 :   if (all < 0)
     815             :   {
     816           0 :     ncyc = dK;
     817           0 :     mat = Flm_init(dK, ncyc);
     818           0 :     if (all == -1) matgrp = Flm_init(lg(bnr_get_cyc(bnr)), ncyc+1);
     819           0 :     rk = 0;
     820             :   }
     821         182 :   xell = pol_xn(ell, 0);
     822             :   do {
     823         182 :     dK = lg(K)-1;
     824         371 :     while (dK)
     825             :     {
     826         189 :       for (i=1; i<dK; i++) y[i] = 0;
     827         189 :       y[i] = 1; /* y = [0,...,0,1,0,...,0], 1 at dK'th position */
     828             :       do
     829             :       {
     830         287 :         pari_sp av = avma;
     831         287 :         GEN be, P=NULL, X;
     832         287 :         if (all < 0)
     833             :         {
     834           0 :           Flv_fill(gel(mat, rk+1), y);
     835           0 :           setlg(mat, rk+2);
     836           0 :           if (Flm_rank(mat, ell) <= rk) continue;
     837             :         }
     838         287 : FOUND:  X = Flm_Flc_mul(K, y, ell);
     839         287 :         if (ok_congruence(X, ell, lW, vecMsup) && ok_sign(X, msign, arch))
     840             :         {/* be satisfies all congruences, x^ell - be is irreducible, signature
     841             :           * and relative discriminant are correct */
     842         182 :           if (all < 0) rk++;
     843         182 :           be = compute_beta(X, vecWB, gell, bnf);
     844         182 :           be = nf_to_scalar_or_alg(nf, be);
     845         182 :           if (typ(be) == t_POL) be = mkpolmod(be, nf_get_pol(nf));
     846         182 :           if (all == -1)
     847             :           {
     848           0 :             pari_sp av2 = avma;
     849           0 :             GEN Kgrp, colgrp = grptocol(rnfnormgroup(bnr, gsub(xell, be)));
     850           0 :             if (ell != 2)
     851             :             {
     852           0 :               if (rk == 1) be1 = be;
     853             :               else
     854             :               { /* Compute the pesky scalar */
     855           0 :                 GEN K2, C = cgetg(4, t_MAT);
     856           0 :                 gel(C,1) = gel(matgrp,1);
     857           0 :                 gel(C,2) = colgrp;
     858           0 :                 gel(C,3) = grptocol(rnfnormgroup(bnr, gsub(xell, gmul(be1,be))));
     859           0 :                 K2 = Flm_ker(C, ell);
     860           0 :                 if (lg(K2) != 2) pari_err_BUG("linear algebra");
     861           0 :                 K2 = gel(K2,1);
     862           0 :                 if (K2[1] != K2[2])
     863           0 :                   Flv_Fl_mul_inplace(colgrp, Fl_div(K2[2],K2[1],ell), ell);
     864             :               }
     865             :             }
     866           0 :             Flv_fill(gel(matgrp,rk), colgrp);
     867           0 :             setlg(matgrp, rk+1);
     868           0 :             Kgrp = Flm_ker(matgrp, ell);
     869           0 :             if (lg(Kgrp) == 2)
     870             :             {
     871           0 :               setlg(gel(Kgrp,1), rk+1);
     872           0 :               y = Flm_Flc_mul(mat, gel(Kgrp,1), ell);
     873           0 :               all = 0; goto FOUND;
     874             :             }
     875           0 :             avma = av2;
     876             :           }
     877             :           else
     878             :           {
     879         182 :             P = gsub(xell, be);
     880         182 :             if (all)
     881           0 :               res = shallowconcat(res, gerepileupto(av, P));
     882             :             else
     883             :             {
     884         182 :               if (ZM_equal(rnfnormgroup(bnr,P),subgroup)) return P; /*DONE*/
     885           0 :               avma = av; continue;
     886             :             }
     887             :           }
     888           0 :           if (all < 0 && rk == ncyc) return res;
     889           0 :           if (firstpass) break;
     890             :         }
     891         105 :         else avma = av;
     892         105 :       } while (increment(y, dK, ell));
     893           7 :       y[dK--] = 0;
     894             :     }
     895           0 :   } while (firstpass--);
     896           0 :   return all? res: gen_0;
     897             : }
     898             : 
     899             : /* alg. 5.3.11 (return only discrete log mod ell) */
     900             : static GEN
     901         826 : isvirtualunit(GEN bnf, GEN v, GEN cycgen, GEN cyc, GEN gell, long rc)
     902             : {
     903         826 :   GEN L, b, eps, y, q, nf = bnf_get_nf(bnf);
     904         826 :   GEN w = idealsqrtn(nf, v, gell, 1);
     905         826 :   long i, l = lg(cycgen);
     906             : 
     907         826 :   L = bnfisprincipal0(bnf, w, nf_GENMAT|nf_FORCE);
     908         826 :   q = gel(L,1);
     909         826 :   if (ZV_equal0(q)) { eps = v; y = q; }
     910             :   else
     911             :   {
     912         140 :     y = cgetg(l,t_COL);
     913         140 :     for (i=1; i<l; i++) gel(y,i) = diviiexact(mulii(gell,gel(q,i)), gel(cyc,i));
     914         140 :     eps = famat_mulpow_shallow(famat_factorback(cycgen,y), gel(L,2), gell);
     915         140 :     eps = famat_mul_shallow(famat_inv(eps), v);
     916             :   }
     917         826 :   setlg(y, rc+1);
     918         826 :   b = bnfisunit(bnf,eps);
     919         826 :   if (lg(b) == 1) pari_err_BUG("isvirtualunit");
     920         826 :   return shallowconcat(lift_shallow(b), y);
     921             : }
     922             : 
     923             : /* J a vector of elements in nfz = relative extension of nf by polrel,
     924             :  * return the Steinitz element attached to the module generated by J */
     925             : static GEN
     926         651 : Stelt(GEN nf, GEN J, GEN polrel)
     927             : {
     928         651 :   long i, l = lg(J), vx = varn(polrel);
     929         651 :   GEN A = cgetg(l, t_VEC), I = cgetg(l, t_VEC);
     930        4487 :   for (i = 1; i < l; i++)
     931             :   {
     932        3836 :     GEN v = gel(J,i);
     933        3836 :     if (typ(v) == t_POL) { v = RgX_rem(v, polrel); setvarn(v,vx); }
     934        3836 :     gel(A,i) = v;
     935        3836 :     gel(I,i) = gen_1;
     936             :   }
     937         651 :   A = RgV_to_RgM(A, degpol(polrel));
     938         651 :   return idealprod(nf, gel(nfhnf(nf, mkvec2(A,I)),2));
     939             : }
     940             : 
     941             : static GEN
     942         126 : polrelKzK(toK_s *T, GEN x)
     943             : {
     944         126 :   GEN P = roots_to_pol(powtau(x, T->m, T->tau), 0);
     945         126 :   long i, l = lg(P);
     946         126 :   for (i=2; i<l; i++) gel(P,i) = downtoK(T, gel(P,i));
     947         126 :   return P;
     948             : }
     949             : 
     950             : /* N: Cl_m(Kz) --> Cl_m(K), lift subgroup from bnr to bnrz using Algo 4.1.11 */
     951             : static GEN
     952         126 : invimsubgroup(GEN bnrz, GEN bnr, GEN subgroup, toK_s *T)
     953             : {
     954             :   long l, j;
     955             :   GEN P, cyc, gen, U, polrel, StZk;
     956         126 :   GEN nf = bnr_get_nf(bnr), nfz = bnr_get_nf(bnrz);
     957         126 :   GEN polz = nf_get_pol(nfz), zkzD = nf_get_zkprimpart(nfz);
     958             : 
     959         126 :   polrel = polrelKzK(T, pol_x(varn(polz)));
     960         126 :   StZk = Stelt(nf, zkzD, polrel);
     961         126 :   cyc = bnr_get_cyc(bnrz); l = lg(cyc);
     962         126 :   gen = bnr_get_gen(bnrz);
     963         126 :   P = cgetg(l,t_MAT);
     964         651 :   for (j=1; j<l; j++)
     965             :   {
     966         525 :     GEN g, id = idealhnf_shallow(nfz, gel(gen,j));
     967         525 :     g = Stelt(nf, RgV_RgM_mul(zkzD, id), polrel);
     968         525 :     g = idealdiv(nf, g, StZk); /* N_{Kz/K}(gen[j]) */
     969         525 :     gel(P,j) = isprincipalray(bnr, g);
     970             :   }
     971         126 :   (void)ZM_hnfall_i(shallowconcat(P, subgroup), &U, 1);
     972         126 :   setlg(U, l); for (j=1; j<l; j++) setlg(U[j], l);
     973         126 :   return ZM_hnfmodid(U, cyc);
     974             : }
     975             : 
     976             : static GEN
     977         252 : pol_from_Newton(GEN S)
     978             : {
     979         252 :   long i, k, l = lg(S);
     980         252 :   GEN C = cgetg(l+1, t_VEC), c = C + 1;
     981         252 :   gel(c,0) = gen_1;
     982         252 :   gel(c,1) = gel(S,1); /* gen_0 in our case */
     983         882 :   for (k = 2; k < l; k++)
     984             :   {
     985         630 :     GEN s = gel(S,k);
     986         630 :     for (i = 2; i < k-1; i++) s = gadd(s, gmul(gel(S,i), gel(c,k-i)));
     987         630 :     gel(c,k) = gdivgs(s, -k);
     988             :   }
     989         252 :   return gtopoly(C, 0);
     990             : }
     991             : 
     992             : /* - mu_b = sum_{0 <= i < m} floor(r_b r_{d-1-i} / ell) tau^i */
     993             : static GEN
     994         602 : get_mmu(long b, GEN r, long ell)
     995             : {
     996         602 :   long i, m = lg(r)-1;
     997         602 :   GEN M = cgetg(m+1, t_VEC);
     998         602 :   for (i = 0; i < m; i++) gel(M,i+1) = stoi((r[b + 1] * r[m - i]) / ell);
     999         602 :   return M;
    1000             : }
    1001             : 
    1002             : /* coeffs(x, a..b) in variable v >= varn(x) */
    1003             : static GEN
    1004        5964 : split_pol(GEN x, long v, long a, long b)
    1005             : {
    1006        5964 :   long i, l = degpol(x);
    1007        5964 :   GEN y = x + a, z;
    1008             : 
    1009        5964 :   if (l < b) b = l;
    1010        5964 :   if (a > b || varn(x) != v) return pol_0(v);
    1011        5320 :   l = b-a + 3;
    1012        5320 :   z = cgetg(l, t_POL); z[1] = x[1];
    1013        5320 :   for (i = 2; i < l; i++) gel(z,i) = gel(y,i);
    1014        5320 :   return normalizepol_lg(z, l);
    1015             : }
    1016             : 
    1017             : /* return (den_a * z) mod (v^ell - num_a/den_a), assuming deg(z) < 2*ell
    1018             :  * allow either num/den to be NULL (= 1) */
    1019             : static GEN
    1020        2982 : mod_Xell_a(GEN z, long v, long ell, GEN num_a, GEN den_a)
    1021             : {
    1022        2982 :   GEN z1 = split_pol(z, v, ell, degpol(z));
    1023        2982 :   GEN z0 = split_pol(z, v, 0,   ell-1); /* z = v^ell z1 + z0*/
    1024        2982 :   if (den_a) z0 = gmul(den_a, z0);
    1025        2982 :   if (num_a) z1 = gmul(num_a, z1);
    1026        2982 :   return gadd(z0, z1);
    1027             : }
    1028             : static GEN
    1029         854 : to_alg(GEN nfz, GEN c, long v)
    1030             : {
    1031             :   GEN z, D;
    1032         854 :   if (typ(c) != t_COL) return c;
    1033         602 :   z = gmul(nf_get_zkprimpart(nfz), c);
    1034         602 :   if (typ(z) == t_POL) setvarn(z, v);
    1035         602 :   D = nf_get_zkden(nfz);
    1036         602 :   if (!equali1(D)) z = RgX_Rg_div(z, D);
    1037         602 :   return z;
    1038             : }
    1039             : 
    1040             : /* th. 5.3.5. and prop. 5.3.9. */
    1041             : static GEN
    1042         252 : compute_polrel(GEN nfz, toK_s *T, GEN be, long g, long ell)
    1043             : {
    1044         252 :   long i, k, m = T->m, vT = fetch_var(), vz = fetch_var();
    1045             :   GEN r, powtaubet, S, p1, root, num_t, den_t, nfzpol, powtau_prim_invbe;
    1046             :   GEN prim_Rk, C_Rk, prim_root, C_root, prim_invbe, C_invbe;
    1047             :   pari_timer ti;
    1048             : 
    1049         252 :   r = cgetg(m+1,t_VECSMALL); /* r[i+1] = g^i mod ell */
    1050         252 :   r[1] = 1;
    1051         252 :   for (i=2; i<=m; i++) r[i] = (r[i-1] * g) % ell;
    1052         252 :   powtaubet = powtau(be, m, T->tau);
    1053         252 :   if (DEBUGLEVEL>1) { err_printf("Computing Newton sums: "); timer_start(&ti); }
    1054         252 :   prim_invbe = Q_primitive_part(nfinv(nfz, be), &C_invbe);
    1055         252 :   powtau_prim_invbe = powtau(prim_invbe, m, T->tau);
    1056             : 
    1057         252 :   root = cgetg(ell + 2, t_POL);
    1058         252 :   root[1] = evalsigne(1) | evalvarn(0);
    1059         252 :   for (i = 0; i < ell; i++) gel(root,2+i) = gen_0;
    1060         854 :   for (i = 0; i < m; i++)
    1061             :   { /* compute (1/be) ^ (-mu) instead of be^mu [mu << 0].
    1062             :      * 1/be = C_invbe * prim_invbe */
    1063         602 :     GEN mmu = get_mmu(i, r, ell);
    1064             :     /* p1 = prim_invbe ^ -mu */
    1065         602 :     p1 = to_alg(nfz, nffactorback(nfz, powtau_prim_invbe, mmu), vz);
    1066         602 :     if (C_invbe) p1 = gmul(p1, powgi(C_invbe, RgV_sumpart(mmu, m)));
    1067             :     /* root += zeta_ell^{r_i} T^{r_i} be^mu_i */
    1068         602 :     gel(root, 2 + r[i+1]) = monomial(p1, r[i+1], vT);
    1069             :   }
    1070             :   /* Other roots are as above with z_ell --> z_ell^j.
    1071             :    * Treat all contents (C_*) and principal parts (prim_*) separately */
    1072         252 :   prim_Rk = prim_root = Q_primitive_part(root, &C_root);
    1073         252 :   C_Rk = C_root;
    1074             : 
    1075         252 :   r = vecsmall_reverse(r); /* theta^ell = be^( sum tau^a r_{d-1-a} ) */
    1076             :   /* Compute modulo X^ell - 1, T^ell - t, nfzpol(vz) */
    1077         252 :   p1 = to_alg(nfz, nffactorback(nfz, powtaubet, r), vz);
    1078         252 :   num_t = Q_remove_denom(p1, &den_t);
    1079             : 
    1080         252 :   nfzpol = leafcopy(nf_get_pol(nfz));
    1081         252 :   setvarn(nfzpol, vz);
    1082         252 :   S = cgetg(ell+1, t_VEC); /* Newton sums */
    1083         252 :   gel(S,1) = gen_0;
    1084         882 :   for (k = 2; k <= ell; k++)
    1085             :   { /* compute the k-th Newton sum */
    1086         630 :     pari_sp av = avma;
    1087         630 :     GEN z, D, Rk = gmul(prim_Rk, prim_root);
    1088         630 :     C_Rk = mul_content(C_Rk, C_root);
    1089         630 :     Rk = mod_Xell_a(Rk, 0, ell, NULL, NULL); /* mod X^ell - 1 */
    1090        3010 :     for (i = 2; i < lg(Rk); i++)
    1091             :     {
    1092        2380 :       if (typ(gel(Rk,i)) != t_POL) continue;
    1093        2352 :       z = mod_Xell_a(gel(Rk,i), vT, ell, num_t,den_t); /* mod T^ell - t */
    1094        2352 :       gel(Rk,i) = RgXQX_red(z, nfzpol); /* mod nfz.pol */
    1095             :     }
    1096         630 :     if (den_t) C_Rk = mul_content(C_Rk, ginv(den_t));
    1097         630 :     prim_Rk = Q_primitive_part(Rk, &D);
    1098         630 :     C_Rk = mul_content(C_Rk, D); /* root^k = prim_Rk * C_Rk */
    1099             : 
    1100             :     /* Newton sum is ell * constant coeff (in X), which has degree 0 in T */
    1101         630 :     z = polcoeff_i(prim_Rk, 0, 0);
    1102         630 :     z = polcoeff_i(z      , 0,vT);
    1103         630 :     z = downtoK(T, gmulgs(z, ell));
    1104         630 :     if (C_Rk) z = gmul(z, C_Rk);
    1105         630 :     gerepileall(av, C_Rk? 3: 2, &z, &prim_Rk, &C_Rk);
    1106         630 :     if (DEBUGLEVEL>1) { err_printf("%ld(%ld) ", k, timer_delay(&ti)); err_flush(); }
    1107         630 :     gel(S,k) = z;
    1108             :   }
    1109         252 :   if (DEBUGLEVEL>1) err_printf("\n");
    1110         252 :   (void)delete_var();
    1111         252 :   (void)delete_var(); return pol_from_Newton(S);
    1112             : }
    1113             : 
    1114             : /* lift elt t in nf to nfz, algebraic form */
    1115             : static GEN
    1116         343 : lifttoKz(GEN nf, GEN t, compo_s *C)
    1117             : {
    1118         343 :   GEN x = nf_to_scalar_or_alg(nf, t);
    1119         343 :   if (typ(x) != t_POL) return x;
    1120         343 :   return RgX_RgXQ_eval(x, C->p, C->R);
    1121             : }
    1122             : /* lift ideal id in nf to nfz */
    1123             : static GEN
    1124         231 : ideallifttoKz(GEN nfz, GEN nf, GEN id, compo_s *C)
    1125             : {
    1126         231 :   GEN I = idealtwoelt(nf,id);
    1127         231 :   GEN x = nf_to_scalar_or_alg(nf, gel(I,2));
    1128         231 :   if (typ(x) != t_POL) return gel(I,1);
    1129         147 :   gel(I,2) = algtobasis(nfz, RgX_RgXQ_eval(x, C->p, C->R));
    1130         147 :   return idealhnf_two(nfz,I);
    1131             : }
    1132             : /* lift ideal pr in nf to ONE prime in nfz (the others are conjugate under tau
    1133             :  * and bring no further information on e_1 W). Assume pr coprime to
    1134             :  * index of both nf and nfz, and unramified in Kz/K (minor simplification) */
    1135             : static GEN
    1136         378 : prlifttoKz(GEN nfz, GEN nf, GEN pr, compo_s *C)
    1137             : {
    1138         378 :   GEN F, p = pr_get_p(pr), t = pr_get_gen(pr), T = nf_get_pol(nfz);
    1139         378 :   if (nf_get_degree(nf) != 1)
    1140             :   { /* restrict to primes above pr */
    1141         343 :     t = Q_primpart( lifttoKz(nf,t,C) );
    1142         343 :     T = FpX_gcd(FpX_red(T,p), FpX_red(t,p), p);
    1143         343 :     T = FpX_normalize(T, p);
    1144             :   }
    1145         378 :   F = FpX_factor(T, p);
    1146         378 :   return idealprimedec_kummer(nfz,gcoeff(F,1,1), pr_get_e(pr), p);
    1147             : }
    1148             : static GEN
    1149         217 : get_przlist(GEN L, GEN nfz, GEN nf, compo_s *C)
    1150             : {
    1151             :   long i, l;
    1152         217 :   GEN M = cgetg_copy(L, &l);
    1153         217 :   for (i = 1; i < l; i++) gel(M,i) = prlifttoKz(nfz, nf, gel(L,i), C);
    1154         217 :   return M;
    1155             : }
    1156             : 
    1157             : static void
    1158         231 : compositum_red(compo_s *C, GEN P, GEN Q)
    1159             : {
    1160         231 :   GEN p, q, a, z = gel(compositum2(P, Q),1);
    1161         231 :   a = gel(z,1);
    1162         231 :   p = gel(gel(z,2), 2);
    1163         231 :   q = gel(gel(z,3), 2);
    1164         231 :   C->k = itos( gel(z,4) );
    1165             :   /* reduce R. FIXME: should be polredbest(a, 1), but breaks rnfkummer bench */
    1166         231 :   z = polredabs0(a, nf_ORIG|nf_PARTIALFACT);
    1167         231 :   C->R = gel(z,1);
    1168         231 :   a = gel(gel(z,2), 2);
    1169         231 :   C->p = RgX_RgXQ_eval(p, a, C->R);
    1170         231 :   C->q = RgX_RgXQ_eval(q, a, C->R);
    1171         231 :   C->rev = QXQ_reverse(a, C->R);
    1172         231 :   if (DEBUGLEVEL>1) err_printf("polred(compositum) = %Ps\n",C->R);
    1173         231 : }
    1174             : 
    1175             : /* replace P->C^(-deg P) P(xC) for the largest integer C such that coefficients
    1176             :  * remain algebraic integers. Lift *rational* coefficients */
    1177             : static void
    1178         252 : nfX_Z_normalize(GEN nf, GEN P)
    1179             : {
    1180             :   long i, l;
    1181         252 :   GEN C, Cj, PZ = cgetg_copy(P, &l);
    1182         252 :   PZ[1] = P[1];
    1183        1386 :   for (i = 2; i < l; i++) /* minor variation on RgX_to_nfX (create PZ) */
    1184             :   {
    1185        1134 :     GEN z = nf_to_scalar_or_basis(nf, gel(P,i));
    1186        1134 :     if (typ(z) == t_INT)
    1187         735 :       gel(PZ,i) = gel(P,i) = z;
    1188             :     else
    1189         399 :       gel(PZ,i) = ZV_content(z);
    1190             :   }
    1191         252 :   (void)ZX_Z_normalize(PZ, &C);
    1192             : 
    1193         504 :   if (C == gen_1) return;
    1194          77 :   Cj = C;
    1195         322 :   for (i = l-2; i > 1; i--)
    1196             :   {
    1197         245 :     if (i != l-2) Cj = mulii(Cj, C);
    1198         245 :     gel(P,i) = gdiv(gel(P,i), Cj);
    1199             :   }
    1200             : }
    1201             : 
    1202             : static GEN
    1203         231 : _rnfkummer_step4(GEN bnfz, GEN gen, GEN cycgen, GEN u, ulong ell, long rc,
    1204             :                  long d, long m, long g, tau_s *tau)
    1205             : {
    1206             :   long i, j;
    1207             :   GEN vecB, vecC, Tc, Q;
    1208         231 :   vecB=cgetg(rc+1,t_VEC);
    1209         231 :   Tc=cgetg(rc+1,t_MAT);
    1210         371 :   for (j=1; j<=rc; j++)
    1211             :   {
    1212         140 :     GEN p1 = tauofideal(gel(gen,j), tau);
    1213         140 :     p1 = isprincipalell(bnfz, p1, cycgen,u,ell,rc);
    1214         140 :     gel(Tc,j)  = gel(p1,1);
    1215         140 :     gel(vecB,j)= gel(p1,2);
    1216             :   }
    1217             : 
    1218         231 :   vecC = cgetg(rc+1,t_VEC);
    1219         231 :   if (rc)
    1220             :   {
    1221             :     GEN p1, p2;
    1222         126 :     for (j=1; j<=rc; j++) gel(vecC,j) = cgetg(1, t_MAT);
    1223         126 :     p1 = Flm_powers(Tc, m-2, ell);
    1224         126 :     p2 = vecB;
    1225         294 :     for (j=1; j<=m-1; j++)
    1226             :     {
    1227         168 :       GEN z = Flm_Fl_mul(gel(p1,m-j), Fl_mul(j,d,ell), ell);
    1228         168 :       p2 = tauofvec(p2, tau);
    1229         364 :       for (i=1; i<=rc; i++)
    1230         392 :         gel(vecC,i) = famat_mul_shallow(gel(vecC,i),
    1231         196 :                                         famat_factorbacks(p2, gel(z,i)));
    1232             :     }
    1233         126 :     for (i=1; i<=rc; i++) gel(vecC,i) = famat_reduce(gel(vecC,i));
    1234             :   }
    1235         231 :   Q = Flm_ker(Flm_Fl_add(Flm_transpose(Tc), Fl_neg(g, ell), ell), ell);
    1236         231 :   return mkvec2(vecC, Q);
    1237             : }
    1238             : 
    1239             : static GEN
    1240         231 : _rnfkummer_step5(GEN bnfz, GEN vselmer, GEN cycgen, GEN gell, long rc,
    1241             :                  long rv, long g, tau_s *tau)
    1242             : {
    1243             :   GEN Tv, P, vecW;
    1244             :   long j, lW;
    1245         231 :   ulong ell = itou(gell);
    1246         231 :   GEN cyc = bnf_get_cyc(bnfz);
    1247         231 :   Tv = cgetg(rv+1,t_MAT);
    1248        1057 :   for (j=1; j<=rv; j++)
    1249             :   {
    1250         826 :     GEN p1 = tauofelt(gel(vselmer,j), tau);
    1251         826 :     if (typ(p1) == t_MAT) /* famat */
    1252         140 :       p1 = nffactorback(bnfz, gel(p1,1), FpC_red(gel(p1,2),gell));
    1253         826 :     gel(Tv,j) = ZV_to_Flv(isvirtualunit(bnfz, p1, cycgen,cyc,gell,rc), ell);
    1254             :   }
    1255         231 :   P = Flm_ker(Flm_Fl_add(Tv, Fl_neg(g, ell), ell), ell);
    1256         231 :   lW = lg(P);
    1257         231 :   vecW = cgetg(lW,t_VEC);
    1258         231 :   for (j=1; j<lW; j++) gel(vecW,j) = famat_factorbacks(vselmer, gel(P,j));
    1259         231 :   return vecW;
    1260             : }
    1261             : 
    1262             : static GEN
    1263         231 : _rnfkummer_step18(toK_s *T, GEN bnr, GEN subgroup, GEN bnfz, GEN M,
    1264             :      GEN vecWB, GEN vecMsup, ulong g, GEN gell, long lW, long all)
    1265             : {
    1266         231 :   GEN K, y, res = NULL, mat = NULL;
    1267         231 :   long i, dK, ncyc = 0;
    1268         231 :   ulong ell = itou(gell);
    1269         231 :   GEN bnf = bnr_get_bnf(bnr);
    1270         231 :   GEN nf  = bnf_get_nf(bnf);
    1271         231 :   GEN polnf = nf_get_pol(nf);
    1272         231 :   GEN nfz = bnf_get_nf(bnfz);
    1273         231 :   long firstpass = all<0;
    1274         231 :   long rk=0;
    1275         231 :   K = Flm_ker(M, ell);
    1276         231 :   if (all < 0)
    1277           0 :     K = fix_kernel(K, M, vecMsup, lW, ell);
    1278         231 :   if (DEBUGLEVEL>2) err_printf("Step 18\n");
    1279         231 :   dK = lg(K)-1;
    1280         231 :   y = cgetg(dK+1,t_VECSMALL);
    1281         231 :   if (all) res = cgetg(1, t_VEC);
    1282         231 :   if (all < 0) { ncyc = dK; rk = 0; mat = zero_Flm(lg(M)-1, ncyc); }
    1283             : 
    1284             :   do {
    1285         231 :     dK = lg(K)-1;
    1286         483 :     while (dK)
    1287             :     {
    1288         238 :       for (i=1; i<dK; i++) y[i] = 0;
    1289         238 :       y[i] = 1; /* y = [0,...,0,1,0,...,0], 1 at dK'th position */
    1290             :       do
    1291             :       { /* cf. algo 5.3.18 */
    1292         252 :         GEN H, be, P, X = Flm_Flc_mul(K, y, ell);
    1293         252 :         if (ok_congruence(X, ell, lW, vecMsup))
    1294             :         {
    1295         252 :           pari_sp av = avma;
    1296         252 :           if (all < 0)
    1297             :           {
    1298           0 :             gel(mat, rk+1) = X;
    1299           0 :             if (Flm_rank(mat,ell) <= rk) continue;
    1300           0 :             rk++;
    1301             :           }
    1302         252 :           be = compute_beta(X, vecWB, gell, bnfz);
    1303         252 :           P = compute_polrel(nfz, T, be, g, ell);
    1304         252 :           nfX_Z_normalize(nf, P);
    1305         252 :           if (DEBUGLEVEL>1) err_printf("polrel(beta) = %Ps\n", P);
    1306         252 :           if (!all) {
    1307         217 :             H = rnfnormgroup(bnr, P);
    1308         217 :             if (ZM_equal(subgroup, H)) return P; /* DONE */
    1309           0 :             avma = av; continue;
    1310             :           } else {
    1311          35 :             GEN P0 = Q_primpart(lift_shallow(P));
    1312          35 :             GEN g = nfgcd(P0, RgX_deriv(P0), polnf, nf_get_index(nf));
    1313          35 :             if (degpol(g)) continue;
    1314          35 :             H = rnfnormgroup(bnr, P);
    1315          35 :             if (!ZM_equal(subgroup,H) && !bnrisconductor(bnr,H)) continue;
    1316             :           }
    1317          35 :           P = gerepilecopy(av, P);
    1318          35 :           res = shallowconcat(res, P);
    1319          35 :           if (all < 0 && rk == ncyc) return res;
    1320          35 :           if (firstpass) break;
    1321             :         }
    1322          35 :       } while (increment(y, dK, ell));
    1323          21 :       y[dK--] = 0;
    1324             :     }
    1325          14 :   } while (firstpass--);
    1326          14 :   return res;
    1327             : }
    1328             : 
    1329             : static GEN
    1330         420 : _rnfkummer(GEN bnr, GEN subgroup, long all, long prec)
    1331             : {
    1332             :   long i, j, m, d, dc, rc, ru, rv, mginv, degK, degKz, vnf;
    1333             :   long lSp, lSml2, lSl2, lW;
    1334             :   ulong g, ell;
    1335             :   GEN polnf,bnf,nf,bnfz,nfz,bid,ideal,cycgen,gell,p1,vselmer;
    1336             :   GEN cyc, gen, step4;
    1337             :   GEN Q,idealz,gothf;
    1338         420 :   GEN res=NULL,u,M,vecMsup,vecW,vecWA,vecWB,vecC,vecAp,vecBp;
    1339             :   GEN matP, Sp, listprSp;
    1340             :   primlist L;
    1341             :   toK_s T;
    1342             :   tau_s tau;
    1343             :   compo_s COMPO;
    1344             :   pari_timer t;
    1345             : 
    1346         420 :   if (DEBUGLEVEL) timer_start(&t);
    1347         420 :   checkbnr(bnr);
    1348         420 :   bnf = bnr_get_bnf(bnr);
    1349         420 :   nf  = bnf_get_nf(bnf);
    1350         420 :   polnf = nf_get_pol(nf); vnf = varn(polnf);
    1351         420 :   if (!vnf) pari_err_PRIORITY("rnfkummer", polnf, "=", 0);
    1352             :   /* step 7 */
    1353         420 :   p1 = bnrconductor_i(bnr, subgroup, 2);
    1354         420 :   if (DEBUGLEVEL) timer_printf(&t, "[rnfkummer] conductor");
    1355         420 :   bnr      = gel(p1,2);
    1356         420 :   subgroup = gel(p1,3);
    1357         420 :   gell = get_gell(bnr,subgroup,all);
    1358         420 :   ell = itou(gell);
    1359         420 :   if (ell == 1) return pol_x(0);
    1360         420 :   if (!uisprime(ell)) pari_err_IMPL("kummer for composite relative degree");
    1361         420 :   if (all && all != -1 && umodiu(bnr_get_no(bnr), ell))
    1362           7 :     return cgetg(1, t_VEC);
    1363         413 :   if (bnf_get_tuN(bnf) % ell == 0)
    1364         182 :     return rnfkummersimple(bnr, subgroup, gell, all);
    1365             : 
    1366         231 :   if (all == -1) all = 0;
    1367         231 :   bid = bnr_get_bid(bnr);
    1368         231 :   ideal = bid_get_ideal(bid);
    1369             :   /* step 1 of alg 5.3.5. */
    1370         231 :   if (DEBUGLEVEL>2) err_printf("Step 1\n");
    1371         231 :   compositum_red(&COMPO, polnf, polcyclo(ell,vnf));
    1372             :   /* step 2 */
    1373         231 :   if (DEBUGLEVEL>2) err_printf("Step 2\n");
    1374         231 :   if (DEBUGLEVEL) timer_printf(&t, "[rnfkummer] compositum");
    1375         231 :   degK  = degpol(polnf);
    1376         231 :   degKz = degpol(COMPO.R);
    1377         231 :   m = degKz / degK;
    1378         231 :   d = (ell-1) / m;
    1379         231 :   g = Fl_powu(pgener_Fl(ell), d, ell);
    1380         231 :   if (Fl_powu(g, m, ell*ell) == 1) g += ell;
    1381             :   /* ord(g) = m in all (Z/ell^k)^* */
    1382             :   /* step 3 */
    1383         231 :   if (DEBUGLEVEL>2) err_printf("Step 3\n");
    1384             :   /* could factor disc(R) using th. 2.1.6. */
    1385         231 :   bnfz = Buchall(COMPO.R, nf_FORCE, maxss(prec,BIGDEFAULTPREC));
    1386         231 :   if (DEBUGLEVEL) timer_printf(&t, "[rnfkummer] bnfinit(Kz)");
    1387         231 :   cycgen = bnf_build_cycgen(bnfz);
    1388         231 :   nfz = bnf_get_nf(bnfz);
    1389         231 :   cyc = bnf_get_cyc(bnfz); rc = prank(cyc,ell);
    1390         231 :   gen = bnf_get_gen(bnfz);
    1391         231 :   u = get_u(ZV_to_Flv(cyc, ell), rc, ell);
    1392             : 
    1393         231 :   vselmer = get_Selmer(bnfz, cycgen, rc);
    1394         231 :   if (DEBUGLEVEL) timer_printf(&t, "[rnfkummer] Selmer group");
    1395         231 :   ru = (degKz>>1)-1;
    1396         231 :   rv = rc+ru+1;
    1397         231 :   get_tau(&tau, nfz, &COMPO, g);
    1398             : 
    1399             :   /* step 4 */
    1400         231 :   if (DEBUGLEVEL>2) err_printf("Step 4\n");
    1401         231 :   step4 = _rnfkummer_step4(bnfz, gen, cycgen, u, ell, rc, d, m, g, &tau);
    1402         231 :   vecC = gel(step4,1);
    1403         231 :   Q    = gel(step4,2);
    1404             :   /* step 5 */
    1405         231 :   if (DEBUGLEVEL>2) err_printf("Step 5\n");
    1406         231 :   vecW = _rnfkummer_step5(bnfz, vselmer, cycgen, gell, rc, rv, g, &tau);
    1407         231 :   lW = lg(vecW);
    1408             :   /* step 8 */
    1409         231 :   if (DEBUGLEVEL>2) err_printf("Step 8\n");
    1410         231 :   p1 = RgXQ_matrix_pow(COMPO.p, degKz, degK, COMPO.R);
    1411         231 :   T.invexpoteta1 = RgM_inv(p1); /* left inverse */
    1412         231 :   T.polnf = polnf;
    1413         231 :   T.tau = &tau;
    1414         231 :   T.m = m;
    1415         231 :   T.powg = Fl_powers(g, m, ell);
    1416             : 
    1417         231 :   idealz = ideallifttoKz(nfz, nf, ideal, &COMPO);
    1418         231 :   if (umodiu(gcoeff(ideal,1,1), ell)) gothf = idealz;
    1419             :   else
    1420             :   { /* ell | N(ideal) */
    1421         126 :     GEN bnrz = Buchray(bnfz, idealz, nf_INIT|nf_GEN);
    1422         126 :     GEN subgroupz = invimsubgroup(bnrz, bnr, subgroup, &T);
    1423         126 :     gothf = bnrconductor_i(bnrz,subgroupz,0);
    1424             :   }
    1425             :   /* step 9, 10, 11 */
    1426         231 :   if (DEBUGLEVEL>2) err_printf("Step 9, 10 and 11\n");
    1427         231 :   i = build_list_Hecke(&L, nfz, NULL, gothf, gell, &tau);
    1428         231 :   if (i) return no_sol(all,i);
    1429             : 
    1430         231 :   lSml2 = lg(L.Sml2);
    1431         231 :   Sp = shallowconcat(L.Sm, L.Sml1); lSp = lg(Sp);
    1432         231 :   listprSp = shallowconcat(L.Sml2, L.Sl); lSl2 = lg(listprSp);
    1433             : 
    1434             :   /* step 12 */
    1435         231 :   if (DEBUGLEVEL>2) err_printf("Step 12\n");
    1436         231 :   vecAp = cgetg(lSp, t_VEC);
    1437         231 :   vecBp = cgetg(lSp, t_VEC);
    1438         231 :   matP  = cgetg(lSp, t_MAT);
    1439             : 
    1440         385 :   for (j = 1; j < lSp; j++)
    1441             :   {
    1442             :     GEN e, a;
    1443         154 :     p1 = isprincipalell(bnfz, gel(Sp,j), cycgen,u,ell,rc);
    1444         154 :     e = gel(p1,1); gel(matP,j) = gel(p1, 1);
    1445         154 :     a = gel(p1,2);
    1446         154 :     gel(vecBp,j) = famat_mul_shallow(famat_factorbacks(vecC, zv_neg(e)), a);
    1447             :   }
    1448         231 :   vecAp = lambdaofvec(vecBp, &T);
    1449             :   /* step 13 */
    1450         231 :   if (DEBUGLEVEL>2) err_printf("Step 13\n");
    1451         231 :   vecWA = shallowconcat(vecW, vecAp);
    1452         231 :   vecWB = shallowconcat(vecW, vecBp);
    1453             : 
    1454             :   /* step 14, 15, and 17 */
    1455         231 :   if (DEBUGLEVEL>2) err_printf("Step 14, 15 and 17\n");
    1456         231 :   mginv = Fl_div(m, g, ell);
    1457         231 :   vecMsup = cgetg(lSml2,t_VEC);
    1458         231 :   M = NULL;
    1459         518 :   for (i = 1; i < lSl2; i++)
    1460             :   {
    1461         287 :     GEN pr = gel(listprSp,i);
    1462         287 :     long e = pr_get_e(pr), z = ell * (e / (ell-1));
    1463             : 
    1464         287 :     if (i < lSml2)
    1465             :     {
    1466         133 :       z += 1 - L.ESml2[i];
    1467         133 :       gel(vecMsup,i) = logall(nfz, vecWA,lW,mginv,ell, pr,z+1);
    1468             :     }
    1469         287 :     M = vconcat(M, logall(nfz, vecWA,lW,mginv,ell, pr,z));
    1470             :   }
    1471         231 :   dc = lg(Q)-1;
    1472         231 :   if (dc)
    1473             :   {
    1474         105 :     GEN QtP = Flm_mul(Flm_transpose(Q), matP, ell);
    1475         105 :     M = vconcat(M, shallowconcat(zero_Flm(dc,lW-1), QtP));
    1476             :   }
    1477         231 :   if (!M) M = zero_Flm(1, lSp-1 + lW-1);
    1478             : 
    1479         231 :   if (!all)
    1480             :   { /* primes landing in subgroup must be totally split */
    1481         217 :     GEN lambdaWB = shallowconcat(lambdaofvec(vecW, &T), vecAp);/*vecWB^lambda*/
    1482         217 :     GEN Lpr = get_prlist(bnr, subgroup, ell, bnfz);
    1483         217 :     GEN Lprz= get_przlist(Lpr, nfz, nf, &COMPO);
    1484         217 :     GEN M2 = subgroup_info(bnfz, Lprz, ell, lambdaWB);
    1485         217 :     M = vconcat(M, M2);
    1486             :   }
    1487         231 :   if (DEBUGLEVEL>2) err_printf("Step 16\n");
    1488             :   /* step 16 && 18 & ff */
    1489         231 :   res = _rnfkummer_step18(&T,bnr,subgroup,bnfz, M, vecWB, vecMsup, g, gell, lW, all);
    1490         231 :   return res? res: gen_0;
    1491             : }
    1492             : 
    1493             : GEN
    1494         420 : rnfkummer(GEN bnr, GEN subgroup, long all, long prec)
    1495             : {
    1496         420 :   pari_sp av = avma;
    1497         420 :   return gerepilecopy(av, _rnfkummer(bnr, subgroup, all, prec));
    1498             : }

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