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 - ellsea.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.12.0 lcov report (development 23724-b3bdf5af3) Lines: 1110 1158 95.9 %
Date: 2019-03-25 05:45:24 Functions: 81 83 97.6 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2008  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             : /* This file is a C version by Bill Allombert of the 'ellsea' GP package
      15             :  * whose copyright statement is as follows:
      16             : Authors:
      17             :   Christophe Doche   <cdoche@math.u-bordeaux.fr>
      18             :   Sylvain Duquesne <duquesne@math.u-bordeaux.fr>
      19             : 
      20             : Universite Bordeaux I, Laboratoire A2X
      21             : For the AREHCC project, see http://www.arehcc.com/
      22             : 
      23             : Contributors:
      24             :   Karim Belabas (code cleanup and package release, faster polynomial arithmetic)
      25             : 
      26             : 'ellsea' is free software; you can redistribute it and/or modify it under the
      27             : terms of the GNU General Public License as published by the Free Software
      28             : Foundation. It is distributed in the hope that it will be useful, but WITHOUT
      29             : ANY WARRANTY WHATSOEVER. */
      30             : 
      31             : /* Extension to non prime finite fields by Bill Allombert 2012 */
      32             : 
      33             : #include "pari.h"
      34             : #include "paripriv.h"
      35             : 
      36             : static THREAD GEN modular_eqn;
      37             : 
      38             : void
      39      123180 : pari_set_seadata(GEN mod)  { modular_eqn = mod; }
      40             : GEN
      41      121765 : pari_get_seadata(void)  { return modular_eqn; }
      42             : 
      43             : static char *
      44          98 : seadata_filename(ulong ell)
      45          98 : { return stack_sprintf("%s/seadata/sea%ld", pari_datadir, ell); }
      46             : 
      47             : static GEN
      48          98 : get_seadata(ulong ell)
      49             : {
      50          98 :   pari_sp av = avma;
      51             :   GEN eqn;
      52          98 :   char *s = seadata_filename(ell);
      53          98 :   pariFILE *F = pari_fopengz(s);
      54          98 :   if (!F) return NULL;
      55          42 :   if (ell) /* large single polynomial */
      56           7 :     eqn = gp_read_stream(F->file);
      57             :   else
      58             :   { /* table of polynomials of small level */
      59          35 :     eqn = gp_readvec_stream(F->file);
      60          35 :     modular_eqn = eqn = gclone(eqn);
      61          35 :     set_avma(av);
      62             :   }
      63          42 :   pari_fclose(F);
      64          42 :   return eqn;
      65             : }
      66             : 
      67             : /*Builds the modular equation corresponding to the vector list. Shallow */
      68             : static GEN
      69        9688 : list_to_pol(GEN list, long vx, long vy)
      70             : {
      71        9688 :   long i, l = lg(list);
      72        9688 :   GEN P = cgetg(l, t_VEC);
      73      195629 :   for (i = 1; i < l; i++)
      74             :   {
      75      185941 :     GEN L = gel(list,i);
      76      185941 :     if (typ(L) == t_VEC) L = RgV_to_RgX_reverse(L, vy);
      77      185941 :     gel(P, i) = L;
      78             :   }
      79        9688 :   return RgV_to_RgX_reverse(P, vx);
      80             : }
      81             : 
      82             : struct meqn {
      83             :   char type;
      84             :   GEN eq, eval;
      85             :   long vx,vy;
      86             : };
      87             : 
      88             : static GEN
      89        9744 : seadata_cache(ulong ell)
      90             : {
      91        9744 :   long n = uprimepi(ell)-1;
      92             :   GEN C;
      93        9744 :   if (!modular_eqn && !get_seadata(0))
      94          56 :     C = NULL;
      95        9688 :   else if (n && n < lg(modular_eqn))
      96        9681 :     C = gel(modular_eqn, n);
      97             :   else
      98           7 :     C = get_seadata(ell);
      99        9744 :   return C;
     100             : }
     101             : /* C = [prime level, type "A" or "C", pol. coeffs] */
     102             : static void
     103        9688 : seadata_parse(struct meqn *M, GEN C, long vx, long vy)
     104             : {
     105        9688 :   M->type = *GSTR(gel(C,2));
     106        9688 :   M->eq = list_to_pol(gel(C,3), vx, vy);
     107        9688 : }
     108             : static void
     109        9723 : get_modular_eqn(struct meqn *M, ulong ell, long vx, long vy)
     110             : {
     111        9723 :   GEN C = seadata_cache(ell);
     112        9723 :   M->vx = vx;
     113        9723 :   M->vy = vy;
     114        9723 :   M->eval = gen_0;
     115        9723 :   if (C) seadata_parse(M, C, vx, vy);
     116             :   else
     117             :   {
     118          56 :     M->type = 'J'; /* j^(1/3) for ell != 3, j for 3 */
     119          56 :     M->eq = polmodular_ZXX(ell, ell==3? 0: 5, vx, vy);
     120             :   }
     121        9723 : }
     122             : 
     123             : GEN
     124          35 : ellmodulareqn(long ell, long vx, long vy)
     125             : {
     126          35 :   pari_sp av = avma;
     127             :   struct meqn meqn;
     128             :   GEN C;
     129          35 :   if (vx < 0) vx = 0;
     130          35 :   if (vy < 0) vy = 1;
     131          35 :   if (varncmp(vx,vy) >= 0)
     132           7 :     pari_err_PRIORITY("ellmodulareqn", pol_x(vx), ">=", vy);
     133          28 :   if (ell < 2 || !uisprime(ell))
     134           7 :     pari_err_PRIME("ellmodulareqn (level)", stoi(ell));
     135          21 :   C = seadata_cache(ell);
     136          21 :   if (!C) pari_err_FILE("seadata file", seadata_filename(ell));
     137          21 :   seadata_parse(&meqn, C, vx, vy);
     138          21 :   return gerepilecopy(av, mkvec2(meqn.eq, meqn.type=='A'? gen_1: gen_0));
     139             : }
     140             : 
     141             : /***********************************************************************/
     142             : /**                                                                   **/
     143             : /**                      n-division polynomial                        **/
     144             : /**                                                                   **/
     145             : /***********************************************************************/
     146             : 
     147             : static GEN divpol(GEN t, GEN r2, long n, void *E, const struct bb_algebra *ff);
     148             : 
     149             : /* f_n^2, return ff->(zero|one) or a clone */
     150             : static GEN
     151      156702 : divpol_f2(GEN t, GEN r2, long n, void *E, const struct bb_algebra *ff)
     152             : {
     153      156702 :   if (n==0) return ff->zero(E);
     154      156702 :   if (n<=2) return ff->one(E);
     155      128534 :   if (gmael(t,2,n)) return gmael(t,2,n);
     156       51149 :   gmael(t,2,n) = gclone(ff->sqr(E,divpol(t,r2,n,E,ff)));
     157       51149 :   return gmael(t,2,n);
     158             : }
     159             : 
     160             : /* f_n f_{n-2}, return ff->zero or a clone */
     161             : static GEN
     162      102186 : divpol_ff(GEN t, GEN r2, long n, void *E, const struct bb_algebra *ff)
     163             : {
     164      102186 :   if (n<=2) return ff->zero(E);
     165      102186 :   if (gmael(t,3,n)) return gmael(t,3,n);
     166       70021 :   if (n<=4) return divpol(t,r2,n,E,ff);
     167       29337 :   gmael(t,3,n) = gclone(ff->mul(E,divpol(t,r2,n,E,ff), divpol(t,r2,n-2,E,ff)));
     168       29337 :   return gmael(t,3,n);
     169             : }
     170             : 
     171             : /* f_n, return ff->zero or a clone */
     172             : static GEN
     173      214641 : divpol(GEN t, GEN r2, long n, void *E, const struct bb_algebra *ff)
     174             : {
     175      214641 :   long m = n/2;
     176      214641 :   pari_sp av = avma;
     177             :   GEN f;
     178      214641 :   if (n==0) return ff->zero(E);
     179      210959 :   if (gmael(t,1,n)) return gmael(t,1,n);
     180       58114 :   switch(n)
     181             :   {
     182             :   case 1:
     183             :   case 2:
     184        7021 :     f = ff->one(E);
     185        7021 :     break;
     186             :   default:
     187       51093 :     if (odd(n))
     188       30926 :       if (odd(m))
     189       25788 :         f = ff->sub(E, ff->mul(E, divpol_ff(t,r2,m+2,E,ff),
     190             :                                   divpol_f2(t,r2,m,E,ff)),
     191       12894 :                        ff->mul(E, r2,
     192       12894 :                                   ff->mul(E,divpol_ff(t,r2,m+1,E,ff),
     193             :                                             divpol_f2(t,r2,m+1,E,ff))));
     194             :       else
     195       54096 :         f = ff->sub(E, ff->mul(E, r2,
     196       18032 :                                   ff->mul(E, divpol_ff(t,r2,m+2,E,ff),
     197             :                                              divpol_f2(t,r2,m,E,ff))),
     198       18032 :                        ff->mul(E, divpol_ff(t,r2,m+1,E,ff),
     199             :                                   divpol_f2(t,r2,m+1,E,ff)));
     200             :     else
     201       40334 :       f = ff->sub(E, ff->mul(E, divpol_ff(t,r2,m+2,E,ff),
     202             :                                 divpol_f2(t,r2,m-1,E,ff)),
     203       20167 :                      ff->mul(E, divpol_ff(t,r2,m,E,ff),
     204             :                                 divpol_f2(t,r2,m+1,E,ff)));
     205             :   }
     206       58114 :   gmael(t,1,n) = f = gclone( ff->red(E, f) );
     207       58114 :   set_avma(av); return f;
     208             : }
     209             : 
     210             : static void
     211       15932 : divpol_free(GEN t)
     212             : {
     213       15932 :   long i, l = lg(gel(t,1));
     214      251062 :   for (i=1; i<l; i++)
     215             :   {
     216      235130 :     guncloneNULL(gmael(t,1,i));
     217      235130 :     guncloneNULL(gmael(t,2,i));
     218      235130 :     guncloneNULL(gmael(t,3,i));
     219             :   }
     220       15932 : }
     221             : 
     222             : static GEN
     223         438 : Flxq_elldivpol34(long n, GEN a4, GEN a6, GEN S, GEN T, ulong p)
     224             : {
     225             :   GEN res;
     226         438 :   long vs = T[1];
     227         438 :   switch(n)
     228             :   {
     229             :   case 3:
     230         219 :     res = mkpoln(5, Fl_to_Flx(3%p,vs), pol0_Flx(vs), Flx_mulu(a4, 6, p),
     231             :                     Flx_mulu(a6, 12, p), Flx_neg(Flxq_sqr(a4, T, p), p));
     232         219 :     break;
     233             :   case 4:
     234             :     {
     235         219 :       GEN a42 = Flxq_sqr(a4, T, p);
     236         438 :       res = mkpoln(7, pol1_Flx(vs), pol0_Flx(vs), Flx_mulu(a4, 5, p),
     237             :           Flx_mulu(a6, 20, p), Flx_mulu(a42,p-5, p),
     238             :           Flx_mulu(Flxq_mul(a4, a6, T, p), p-4, p),
     239         219 :           Flx_sub(Flx_mulu(Flxq_sqr(a6, T, p), p-8%p, p),
     240             :             Flxq_mul(a4, a42, T, p), p));
     241         219 :       res = FlxX_double(res, p);
     242             :     }
     243         219 :     break;
     244             :     default:
     245           0 :       pari_err_BUG("Flxq_elldivpol34");
     246             :       return NULL;/*LCOV_EXCL_LINE*/
     247             :   }
     248         438 :   setvarn(res, get_FlxqX_var(S));
     249         438 :   return FlxqX_rem(res, S, T, p);
     250             : }
     251             : 
     252             : static GEN
     253       31426 : Fq_elldivpol34(long n, GEN a4, GEN a6, GEN S, GEN T, GEN p)
     254             : {
     255             :   GEN res;
     256       31426 :   switch(n)
     257             :   {
     258             :   case 3:
     259       15713 :     res = mkpoln(5, utoi(3), gen_0, Fq_mulu(a4, 6, T, p),
     260             :         Fq_mulu(a6, 12, T, p), Fq_neg(Fq_sqr(a4, T, p), T, p));
     261       15713 :     break;
     262             :   case 4:
     263             :     {
     264       15713 :       GEN a42 = Fq_sqr(a4, T, p);
     265       15713 :       res = mkpoln(7, gen_1, gen_0, Fq_mulu(a4, 5, T, p),
     266             :           Fq_mulu(a6, 20, T, p), Fq_Fp_mul(a42,stoi(-5), T, p),
     267             :           Fq_Fp_mul(Fq_mul(a4, a6, T, p), stoi(-4), T, p),
     268             :           Fq_sub(Fq_Fp_mul(Fq_sqr(a6, T, p), stoi(-8), T, p),
     269             :             Fq_mul(a4,a42, T, p), T, p));
     270       15713 :       res = FqX_mulu(res, 2, T, p);
     271             :     }
     272       15713 :     break;
     273             :     default:
     274           0 :       pari_err_BUG("Fq_elldivpol34");
     275             :       return NULL;/*LCOV_EXCL_LINE*/
     276             :   }
     277       31426 :   if (S)
     278             :   {
     279       31342 :     setvarn(res, get_FpXQX_var(S));
     280       31342 :     res = FqX_rem(res, S, T, p);
     281             :   }
     282       31426 :   return res;
     283             : }
     284             : 
     285             : static GEN
     286       22816 : rhs(GEN a4, GEN a6, long v)
     287             : {
     288       22816 :   GEN RHS = mkpoln(4, gen_1, gen_0, a4, a6);
     289       22816 :   setvarn(RHS, v); return RHS;
     290             : }
     291             : 
     292             : static GEN
     293         438 : Flxq_rhs(GEN a4, GEN a6, long v, long vs)
     294             : {
     295         438 :   GEN RHS = mkpoln(4, pol1_Flx(vs),  pol0_Flx(vs), a4, a6);
     296         438 :   setvarn(RHS, v); return RHS;
     297             : }
     298             : 
     299             : struct divpolmod_red
     300             : {
     301             :   const struct bb_algebra *ff;
     302             :   void *E;
     303             :   GEN t, r2;
     304             : };
     305             : 
     306             : static void
     307       15932 : divpolmod_init(struct divpolmod_red *d, GEN D3, GEN D4, GEN RHS, long n,
     308             :                void *E, const struct bb_algebra *ff)
     309             : {
     310       15932 :   long k = n+2;
     311       15932 :   d->ff = ff; d->E = E;
     312       15932 :   d->t  = mkvec3(const_vec(k, NULL),const_vec(k, NULL),const_vec(k, NULL));
     313       15932 :   if (k>=3) gmael(d->t,1,3) = gclone(D3);
     314       15932 :   if (k>=4) gmael(d->t,1,4) = gclone(D4);
     315       15932 :   d->r2 = ff->sqr(E, RHS);
     316       15932 : }
     317             : 
     318             : static void
     319       15713 : Fq_elldivpolmod_init(struct divpolmod_red *d, GEN a4, GEN a6, long n, GEN h, GEN T, GEN p)
     320             : {
     321             :   void *E;
     322             :   const struct bb_algebra *ff;
     323       15713 :   GEN RHS, D3 = NULL, D4 = NULL;
     324       15713 :   long v = h ? get_FpXQX_var(h): 0;
     325       15713 :   D3 = n>=0 ? Fq_elldivpol34(3, a4, a6, h, T, p): NULL;
     326       15713 :   D4 = n>=1 ? Fq_elldivpol34(4, a4, a6, h, T, p): NULL;
     327       15713 :   RHS = rhs(a4, a6, v);
     328       15713 :   RHS = h ? FqX_rem(RHS, h, T, p): RHS;
     329       15713 :   RHS = FqX_mulu(RHS, 4, T, p);
     330       15755 :   ff = h ? T ? get_FpXQXQ_algebra(&E, h, T, p): get_FpXQ_algebra(&E, h, p):
     331          42 :            T ? get_FpXQX_algebra(&E, T, p, v): get_FpX_algebra(&E, p, v);
     332       15713 :   divpolmod_init(d, D3, D4, RHS, n, E, ff);
     333       15713 : }
     334             : 
     335             : static void
     336         219 : Flxq_elldivpolmod_init(struct divpolmod_red *d, GEN a4, GEN a6, long n, GEN h, GEN T, ulong p)
     337             : {
     338             :   void *E;
     339             :   const struct bb_algebra *ff;
     340         219 :   GEN RHS, D3 = NULL, D4 = NULL;
     341         219 :   long v = get_FlxqX_var(h), vT = get_Flx_var(T);
     342         219 :   D3 = n>=0 ? Flxq_elldivpol34(3, a4, a6, h, T, p): NULL;
     343         219 :   D4 = n>=1 ? Flxq_elldivpol34(4, a4, a6, h, T, p): NULL;
     344         219 :   RHS = FlxX_Fl_mul(FlxqX_rem(Flxq_rhs(a4, a6, v, vT), h, T, p), 4, p);
     345         219 :   ff = get_FlxqXQ_algebra(&E, h, T, p);
     346         219 :   divpolmod_init(d, D3, D4, RHS, n, E, ff);
     347         219 : }
     348             : 
     349             : /*Computes the n-division polynomial modulo the polynomial h \in Fq[x] */
     350             : GEN
     351        9618 : Fq_elldivpolmod(GEN a4, GEN a6, long n, GEN h, GEN T, GEN p)
     352             : {
     353             :   struct divpolmod_red d;
     354        9618 :   pari_sp ltop = avma;
     355             :   GEN res;
     356        9618 :   Fq_elldivpolmod_init(&d, a4, a6, n, h, T, p);
     357        9618 :   res = gcopy(divpol(d.t,d.r2,n,d.E,d.ff));
     358        9618 :   divpol_free(d.t);
     359        9618 :   return gerepileupto(ltop, res);
     360             : }
     361             : 
     362             : GEN
     363          42 : FpXQ_elldivpol(GEN a4, GEN a6, long n, GEN T, GEN p)
     364          42 : { return Fq_elldivpolmod(a4,a6,n,NULL,T,p); }
     365             : 
     366             : GEN
     367           0 : Fp_elldivpol(GEN a4, GEN a6, long n, GEN p)
     368           0 : { return Fq_elldivpolmod(a4,a6,n,NULL,NULL,p); }
     369             : 
     370             : static GEN
     371       23506 : Fq_ellyn(struct divpolmod_red *d, long k)
     372             : {
     373       23506 :   pari_sp av = avma;
     374       23506 :   void *E = d->E;
     375       23506 :   const struct bb_algebra *ff = d->ff;
     376       23506 :   if (k==1) return mkvec2(ff->one(E), ff->one(E));
     377             :   else
     378             :   {
     379       18172 :     GEN t = d->t, r2 = d->r2;
     380       18172 :     GEN pn2 = divpol(t,r2,k-2,E,ff);
     381       18172 :     GEN pp2 = divpol(t,r2,k+2,E,ff);
     382       18172 :     GEN pn12 = divpol_f2(t,r2,k-1,E,ff);
     383       18172 :     GEN pp12 = divpol_f2(t,r2,k+1,E,ff);
     384       18172 :     GEN on = ff->red(E,ff->sub(E, ff->mul(E,pp2,pn12), ff->mul(E,pn2,pp12)));
     385       18172 :     GEN f  = divpol(t,r2,k,E,ff);
     386       18172 :     GEN f2 = divpol_f2(t,r2,k,E,ff);
     387       18172 :     GEN f3 = ff->mul(E,f,f2);
     388       18172 :     if (!odd(k)) f3 = ff->mul(E,f3,r2);
     389       18172 :     return gerepilecopy(av,mkvec2(on, f3));
     390             :   }
     391             : }
     392             : 
     393             : static void
     394        6314 : Fq_elldivpolmod_close(struct divpolmod_red *d)
     395        6314 : { divpol_free(d->t); }
     396             : static GEN
     397       10255 : Fq_elldivpol2(GEN a4, GEN a6, GEN T, GEN p)
     398       10255 : { return mkpoln(4, utoi(4), gen_0, Fq_mulu(a4, 4, T, p), Fq_mulu(a6, 4, T, p)); }
     399             : 
     400             : static GEN
     401       10255 : Fq_elldivpol2d(GEN a4, GEN T, GEN p)
     402       10255 : { return mkpoln(3, utoi(6), gen_0, Fq_mulu(a4, 2, T, p)); }
     403             : 
     404             : static GEN
     405        1533 : FqX_numer_isog_abscissa(GEN h, GEN a4, GEN a6, GEN T, GEN p, long vx)
     406             : {
     407             :   GEN mp1, dh, ddh, t, u, t1, t2, t3, t4, f0;
     408        1533 :   long m = degpol(h);
     409        1533 :   mp1 = gel(h, m + 1); /* negative of first power sum */
     410        1533 :   dh = FqX_deriv(h, T, p);
     411        1533 :   ddh = FqX_deriv(dh, T, p);
     412        1533 :   t  = Fq_elldivpol2(a4, a6, T, p);
     413        1533 :   u  = Fq_elldivpol2d(a4, T, p);
     414        1533 :   t1 = FqX_sub(FqX_sqr(dh, T, p), FqX_mul(ddh, h, T, p), T, p);
     415        1533 :   t2 = FqX_mul(u, FqX_mul(h, dh, T, p), T, p);
     416        1533 :   t3 = FqX_mul(FqX_sqr(h, T, p),
     417             :                deg1pol_shallow(stoi(2*m), Fq_mulu(mp1, 2, T, p), vx), T, p);
     418        1533 :   f0 = FqX_add(FqX_sub(FqX_mul(t, t1, T, p), t2, T, p), t3, T, p);
     419        1533 :   t4 = FqX_mul(pol_x(vx),  FqX_sqr(h, T, p), T, p);
     420        1533 :   return FqX_add(t4, f0, T, p);
     421             : }
     422             : 
     423             : static GEN
     424        1092 : Zq_inv(GEN b, GEN T, GEN q, GEN p, long e)
     425             : {
     426        2135 :   return e==1 ? Fq_inv(b, T, p):
     427        1043 :          typ(b)==t_INT ? Fp_inv(b, q):  ZpXQ_inv(b, T, p, e);
     428             : }
     429             : 
     430             : static GEN
     431      248262 : Zq_div(GEN a, GEN b, GEN T, GEN q, GEN p, long e)
     432             : {
     433      248262 :   if (e==1) return Fq_div(a, b, T, q);
     434        1043 :   return Fq_mul(a, Zq_inv(b, T, q, p, e), T, q);
     435             : }
     436             : 
     437             : static GEN
     438           0 : Zq_sqrt(GEN b, GEN T, GEN q, GEN p, long e)
     439             : {
     440           0 :   return e==1 ? Fq_sqrt(b, T, q):
     441           0 :          typ(b)==t_INT ? Zp_sqrt(b, p, e):  ZpXQ_sqrt(b, T, p, e);
     442             : }
     443             : 
     444             : static GEN
     445       91119 : Zq_divexact(GEN a, GEN b)
     446       91119 : { return typ(a)==t_INT ? diviiexact(a, b): ZX_Z_divexact(a, b); }
     447             : 
     448             : static long
     449       91084 : Zq_pval(GEN a, GEN p)
     450       91084 : { return typ(a)==t_INT ? Z_pval(a, p): ZX_pval(a, p); }
     451             : 
     452             : static GEN
     453      147280 : Zq_Z_div_safe(GEN a, GEN b, GEN T, GEN q, GEN p, long e)
     454             : {
     455             :   long v;
     456      147280 :   if (e==1) return Fq_div(a, b, T, q);
     457         770 :   v = Z_pvalrem(b, p, &b);
     458         770 :   if (v>0)
     459             :   {
     460          35 :     long w = Z_pval(Q_content(a), p);
     461          35 :     if (v>w) pari_err_INV("Zq_div",b);
     462          35 :     a = Zq_divexact(a, powiu(p,v));
     463             :   }
     464         770 :   return Fq_Fp_mul(a, Fp_inv(b, q), T, q);
     465             : }
     466             : 
     467             : /*Gives the first precS terms of the Weierstrass series related to */
     468             : /*E: y^2 = x^3 + a4x + a6.  Assumes (precS-2)*(2precS+3) < ULONG_MAX, i.e.
     469             :  * precS < 46342 in 32-bit machines */
     470             : static GEN
     471       17444 : find_coeff(GEN a4, GEN a6, GEN T, GEN p, long precS, GEN pp, long e)
     472             : {
     473             :   GEN res, den;
     474             :   long k, h;
     475       17444 :   if (e > 1) { p = sqri(p); e *= 2; }
     476       17444 :   res = cgetg(precS+1, t_VEC);
     477       17444 :   den = cgetg(precS+1, t_VECSMALL);
     478       17444 :   if (precS == 0) return res;
     479       17444 :   gel(res, 1) = Fq_div(a4, stoi(-5), T, p);
     480       17444 :   den[1] = 0;
     481       17444 :   if (precS == 1) return res;
     482       17444 :   gel(res, 2) = Fq_div(a6, stoi(-7), T, p);
     483       17444 :   den[2] = 0;
     484      164724 :   for (k = 3; k <= precS; ++k)
     485             :   {
     486      147280 :     pari_sp btop = avma;
     487      147280 :     GEN a = gen_0, d;
     488      147280 :     long v=0;
     489      147280 :     if (e > 1)
     490        8358 :       for (h = 1; h <= k-2; h++)
     491        7588 :         v = maxss(v, den[h]+den[k-1-h]);
     492     1209516 :     for (h = 1; h <= k-2; h++)
     493             :     {
     494     1062236 :       GEN b = Fq_mul(gel(res, h), gel(res, k-1-h), T, p);
     495     1062236 :       if (v)
     496        1876 :         b = Fq_Fp_mul(b, powiu(pp, v-(den[h]+den[k-1-h])), T, p);
     497     1062236 :       a = Fq_add(a, b, T, p);
     498             :     }
     499      147280 :     v += Z_pvalrem(utoi((k-2) * (2*k + 3)), pp, &d);
     500      147280 :     a = Zq_div(gmulgs(a, 3), d, T, p, pp, e);
     501      147280 :     gel(res, k) = gerepileupto(btop, a);
     502      147280 :     den[k] = v;
     503             :   }
     504       17444 :   return mkvec2(res, den);
     505             : }
     506             : 
     507             : /****************************************************************************/
     508             : /*               SIMPLE ELLIPTIC CURVE OVER Fq                              */
     509             : /****************************************************************************/
     510             : 
     511             : static GEN
     512        2555 : Fq_ellj(GEN a4, GEN a6, GEN T, GEN p)
     513             : {
     514        2555 :   pari_sp ltop=avma;
     515        2555 :   GEN a43 = Fq_mulu(Fq_powu(a4, 3, T, p), 4, T, p);
     516        2555 :   GEN j   = Fq_div(Fq_mulu(a43, 1728, T, p),
     517             :                    Fq_add(a43, Fq_mulu(Fq_sqr(a6, T, p), 27, T, p), T, p), T, p);
     518        2555 :   return gerepileupto(ltop, j);
     519             : }
     520             : 
     521             : static GEN
     522        2534 : Zq_ellj(GEN a4, GEN a6, GEN T, GEN p, GEN pp, long e)
     523             : {
     524        2534 :   pari_sp ltop=avma;
     525        2534 :   GEN a43 = Fq_mulu(Fq_powu(a4, 3, T, p), 4, T, p);
     526        2534 :   GEN j   = Zq_div(Fq_mulu(a43, 1728, T, p),
     527             :                    Fq_add(a43, Fq_mulu(Fq_sqr(a6, T, p), 27, T, p), T, p), T, p, pp, e);
     528        2534 :   return gerepileupto(ltop, j);
     529             : }
     530             : /****************************************************************************/
     531             : /*                              EIGENVALUE                                  */
     532             : /****************************************************************************/
     533             : 
     534             : static GEN
     535          68 : Fq_to_Flx(GEN a4, GEN T, ulong p)
     536          68 : { return typ(a4)==t_INT ? Z_to_Flx(a4, p, get_Flx_var(T)): ZX_to_Flx(a4, p); }
     537             : 
     538             : static GEN
     539         219 : Flxq_find_eigen_Frobenius(GEN a4, GEN a6, GEN h, GEN T, ulong p)
     540             : {
     541         219 :   long v = get_FlxqX_var(h), vT = get_Flx_var(T);
     542         219 :   GEN RHS = FlxqX_rem(Flxq_rhs(a4, a6, v, vT), h, T, p);
     543         219 :   return FlxqXQ_halfFrobenius(RHS, h, T, p);
     544             : }
     545             : 
     546             : static GEN
     547        6095 : Fq_find_eigen_Frobenius(GEN a4, GEN a6, GEN h, GEN T, GEN p)
     548             : {
     549        6095 :   long v = T ? get_FpXQX_var(h): get_FpX_var(h);
     550        6095 :   GEN RHS  = FqX_rem(rhs(a4, a6, v), h, T, p);
     551       12045 :   return T ? FpXQXQ_halfFrobenius(RHS, h, T, p):
     552        5950 :              FpXQ_pow(RHS, shifti(p, -1), h, p);
     553             : }
     554             : /*Finds the eigenvalue of the Frobenius given E, ell odd prime, h factor of the
     555             :  *ell-division polynomial, p and tr the possible values for the trace
     556             :  *(useful for primes with one root)*/
     557             : static ulong
     558         483 : find_eigen_value_oneroot(GEN a4, GEN a6, ulong ell, GEN tr, GEN h, GEN T, GEN p)
     559             : {
     560         483 :   pari_sp ltop = avma;
     561             :   ulong t;
     562             :   struct divpolmod_red d;
     563             :   GEN f, Dy, Gy;
     564         483 :   h = FqX_get_red(h, T, p);
     565         483 :   Gy = Fq_find_eigen_Frobenius(a4, a6, h, T, p);
     566         483 :   t = Fl_div(tr[1], 2, ell);
     567         483 :   if (t < (ell>>1)) t = ell - t;
     568         483 :   Fq_elldivpolmod_init(&d, a4, a6, t, h, T, p);
     569         483 :   f = Fq_ellyn(&d, t);
     570         483 :   Dy = FqXQ_mul(Gy, gel(f,2), h, T, p);
     571         483 :   if (!gequal(gel(f,1), Dy)) t = ell-t;
     572         483 :   Fq_elldivpolmod_close(&d);
     573         483 :   return gc_ulong(ltop, t);
     574             : }
     575             : 
     576             : static ulong
     577         219 : Flxq_find_eigen_value_power(GEN a4, GEN a6, ulong ell, long k, ulong lambda,
     578             :                             GEN h, GEN T, ulong p)
     579             : {
     580         219 :   pari_sp ltop = avma;
     581         219 :   ulong t, ellk1 = upowuu(ell, k-1), ellk = ell*ellk1;
     582             :   pari_timer ti;
     583             :   struct divpolmod_red d;
     584             :   GEN Gy;
     585         219 :   timer_start(&ti);
     586         219 :   h = FlxqX_get_red(h, T, p);
     587         219 :   Gy = Flxq_find_eigen_Frobenius(a4, a6, h, T, p);
     588         219 :   if (DEBUGLEVEL>2) err_printf(" (%ld ms)",timer_delay(&ti));
     589         219 :   Flxq_elldivpolmod_init(&d, a4, a6, ellk, h, T, p);
     590        1158 :   for (t = lambda; t < ellk; t += ellk1)
     591             :   {
     592        1158 :     GEN f = Fq_ellyn(&d, t);
     593        1158 :     GEN Dr = FlxqXQ_mul(Gy, gel(f,2), h, T, p);
     594        1158 :     if (varn(gel(f,1))!=varn(Dr)) pari_err_BUG("find_eigen_value_power");
     595        1158 :     if (gequal(gel(f,1), Dr)) break;
     596        1013 :     if (gequal(gel(f,1), FlxX_neg(Dr,p))) { t = ellk-t; break; }
     597             :   }
     598         219 :   if (DEBUGLEVEL>2) err_printf(" (%ld ms)",timer_delay(&ti));
     599         219 :   Fq_elldivpolmod_close(&d);
     600         219 :   return gc_ulong(ltop, t);
     601             : }
     602             : 
     603             : /*Finds the eigenvalue of the Frobenius modulo ell^k given E, ell, k, h factor
     604             :  *of the ell-division polynomial, lambda the previous eigen value and p */
     605             : static ulong
     606        5612 : Fq_find_eigen_value_power(GEN a4, GEN a6, ulong ell, long k, ulong lambda, GEN h, GEN T, GEN p)
     607             : {
     608        5612 :   pari_sp ltop = avma;
     609        5612 :   ulong t, ellk1 = upowuu(ell, k-1), ellk = ell*ellk1;
     610             :   pari_timer ti;
     611             :   struct divpolmod_red d;
     612             :   GEN Gy;
     613        5612 :   timer_start(&ti);
     614        5612 :   h = FqX_get_red(h, T, p);
     615        5612 :   Gy = Fq_find_eigen_Frobenius(a4, a6, h, T, p);
     616        5612 :   if (DEBUGLEVEL>2) err_printf(" (%ld ms)",timer_delay(&ti));
     617        5612 :   Fq_elldivpolmod_init(&d, a4, a6, ellk, h, T, p);
     618       21865 :   for (t = lambda; t < ellk; t += ellk1)
     619             :   {
     620       21865 :     GEN f = Fq_ellyn(&d, t);
     621       21865 :     GEN Dr = FqXQ_mul(Gy, gel(f,2), h, T, p);
     622       21865 :     if (varn(gel(f,1))!=varn(Dr)) pari_err_BUG("find_eigen_value_power");
     623       21865 :     if (gequal(gel(f,1), Dr)) break;
     624       17404 :     if (gequal(gel(f,1), FqX_neg(Dr,T,p))) { t = ellk-t; break; }
     625             :   }
     626        5612 :   if (DEBUGLEVEL>2) err_printf(" (%ld ms)",timer_delay(&ti));
     627        5612 :   Fq_elldivpolmod_close(&d);
     628        5612 :   return gc_ulong(ltop, t);
     629             : }
     630             : 
     631             : static ulong
     632        5831 : find_eigen_value_power(GEN a4, GEN a6, ulong ell, long k, ulong lambda, GEN hq, GEN T, GEN p)
     633             : {
     634        5831 :   ulong pp = itou_or_0(p);
     635        5831 :   if (pp && T)
     636             :   {
     637         219 :     GEN a4p = ZX_to_Flx(a4, pp);
     638         219 :     GEN a6p = ZX_to_Flx(a6, pp);
     639         219 :     GEN hp = ZXXT_to_FlxXT(hq, pp,varn(a4));
     640         219 :     GEN Tp = ZXT_to_FlxT(T, pp);
     641         219 :     return Flxq_find_eigen_value_power(a4p, a6p, ell, k, lambda, hp, Tp, pp);
     642             :   }
     643        5612 :   return Fq_find_eigen_value_power(a4, a6, ell, k, lambda, hq, T, p);
     644             : }
     645             : 
     646             : /*Finds the kernel polynomial h, dividing the ell-division polynomial from the
     647             :   isogenous curve Eb and trace term pp1. Uses CCR algorithm and returns h.
     648             :   Return NULL if E and Eb are *not* isogenous. */
     649             : static GEN
     650        8722 : find_kernel(GEN a4, GEN a6, ulong ell, GEN a4t, GEN a6t, GEN pp1, GEN T, GEN p, GEN pp, long e)
     651             : {
     652        8722 :   const long ext = 2;
     653        8722 :   pari_sp ltop = avma, btop;
     654             :   GEN P, v, tlist, h;
     655             :   long i, j, k;
     656        8722 :   long deg = (ell - 1)/2, dim = 2 + deg + ext;
     657        8722 :   GEN psi2 = Fq_elldivpol2(a4, a6, T, p);
     658        8722 :   GEN Dpsi2 = Fq_elldivpol2d(a4, T, p);
     659        8722 :   GEN C  = find_coeff(a4, a6, T, p, dim, pp, e);
     660        8722 :   GEN Ct = find_coeff(a4t, a6t, T, p, dim, pp, e);
     661        8722 :   GEN V = cgetg(dim+1, t_VEC);
     662       99806 :   for (k = 1; k <= dim; k++)
     663             :   {
     664       91084 :     long v = mael(C,2,k);
     665       91084 :     GEN z = gmul(gsub(gmael(Ct,1,k), gmael(C,1,k)), shifti(mpfact(2*k), -1));
     666       91084 :     if (signe(z) && Zq_pval(z, pp) < v) return NULL;
     667       91084 :     gel(V, k) = Zq_divexact(z, powiu(pp, v));
     668             :   }
     669        8722 :   btop = avma;
     670        8722 :   v = zerovec(dim);
     671        8722 :   gel(v, 1) = utoi(deg);
     672        8722 :   gel(v, 2) = pp1;
     673        8722 :   P = pol_x(0);
     674       82362 :   for (k = 3; k <= dim; k++)
     675             :   {
     676       73640 :     GEN s, r = FqX_Fq_mul(Dpsi2, gel(P, 3), T, p);
     677      531118 :     for (j = 4; j < lg(P); j++)
     678             :     {
     679      457478 :       long o = j - 2;
     680      457478 :       GEN D = FqX_add(RgX_shift_shallow(Dpsi2, 1), FqX_mulu(psi2, o-1, T, p), T, p);
     681      457478 :       GEN E = FqX_Fq_mul(D, Fq_mulu(gel(P, j), o, T, p), T, p);
     682      457478 :       r = FqX_add(r, RgX_shift_shallow(E, o-2), T, p);
     683             :     }
     684       73640 :     P = r;
     685       73640 :     s = Fq_mul(gel(P, 2), gel(v, 1), T, p);
     686      604758 :     for (j = 3; j < lg(P)-1; j++)
     687      531118 :       s = Fq_add(s, Fq_mul(gel(P, j), gel(v, j-1), T, p), T, p);
     688       73640 :     gel(v, k) = Zq_Z_div_safe(Fq_sub(gel(V, k-2), s, T, p), gel(P, j), T, p, pp, e);
     689       73640 :     if (gc_needed(btop, 1))
     690             :     {
     691           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"find_kernel");
     692           0 :       gerepileall(btop, 2, &v, &P);
     693             :     }
     694             :   }
     695        8722 :   tlist = cgetg(dim, t_VEC);
     696        8722 :   gel(tlist, dim-1) = gen_1;
     697       82362 :   for (k = 1; k <= dim-2; k++)
     698             :   {
     699       73640 :     pari_sp btop = avma;
     700       73640 :     GEN s = gel(v, k+1);
     701      531118 :     for (i = 1; i < k; i++)
     702      457478 :       s = Fq_add(s, Fq_mul(gel(tlist, dim-i-1), gel(v, k-i+1), T, p), T, p);
     703       73640 :     gel(tlist, dim-k-1) = gerepileupto(btop, Zq_Z_div_safe(s, stoi(-k), T, p, pp, e));
     704             :   }
     705       23324 :   for (i = 1; i <= ext; i++)
     706       16023 :     if (signe(Fq_red(gel(tlist, i),T, pp))) return gc_NULL(ltop);
     707        7301 :   h = FqX_red(RgV_to_RgX(vecslice(tlist, ext+1, dim-1), 0),T,p);
     708        7301 :   return signe(Fq_elldivpolmod(a4, a6, ell, h, T, pp)) ? NULL: h;
     709             : }
     710             : 
     711             : static GEN
     712        6279 : compute_u(GEN gprime, GEN Dxxg, GEN DxJg, GEN DJJg, GEN j, GEN pJ, GEN px, ulong q, GEN E4, GEN E6, GEN T, GEN p, GEN pp, long e)
     713             : {
     714        6279 :   pari_sp ltop = avma;
     715        6279 :   GEN dxxgj = FqX_eval(Dxxg, j, T, p);
     716        6279 :   GEN dxJgj = FqX_eval(DxJg, j, T, p);
     717        6279 :   GEN dJJgj = FqX_eval(DJJg, j, T, p);
     718        6279 :   GEN E42 = Fq_sqr(E4, T, p), E6ovE4 = Zq_div(E6, E4, T, p, pp, e);
     719        6279 :   GEN a = Fq_mul(gprime, dxxgj, T, p);
     720        6279 :   GEN b = Fq_mul(Fq_mul(Fq_mulu(j,2*q, T, p), dxJgj, T, p), E6ovE4, T, p);
     721        6279 :   GEN c = Fq_mul(Zq_div(Fq_sqr(E6ovE4, T, p), gprime, T, p, pp, e), j, T, p);
     722        6279 :   GEN d = Fq_mul(Fq_mul(c,sqru(q), T, p), Fq_add(pJ, Fq_mul(j, dJJgj, T, p), T, p), T, p);
     723        6279 :   GEN f = Fq_sub(Fq_div(E6ovE4,utoi(3), T, p),
     724             :                  Zq_div(E42, Fq_mulu(E6,2,T, p), T, p, pp, e), T, p);
     725        6279 :   GEN g = Fq_sub(Fq_sub(b,a,T,p), d, T, p);
     726        6279 :   return gerepileupto(ltop, Fq_add(Zq_div(g,px,T,p,pp,e), Fq_mulu(f,q,T,p), T, p));
     727             : }
     728             : 
     729             : /* Finds the isogenous EC, and the sum of the x-coordinates of the points in
     730             :  * the kernel of the isogeny E -> Eb
     731             :  * E: elliptic curve, ell: a prime, meqn: Atkin modular equation
     732             :  * g: root of meqn defining isogenous curve Eb. */
     733             : static GEN
     734        2464 : find_isogenous_from_Atkin(GEN a4, GEN a6, ulong ell, struct meqn *MEQN, GEN g, GEN T, GEN pp, long e)
     735             : {
     736        2464 :   pari_sp ltop = avma, btop;
     737        2464 :   GEN meqn = MEQN->eq, meqnx, Dmeqnx, Roots, gprime, u1;
     738        2464 :   long k, vJ = MEQN->vy;
     739        2464 :   GEN p = e==1 ? pp: powiu(pp, e);
     740        2464 :   GEN j = Zq_ellj(a4, a6, T, p, pp, e);
     741        2464 :   GEN E4 = Fq_div(a4, stoi(-3), T, p);
     742        2464 :   GEN E6 = Fq_neg(Fq_halve(a6, T, p), T, p);
     743        2464 :   GEN Dx = RgX_deriv(meqn);
     744        2464 :   GEN DJ = deriv(meqn, vJ);
     745        2464 :   GEN Dxg = FpXY_Fq_evaly(Dx, g, T, p, vJ);
     746        2464 :   GEN px = FqX_eval(Dxg, j, T, p), dx = Fq_mul(px, g, T, p);
     747        2464 :   GEN DJg = FpXY_Fq_evaly(DJ, g, T, p, vJ);
     748        2464 :   GEN pJ = FqX_eval(DJg, j, T, p), dJ = Fq_mul(pJ, j, T, p);
     749        2464 :   GEN Dxx = RgX_deriv(Dx);
     750        2464 :   GEN DxJg = FqX_deriv(Dxg, T, p);
     751             : 
     752        2464 :   GEN Dxxg = FpXY_Fq_evaly(Dxx, g, T, p, vJ);
     753        2464 :   GEN DJJg = FqX_deriv(DJg, T, p);
     754             :   GEN a, b;
     755        2464 :   if (!signe(Fq_red(dJ,T,pp)) || !signe(Fq_red(dx,T,pp)))
     756             :   {
     757          28 :     if (DEBUGLEVEL>0) err_printf("[A: d%c=0]",signe(dJ)? 'x': 'J');
     758          28 :     return gc_NULL(ltop);
     759             :   }
     760        2436 :   a = Fq_mul(dJ, Fq_mul(g, E6, T, p), T, p);
     761        2436 :   b = Fq_mul(E4, dx, T, p);
     762        2436 :   gprime = Zq_div(a, b, T, p, pp, e);
     763             : 
     764        2436 :   u1 = compute_u(gprime, Dxxg, DxJg, DJJg, j, pJ, px, 1, E4, E6, T, p, pp, e);
     765        2436 :   meqnx = FpXY_Fq_evaly(meqn, g, T, p, vJ);
     766        2436 :   Dmeqnx = FqX_deriv(meqnx, T, pp);
     767        2436 :   Roots = FqX_roots(meqnx, T, pp);
     768             : 
     769        2436 :   btop = avma;
     770        3857 :   for (k = lg(Roots)-1; k >= 1; k--, set_avma(btop))
     771             :   {
     772        3857 :     GEN jt = gel(Roots, k);
     773        3857 :     if (signe(FqX_eval(Dmeqnx, jt, T, pp))==0)
     774           0 :       continue;
     775        3857 :     if (e > 1)
     776          21 :       jt = ZqX_liftroot(meqnx, gel(Roots, k), T, pp, e);
     777        3857 :     if (signe(Fq_red(jt, T, pp)) == 0 || signe(Fq_sub(jt, utoi(1728), T, pp)) == 0)
     778             :     {
     779          14 :       if (DEBUGLEVEL>0) err_printf("[A: jt=%ld]",signe(Fq_red(jt,T,p))? 1728: 0);
     780          14 :       return gc_NULL(ltop);
     781             :     }
     782             :     else
     783             :     {
     784        3843 :       GEN pxstar = FqX_eval(Dxg, jt, T, p);
     785        3843 :       GEN dxstar = Fq_mul(pxstar, g, T, p);
     786        3843 :       GEN pJstar = FqX_eval(DJg, jt, T, p);
     787        3843 :       GEN dJstar = Fq_mul(Fq_mulu(jt, ell, T, p), pJstar, T, p);
     788        3843 :       GEN u = Fq_mul(Fq_mul(dxstar, dJ, T, p), E6, T, p);
     789        3843 :       GEN v = Fq_mul(Fq_mul(dJstar, dx, T, p), E4, T, p);
     790        3843 :       GEN E4t = Zq_div(Fq_mul(Fq_sqr(u, T, p), jt, T, p), Fq_mul(Fq_sqr(v, T, p), Fq_sub(jt, utoi(1728), T, p), T, p), T, p, pp, e);
     791        3843 :       GEN E6t = Zq_div(Fq_mul(u, E4t, T, p), v, T, p, pp, e);
     792        3843 :       GEN u2 = compute_u(gprime, Dxxg, DxJg, DJJg, jt, pJstar, pxstar, ell, E4t, E6t, T, p, pp, e);
     793        3843 :       GEN pp1 = Fq_mulu(Fq_sub(u1, u2, T, p), 3*ell, T, p);
     794        3843 :       GEN a4t = Fq_mul(mulsi(-3, powuu(ell,4)), E4t, T, p);
     795        3843 :       GEN a6t = Fq_mul(mulsi(-2, powuu(ell,6)), E6t, T, p);
     796        3843 :       GEN h = find_kernel(a4, a6, ell, a4t, a6t, pp1, T, p, pp, e);
     797        3843 :       if (h) return gerepilecopy(ltop, mkvec3(a4t, a6t, h));
     798             :     }
     799             :   }
     800           0 :   pari_err_BUG("find_isogenous_from_Atkin, kernel not found");
     801             :   return NULL;/*LCOV_EXCL_LINE*/
     802             : }
     803             : 
     804             : /* Finds E' ell-isogenous to E and the trace term p1 from canonical modular
     805             :  *   equation meqn
     806             :  * E: elliptic curve, ell: a prime, meqn: canonical modular equation
     807             :  * g: root of meqn defining isogenous curve Eb. */
     808             : static GEN
     809        4844 : find_isogenous_from_canonical(GEN a4, GEN a6, ulong ell, struct meqn *MEQN, GEN g, GEN T, GEN pp, long e)
     810             : {
     811        4844 :   pari_sp ltop = avma;
     812        4844 :   GEN meqn = MEQN->eq;
     813        4844 :   long vJ = MEQN->vy;
     814        4844 :   GEN p = e==1 ? pp: powiu(pp, e);
     815             :   GEN h;
     816        4844 :   GEN E4 = Fq_div(a4, stoi(-3), T, p);
     817        4844 :   GEN E6 = Fq_neg(Fq_halve(a6, T, p), T, p);
     818        4844 :   GEN E42 = Fq_sqr(E4, T, p);
     819        4844 :   GEN E43 = Fq_mul(E4, E42, T, p);
     820        4844 :   GEN E62 = Fq_sqr(E6, T, p);
     821        4844 :   GEN delta = Fq_div(Fq_sub(E43, E62, T, p), utoi(1728), T, p);
     822        4844 :   GEN j = Zq_div(E43, delta, T, p, pp, e);
     823        4844 :   GEN Dx = RgX_deriv(meqn);
     824        4844 :   GEN DJ = deriv(meqn, vJ);
     825        4844 :   GEN Dxg = FpXY_Fq_evaly(Dx, g, T, p, vJ);
     826        4844 :   GEN px  = FqX_eval(Dxg, j, T, p), dx  = Fq_mul(px, g, T, p);
     827        4844 :   GEN DJg = FpXY_Fq_evaly(DJ, g, T, p, vJ);
     828        4844 :   GEN pJ = FqX_eval(DJg, j, T, p), dJ = Fq_mul(j, pJ, T, p);
     829        4844 :   GEN Dxx = RgX_deriv(Dx);
     830        4844 :   GEN DxJg = FqX_deriv(Dxg, T, p);
     831             : 
     832        4844 :   GEN ExJ = FqX_eval(DxJg, j, T, p);
     833        4844 :   ulong tis = ugcd(12, ell-1), is = 12 / tis;
     834        4844 :   GEN itis = Fq_inv(stoi(-tis), T, p);
     835        4844 :   GEN deltal = Fq_div(Fq_mul(delta, Fq_powu(g, tis, T, p), T, p), powuu(ell, 12), T, p);
     836             :   GEN E4l, E6l, a4tilde, a6tilde, p_1;
     837        4844 :   if (signe(Fq_red(dx,T, pp))==0)
     838             :   {
     839           7 :     if (DEBUGLEVEL>0) err_printf("[C: dx=0]");
     840           7 :     return gc_NULL(ltop);
     841             :   }
     842        4837 :   if (signe(Fq_red(dJ, T, pp))==0)
     843             :   {
     844             :     GEN jl;
     845           0 :     if (DEBUGLEVEL>0) err_printf("[C: dJ=0]");
     846           0 :     E4l = Fq_div(E4, sqru(ell), T, p);
     847           0 :     jl  = Zq_div(Fq_powu(E4l, 3, T, p), deltal, T, p, pp, e);
     848           0 :     E6l = Zq_sqrt(Fq_mul(Fq_sub(jl, utoi(1728), T, p), deltal, T, p), T, p, pp, e);
     849           0 :     p_1 = gen_0;
     850             :   }
     851             :   else
     852             :   {
     853             :     GEN jl, f, fd, Dgs, Djs, jld;
     854        4837 :     GEN E2s = Zq_div(Fq_mul(Fq_neg(Fq_mulu(E6, 12, T, p), T, p), dJ, T, p), Fq_mul(Fq_mulu(E4, is, T, p), dx, T, p), T, p, pp, e);
     855        4837 :     GEN gd = Fq_mul(Fq_mul(E2s, itis, T, p), g, T, p);
     856        4837 :     GEN jd = Zq_div(Fq_mul(Fq_neg(E42, T, p), E6, T, p), delta, T, p, pp, e);
     857        4837 :     GEN E0b = Zq_div(E6, Fq_mul(E4, E2s, T, p), T, p, pp, e);
     858        4837 :     GEN Dxxgj = FqXY_eval(Dxx, g, j, T, p);
     859        4837 :     GEN Dgd = Fq_add(Fq_mul(gd, px, T, p), Fq_mul(g, Fq_add(Fq_mul(gd, Dxxgj, T, p), Fq_mul(jd, ExJ, T, p), T, p), T, p), T, p);
     860        4837 :     GEN DJgJj = FqX_eval(FqX_deriv(DJg, T, p), j, T, p);
     861        4837 :     GEN Djd = Fq_add(Fq_mul(jd, pJ, T, p), Fq_mul(j, Fq_add(Fq_mul(jd, DJgJj, T, p), Fq_mul(gd, ExJ, T, p), T, p), T, p), T, p);
     862        4837 :     GEN E0bd = Zq_div(Fq_sub(Fq_mul(Dgd, itis, T, p), Fq_mul(E0b, Djd, T, p), T, p), dJ, T, p, pp, e);
     863        4837 :     E4l = Zq_div(Fq_sub(E4, Fq_mul(E2s, Fq_sub(Fq_sub(Fq_add(Zq_div(Fq_mulu(E0bd, 12, T, p), E0b, T, p, pp, e), Zq_div(Fq_mulu(E42, 6, T, p), E6, T, p, pp, e), T, p), Zq_div(Fq_mulu(E6, 4, T, p), E4, T, p, pp, e), T, p), E2s, T, p), T, p), T, p), sqru(ell), T, p, pp, e);
     864        4837 :     jl = Zq_div(Fq_powu(E4l, 3, T, p), deltal, T, p, pp, e);
     865        4837 :     if (signe(Fq_red(jl,T,pp))==0)
     866             :     {
     867           7 :       if (DEBUGLEVEL>0) err_printf("[C: jl=0]");
     868           7 :       return gc_NULL(ltop);
     869             :     }
     870        4830 :     f =  Zq_div(powuu(ell, is), g, T, p, pp, e);
     871        4830 :     fd = Fq_neg(Fq_mul(Fq_mul(E2s, f, T, p), itis, T, p), T, p);
     872        4830 :     Dgs = FqXY_eval(Dx, f, jl, T, p);
     873        4830 :     Djs = FqXY_eval(DJ, f, jl, T, p);
     874        4830 :     jld = Zq_div(Fq_mul(Fq_neg(fd, T, p), Dgs, T, p), Fq_mulu(Djs, ell, T, p), T, p, pp, e);
     875        4830 :     E6l = Zq_div(Fq_mul(Fq_neg(E4l, T, p), jld, T, p), jl, T, p, pp, e);
     876        4830 :     p_1 = Fq_neg(Fq_halve(Fq_mulu(E2s, ell, T, p), T, p),T,p);
     877             :   }
     878        4830 :   a4tilde = Fq_mul(Fq_mul(stoi(-3), powuu(ell,4), T, p), E4l, T, p);
     879        4830 :   a6tilde = Fq_mul(Fq_mul(stoi(-2), powuu(ell,6), T, p), E6l, T, p);
     880        4830 :   h = find_kernel(a4, a6, ell, a4tilde, a6tilde, p_1, T, p, pp, e);
     881        4830 :   if (!h) return NULL;
     882        4830 :   return gerepilecopy(ltop, mkvec3(a4tilde, a6tilde, h));
     883             : }
     884             : 
     885             : static GEN
     886          98 : corr(GEN c4, GEN c6, GEN T, GEN p, GEN pp, long e)
     887             : {
     888          98 :   GEN c46 = Zq_div(Fq_sqr(c4, T, p), c6, T, p, pp, e);
     889          98 :   GEN c64 = Zq_div(c6, c4, T, p, pp, e);
     890          98 :   GEN a = Fp_divu(gen_2, 3, p);
     891          98 :   return Fq_add(Fq_halve(c46, T, p), Fq_mul(a, c64, T, p), T, p);
     892             : }
     893             : 
     894             : static GEN
     895         168 : RgXY_deflatex(GEN H, long n, long d)
     896             : {
     897         168 :   long i, l = lg(H);
     898         168 :   GEN R = cgetg(l, t_POL);
     899         168 :   R[1] = H[1];
     900         980 :   for(i = 2; i < l; i++)
     901             :   {
     902         812 :     GEN Hi = gel(H, i);
     903         812 :     gel(R,i) = typ(Hi)==t_POL? RgX_deflate(RgX_shift_shallow(Hi, d), n): Hi;
     904             :   }
     905         168 :   return RgX_renormalize_lg(R, l);
     906             : }
     907             : 
     908             : static GEN
     909          70 : Fq_polmodular_eval(GEN meqn, GEN j, long N, GEN T, GEN p, long vJ)
     910             : {
     911          70 :   pari_sp av = avma;
     912             :   GEN R, dR, ddR;
     913          70 :   long t0 = N%3 == 1 ? 2: 0;
     914          70 :   long t2 = N%3 == 1 ? 0: 2;
     915          70 :   if (N == 3)
     916             :   {
     917          14 :     GEN P = FpXX_red(meqn, p);
     918          14 :     GEN dP = deriv(P, -1), ddP = deriv(dP, -1);
     919          14 :     R = FpXY_Fq_evaly(P, j, T, p, vJ);
     920          14 :     dR = FpXY_Fq_evaly(dP, j, T, p, vJ);
     921          14 :     ddR = FpXY_Fq_evaly(ddP, j, T, p, vJ);
     922          14 :     return gerepilecopy(av, mkvec3(R,dR,ddR));
     923             :   }
     924             :   else
     925             :   {
     926          56 :     GEN P5 = FpXX_red(meqn, p);
     927          56 :     GEN H = RgX_splitting(P5, 3);
     928          56 :     GEN H0 = RgXY_deflatex(gel(H,1), 3, -t0);
     929          56 :     GEN H1 = RgXY_deflatex(gel(H,2), 3, -1);
     930          56 :     GEN H2 = RgXY_deflatex(gel(H,3), 3, -t2);
     931          56 :     GEN h0 = FpXY_Fq_evaly(H0, j, T, p, vJ);
     932          56 :     GEN h1 = FpXY_Fq_evaly(H1, j, T, p, vJ);
     933          56 :     GEN h2 = FpXY_Fq_evaly(H2, j, T, p, vJ);
     934          56 :     GEN dH0 = RgX_deriv(H0);
     935          56 :     GEN dH1 = RgX_deriv(H1);
     936          56 :     GEN dH2 = RgX_deriv(H2);
     937          56 :     GEN ddH0 = RgX_deriv(dH0);
     938          56 :     GEN ddH1 = RgX_deriv(dH1);
     939          56 :     GEN ddH2 = RgX_deriv(dH2);
     940          56 :     GEN d0 = FpXY_Fq_evaly(dH0, j, T, p, vJ);
     941          56 :     GEN d1 = FpXY_Fq_evaly(dH1, j, T, p, vJ);
     942          56 :     GEN d2 = FpXY_Fq_evaly(dH2, j, T, p, vJ);
     943          56 :     GEN dd0 = FpXY_Fq_evaly(ddH0, j, T, p, vJ);
     944          56 :     GEN dd1 = FpXY_Fq_evaly(ddH1, j, T, p, vJ);
     945          56 :     GEN dd2 = FpXY_Fq_evaly(ddH2, j, T, p, vJ);
     946             :     GEN h02, h12, h22, h03, h13, h23, h012, dh03, dh13, dh23, dh012;
     947             :     GEN ddh03, ddh13, ddh23, ddh012;
     948             :     GEN R1, dR1, ddR1, ddR2;
     949          56 :     h02 = FqX_sqr(h0, T, p);
     950          56 :     h12 = FqX_sqr(h1, T, p);
     951          56 :     h22 = FqX_sqr(h2, T, p);
     952          56 :     h03 = FqX_mul(h0, h02, T, p);
     953          56 :     h13 = FqX_mul(h1, h12, T, p);
     954          56 :     h23 = FqX_mul(h2, h22, T, p);
     955          56 :     h012 = FqX_mul(FqX_mul(h0, h1, T, p), h2, T, p);
     956          56 :     dh03 = FqX_mul(FqX_mulu(d0, 3, T, p), h02, T, p);
     957          56 :     dh13 = FqX_mul(FqX_mulu(d1, 3, T, p), h12, T, p);
     958          56 :     dh23 = FqX_mul(FqX_mulu(d2, 3, T, p), h22, T, p);
     959          56 :     dh012 = FqX_add(FqX_add(FqX_mul(FqX_mul(d0, h1, T, p), h2, T, p), FqX_mul(FqX_mul(h0, d1, T, p), h2, T, p), T, p), FqX_mul(FqX_mul(h0, h1, T, p), d2, T, p), T, p);
     960          56 :     R1 = FqX_sub(h13, FqX_mulu(h012, 3, T, p), T, p);
     961          56 :     R = FqX_add(FqX_add(FqX_Fq_mul(RgX_shift_shallow(h23, t2), Fq_sqr(j, T, p), T, p), FqX_Fq_mul(RgX_shift_shallow(R1, 1), j, T, p), T, p), RgX_shift_shallow(h03, t0), T, p);
     962          56 :     dR1 = FqX_sub(dh13, FqX_mulu(dh012, 3, T, p), T, p);
     963          56 :     dR = FqX_add(FqX_add(RgX_shift_shallow(FqX_add(FqX_Fq_mul(dh23, Fq_sqr(j, T, p), T, p), FqX_Fq_mul(h23, Fq_mulu(j, 2, T, p), T, p), T, p), t2), RgX_shift_shallow(FqX_add(FqX_Fq_mul(dR1, j, T, p), R1, T, p), 1), T, p), RgX_shift_shallow(dh03, t0), T, p);
     964          56 :     ddh03 = FqX_mulu(FqX_add(FqX_mul(dd0, h02, T, p), FqX_mul(FqX_mulu(FqX_sqr(d0, T, p), 2, T, p), h0, T, p), T, p), 3, T, p);
     965          56 :     ddh13 = FqX_mulu(FqX_add(FqX_mul(dd1, h12, T, p), FqX_mul(FqX_mulu(FqX_sqr(d1, T, p), 2, T, p), h1, T, p), T, p), 3, T, p);
     966          56 :     ddh23 = FqX_mulu(FqX_add(FqX_mul(dd2, h22, T, p), FqX_mul(FqX_mulu(FqX_sqr(d2, T, p), 2, T, p), h2, T, p), T, p), 3, T, p);
     967          56 :     ddh012 = FqX_add(FqX_add(FqX_add(FqX_mul(FqX_mul(dd0, h1, T, p), h2, T, p), FqX_mul(FqX_mul(h0, dd1, T, p), h2, T, p), T, p), FqX_mul(FqX_mul(h0, h1, T, p), dd2, T, p), T, p), FqX_mulu(FqX_add(FqX_add(FqX_mul(FqX_mul(d0, d1, T, p), h2, T, p), FqX_mul(FqX_mul(d0, h1, T, p), d2, T, p), T, p), FqX_mul(FqX_mul(h0, d1, T, p), d2, T, p), T, p), 2, T, p), T, p);
     968          56 :     ddR1 = FqX_sub(ddh13, FqX_mulu(ddh012, 3, T, p), T, p);
     969          56 :     ddR2 = FqX_add(FqX_add(FqX_Fq_mul(ddh23, Fq_sqr(j, T, p), T, p), FqX_Fq_mul(dh23, Fq_mulu(j, 4, T, p), T, p), T, p), FqX_mulu(h23, 2, T, p), T, p);
     970          56 :     ddR = FqX_add(FqX_add(RgX_shift_shallow(ddR2, t2), RgX_shift_shallow(FqX_add(FqX_mulu(dR1, 2, T, p), FqX_Fq_mul(ddR1, j, T, p), T, p), 1), T, p), RgX_shift_shallow(ddh03, t0), T, p);
     971          56 :     return gerepilecopy(av, mkvec3(R ,dR ,ddR));
     972             :   }
     973             : }
     974             : 
     975             : static GEN
     976       11277 : meqn_j(struct meqn *MEQN, GEN j, long ell, GEN T, GEN p)
     977             : {
     978       11277 :   if (MEQN->type=='J')
     979             :   {
     980          70 :     MEQN->eval = Fq_polmodular_eval(MEQN->eq, j, ell, T, p, MEQN->vy);
     981          70 :     return gel(MEQN->eval, 1);
     982             :   }
     983             :   else
     984       11207 :     return FqXY_evalx(MEQN->eq, j, T, p);
     985             : }
     986             : 
     987             : static GEN
     988          49 : find_isogenous_from_J(GEN a4, GEN a6, ulong ell, struct meqn *MEQN, GEN g, GEN T, GEN pp, long e)
     989             : {
     990          49 :   pari_sp ltop = avma;
     991          49 :   GEN meqn = MEQN->eval;
     992          49 :   GEN p = e==1 ? pp: powiu(pp, e);
     993             :   GEN h;
     994             :   GEN C4, C6, C4t, C6t;
     995             :   GEN j, jp, jtp, jtp2, jtp3;
     996             :   GEN Py, Pxy, Pyy, Pxj, Pyj, Pxxj, Pxyj, Pyyj;
     997             :   GEN s0, s1, s2, s3;
     998             :   GEN den, D, co, cot, c0, p_1, a4tilde, a6tilde;
     999          49 :   if (signe(g) == 0 || signe(Fq_sub(g, utoi(1728), T, p)) == 0)
    1000             :   {
    1001           0 :     if (DEBUGLEVEL>0) err_printf("[J: g=%ld]",signe(g)==0 ?0: 1728);
    1002           0 :     return gc_NULL(ltop);
    1003             :   }
    1004          49 :   C4 = Fq_mul(a4, stoi(-48), T, p);
    1005          49 :   C6 = Fq_mul(a6, stoi(-864), T, p);
    1006          49 :   if (signe(C4)==0 || signe(C6)==0)
    1007             :   {
    1008           0 :     if (DEBUGLEVEL>0) err_printf("[J: C%ld=0]",signe(C4)==0 ?4: 6);
    1009           0 :     return gc_NULL(ltop);
    1010             :   }
    1011          49 :   j = Zq_ellj(a4, a6, T, p, pp, e);
    1012          49 :   jp = Fq_mul(j, Zq_div(C6, C4, T, p, pp, e), T, p);
    1013          49 :   co = corr(C4, C6, T, p, pp, e);
    1014          49 :   Py = RgX_deriv(gel(meqn, 1));
    1015          49 :   Pxy = RgX_deriv(gel(meqn,2));
    1016          49 :   Pyy = RgX_deriv(Py);
    1017          49 :   Pxj = FqX_eval(gel(meqn, 2), g, T, p);
    1018          49 :   if (signe(Pxj)==0)
    1019             :   {
    1020           0 :     if (DEBUGLEVEL>0) err_printf("[J: Pxj=0]");
    1021           0 :     return gc_NULL(ltop);
    1022             :   }
    1023          49 :   Pyj = FqX_eval(Py, g, T, p);
    1024          49 :   Pxxj = FqX_eval(gel(meqn, 3), g, T, p);
    1025          49 :   Pxyj = FqX_eval(Pxy, g, T, p);
    1026          49 :   Pyyj = FqX_eval(Pyy, g, T, p);
    1027          49 :   jtp = Fq_div(Fq_mul(jp, Zq_div(Pxj, Pyj, T, p, pp, e), T, p), negi(utoi(ell)), T, p);
    1028          49 :   jtp2 = Fq_sqr(jtp,T,p);
    1029          49 :   jtp3 = Fq_mul(jtp,jtp2,T,p);
    1030          49 :   den = Fq_mul(Fq_sqr(g,T,p),Fq_sub(g,utoi(1728),T,p),T, p);
    1031          49 :   D  =  Zq_inv(den,T,p,pp, e);
    1032          49 :   C4t = Fq_mul(jtp2,Fq_mul(g, D, T, p), T, p);
    1033          49 :   C6t = Fq_mul(jtp3, D, T, p);
    1034          49 :   s0 = Fq_mul(Fq_sqr(jp, T, p), Pxxj, T, p);
    1035          49 :   s1 = Fq_mul(Fq_mulu(Fq_mul(jp,jtp,T,p),2*ell,T,p), Pxyj, T, p);
    1036          49 :   s2 = Fq_mul(Fq_mulu(jtp2,ell*ell,T,p), Pyyj, T, p);
    1037          49 :   s3 = Zq_div(Fq_add(s0, Fq_add(s1, s2, T, p), T, p),Fq_mul(jp, Pxj, T, p),T,p,pp,e);
    1038          49 :   cot = corr(C4t, C6t, T, p, pp, e);
    1039          49 :   c0 = Fq_sub(co,Fq_mulu(cot,ell,T,p),T,p);
    1040          49 :   p_1 = Fq_div(Fq_mulu(Fq_add(s3, c0, T, p),ell,T,p),stoi(-4),T,p);
    1041          49 :   a4tilde = Fq_mul(Fq_div(C4t, stoi(-48), T, p),powuu(ell,4), T, p);
    1042          49 :   a6tilde = Fq_mul(Fq_div(C6t, stoi(-864), T, p),powuu(ell,6), T, p);
    1043          49 :   h = find_kernel(a4, a6, ell, a4tilde, a6tilde, p_1, T, p, pp, e);
    1044          49 :   if (!h) return NULL;
    1045          49 :   return gerepilecopy(ltop, mkvec3(a4tilde, a6tilde, h));
    1046             : }
    1047             : 
    1048             : static GEN
    1049        7357 : find_isogenous(GEN a4,GEN a6, ulong ell, struct meqn *MEQN, GEN g, GEN T,GEN p)
    1050             : {
    1051        7357 :   ulong pp = itou_or_0(p);
    1052        7357 :   long e = (pp && pp <= 2*ell+3) ? 2+factorial_lval(ell, pp): 1;
    1053        7357 :   if (e > 1)
    1054             :   {
    1055          21 :     GEN pe = powiu(p, e);
    1056          21 :     GEN meqnj = meqn_j(MEQN, Zq_ellj(a4, a6, T, pe, p, e), ell, T, pe);
    1057          21 :     g = ZqX_liftroot(meqnj, g, T, p, e);
    1058             :   }
    1059        7357 :   switch(MEQN->type)
    1060             :   {
    1061        4844 :     case 'C': return find_isogenous_from_canonical(a4,a6,ell, MEQN, g, T,p,e);
    1062        2464 :     case 'A': return find_isogenous_from_Atkin(a4,a6,ell, MEQN, g, T,p,e);
    1063          49 :     default:  return find_isogenous_from_J(a4,a6,ell, MEQN, g, T,p,e);
    1064             :   }
    1065             : }
    1066             : 
    1067             : static GEN
    1068        6139 : FqX_homogenous_eval(GEN P, GEN A, GEN B, GEN T, GEN p)
    1069             : {
    1070        6139 :   long d = degpol(P), i, v = varn(A);
    1071        6139 :   GEN s =  scalar_ZX_shallow(gel(P, d+2), v), Bn = pol_1(v);
    1072       20384 :   for (i = d-1; i >= 0; i--)
    1073             :   {
    1074       14245 :     Bn = FqX_mul(Bn, B, T, p);
    1075       14245 :     s = FqX_add(FqX_mul(s, A, T, p), FqX_Fq_mul(Bn, gel(P,i+2), T, p), T, p);
    1076             :   }
    1077        6139 :   return s;
    1078             : }
    1079             : 
    1080             : static GEN
    1081        1288 : FqX_homogenous_div(GEN P, GEN Q, GEN A, GEN B, GEN T, GEN p)
    1082             : {
    1083        1288 :   GEN z = cgetg(3, t_RFRAC);
    1084        1288 :   long d = degpol(Q)-degpol(P);
    1085        1288 :   gel(z, 1) = FqX_homogenous_eval(P, A, B, T, p);
    1086        1288 :   gel(z, 2) = FqX_homogenous_eval(Q, A, B, T, p);
    1087        1288 :   if (d > 0)
    1088           0 :     gel(z, 1) = FqX_mul(gel(z, 1), FqX_powu(B, d, T, p), T, p);
    1089        1288 :   else if (d < 0)
    1090        1288 :     gel(z, 2) = FqX_mul(gel(z, 2), FqX_powu(B, -d, T, p), T, p);
    1091        1288 :   return z;
    1092             : }
    1093             : 
    1094             : static GEN
    1095        1533 : find_kernel_power(GEN Eba4, GEN Eba6, GEN Eca4, GEN Eca6, ulong ell, struct meqn *MEQN, GEN kpoly, GEN Ib, GEN T, GEN p)
    1096             : {
    1097        1533 :   pari_sp ltop = avma, btop;
    1098             :   GEN a4t, a6t, gtmp;
    1099        1533 :   GEN num_iso = FqX_numer_isog_abscissa(kpoly, Eba4, Eba6, T, p, 0);
    1100        1533 :   GEN mpoly = meqn_j(MEQN, Fq_ellj(Eca4, Eca6, T, p), ell, T, p);
    1101        1533 :   GEN mroots = FqX_roots(mpoly, T, p);
    1102        1533 :   GEN kpoly2 = FqX_sqr(kpoly, T, p);
    1103        1533 :   long i, l1 = lg(mroots);
    1104        1533 :   btop = avma;
    1105        2520 :   for (i = 1; i < l1; i++)
    1106             :   {
    1107             :     GEN h;
    1108        2282 :     GEN tmp = find_isogenous(Eca4, Eca6, ell, MEQN, gel(mroots, i), T, p);
    1109        2282 :     if (!tmp) return gc_NULL(ltop);
    1110        2275 :     a4t =  gel(tmp, 1);
    1111        2275 :     a6t =  gel(tmp, 2);
    1112        2275 :     gtmp = gel(tmp, 3);
    1113             : 
    1114             :     /*check that the kernel kpoly is the good one */
    1115        2275 :     h = FqX_homogenous_eval(gtmp, num_iso, kpoly2, T, p);
    1116        2275 :     if (signe(Fq_elldivpolmod(Eba4, Eba6, ell, h, T, p)))
    1117             :     {
    1118        1288 :       GEN Ic = FqX_homogenous_div(num_iso,kpoly2, numer_i(Ib),denom_i(Ib), T,p);
    1119        1288 :       GEN kpoly_new = FqX_homogenous_eval(gtmp,   numer_i(Ic),denom_i(Ic), T,p);
    1120        1288 :       return gerepilecopy(ltop, mkvecn(5, a4t, a6t, kpoly_new, gtmp, Ic));
    1121             :     }
    1122         987 :     set_avma(btop);
    1123             :   }
    1124         238 :   return gc_NULL(ltop);
    1125             : }
    1126             : 
    1127             : /****************************************************************************/
    1128             : /*                                  TRACE                                   */
    1129             : /****************************************************************************/
    1130             : enum mod_type {MTpathological, MTAtkin, MTElkies, MTone_root, MTroots};
    1131             : 
    1132             : static GEN
    1133         389 : Flxq_study_eqn(GEN mpoly, GEN T, ulong p, long *pt_dG, long *pt_r)
    1134             : {
    1135         389 :   GEN Xq = FlxqX_Frobenius(mpoly, T, p);
    1136         389 :   GEN G  = FlxqX_gcd(FlxX_sub(Xq, pol_x(0), p), mpoly, T, p);
    1137         389 :   *pt_dG = degpol(G);
    1138         389 :   if (!*pt_dG) { *pt_r = FlxqX_ddf_degree(mpoly, Xq, T, p); return NULL; }
    1139         257 :   return gel(FlxqX_roots(G, T, p), 1);
    1140             : }
    1141             : 
    1142             : static GEN
    1143        9205 : Fp_study_eqn(GEN mpoly, GEN p, long *pt_dG, long *pt_r)
    1144             : {
    1145        9205 :   GEN T  = FpX_get_red(mpoly, p);
    1146        9205 :   GEN XP = FpX_Frobenius(T, p);
    1147        9205 :   GEN G  = FpX_gcd(FpX_sub(XP, pol_x(0), p), mpoly, p);
    1148        9205 :   *pt_dG = degpol(G);
    1149        9205 :   if (!*pt_dG) { *pt_r = FpX_ddf_degree(T, XP, p); return NULL; }
    1150        4816 :   return FpX_oneroot(G, p);
    1151             : }
    1152             : 
    1153             : static GEN
    1154        9716 : Fq_study_eqn(GEN mpoly, GEN T, GEN p, long *pt_dG, long *pt_r)
    1155             : {
    1156             :   GEN G;
    1157        9716 :   if (!T) return Fp_study_eqn(mpoly, p, pt_dG, pt_r);
    1158         511 :   if (lgefint(p)==3)
    1159             :   {
    1160         389 :     ulong pp = p[2];
    1161         389 :     GEN Tp = ZXT_to_FlxT(T,pp);
    1162         389 :     GEN mpolyp = ZXX_to_FlxX(mpoly,pp,get_FpX_var(T));
    1163         389 :     G = Flxq_study_eqn(mpolyp, Tp, pp, pt_dG, pt_r);
    1164         389 :     return G ? Flx_to_ZX(G): NULL;
    1165             :   }
    1166             :   else
    1167             :   {
    1168         122 :     GEN Xq = FpXQX_Frobenius(mpoly, T, p);
    1169         122 :     G  = FpXQX_gcd(FpXX_sub(Xq, pol_x(0), p), mpoly, T, p);
    1170         122 :     *pt_dG = degpol(G);
    1171         122 :     if (!*pt_dG) { *pt_r = FpXQX_ddf_degree(mpoly, Xq, T, p); return NULL; }
    1172          72 :     return gel(FpXQX_roots(G, T, p), 1);
    1173             :   }
    1174             : }
    1175             : 
    1176             : /* Berlekamp variant */
    1177             : static GEN
    1178        9723 : study_modular_eqn(long ell, GEN mpoly, GEN T, GEN p, enum mod_type *mt, long *ptr_r)
    1179             : {
    1180        9723 :   pari_sp ltop = avma;
    1181        9723 :   GEN g = gen_0;
    1182        9723 :   *ptr_r = 0; /*gcc -Wall*/
    1183        9723 :   if (!FqX_is_squarefree(mpoly, T, p)) *mt = MTpathological;
    1184             :   else
    1185             :   {
    1186             :     long dG;
    1187        9716 :     g = Fq_study_eqn(mpoly, T, p, &dG, ptr_r);
    1188        9716 :     switch(dG)
    1189             :     {
    1190        4571 :       case 0:  *mt = MTAtkin; break;
    1191         518 :       case 1:  *mt = MTone_root; break;
    1192        4557 :       case 2:  *mt = MTElkies;   break;
    1193          70 :       default: *mt = (dG == ell + 1)? MTroots: MTpathological;
    1194             :     }
    1195             :   }
    1196        9723 :   if (DEBUGLEVEL) switch(*mt)
    1197             :   {
    1198           0 :     case MTone_root: err_printf("One root\t"); break;
    1199           0 :     case MTElkies: err_printf("Elkies\t"); break;
    1200           0 :     case MTroots: err_printf("l+1 roots\t"); break;
    1201           0 :     case MTAtkin: err_printf("Atkin\t"); break;
    1202           0 :     case MTpathological: err_printf("Pathological\n"); break;
    1203             :   }
    1204        9723 :   return g ? gerepilecopy(ltop, g): NULL;
    1205             : }
    1206             : 
    1207             : /*Returns the trace modulo ell^k when ell is an Elkies prime */
    1208             : static GEN
    1209        5075 : find_trace_Elkies_power(GEN a4, GEN a6, ulong ell, long *pt_k, struct meqn *MEQN, GEN g, GEN tr, GEN q, GEN T, GEN p, long smallfact, pari_timer *ti)
    1210             : {
    1211        5075 :   pari_sp ltop = avma, btop;
    1212             :   GEN tmp, Eba4, Eba6, Eca4, Eca6, Ib, kpoly;
    1213        5075 :   long k = *pt_k;
    1214        5075 :   ulong lambda, ellk = upowuu(ell, k), pellk = umodiu(q, ellk);
    1215             :   long cnt;
    1216             : 
    1217        5075 :   if (DEBUGLEVEL) { err_printf("mod %ld", ell); }
    1218        5075 :   Eba4 = a4;
    1219        5075 :   Eba6 = a6;
    1220        5075 :   tmp = find_isogenous(a4,a6, ell, MEQN, g, T, p);
    1221        5075 :   if (!tmp) return gc_NULL(ltop);
    1222        5026 :   Eca4 =  gel(tmp, 1);
    1223        5026 :   Eca6 =  gel(tmp, 2);
    1224        5026 :   kpoly = gel(tmp, 3);
    1225        5026 :   Ib = pol_x(0);
    1226        9569 :   lambda = tr ? find_eigen_value_oneroot(a4, a6, ell, tr, kpoly, T, p):
    1227        4543 :                 find_eigen_value_power(a4, a6, ell, 1, 1, kpoly, T, p);
    1228        5026 :   if (DEBUGLEVEL>1) err_printf(" [%ld ms]", timer_delay(ti));
    1229        5026 :   if (smallfact && smallfact%(long)ell!=0)
    1230             :   {
    1231         378 :     ulong pell = pellk%ell;
    1232         378 :     ulong ap = Fl_add(lambda, Fl_div(pell, lambda, ell), ell);
    1233         378 :     if (Fl_sub(pell, ap, ell)==ell-1) { set_avma(ltop); return mkvecsmall(ap); }
    1234         364 :     if (smallfact < 0 && Fl_add(pell, ap, ell)==ell-1) { set_avma(ltop); return mkvecsmall(ap); }
    1235             :   }
    1236        4998 :   btop = avma;
    1237        6286 :   for (cnt = 2; cnt <= k; cnt++)
    1238             :   {
    1239        1533 :     GEN tmp = find_kernel_power(Eba4, Eba6, Eca4, Eca6, ell, MEQN, kpoly, Ib, T, p);
    1240        1533 :     if (!tmp) { k = cnt-1; break; }
    1241        1288 :     if (DEBUGLEVEL) err_printf(", %Ps", powuu(ell, cnt));
    1242        1288 :     lambda = find_eigen_value_power(a4, a6, ell, cnt, lambda, gel(tmp,3), T, p);
    1243        1288 :     Eba4 = Eca4;
    1244        1288 :     Eba6 = Eca6;
    1245        1288 :     Eca4 = gel(tmp,1);
    1246        1288 :     Eca6 = gel(tmp,2);
    1247        1288 :     kpoly = gel(tmp,4);
    1248        1288 :     Ib = gel(tmp, 5);
    1249        1288 :     if (gc_needed(btop, 1))
    1250             :     {
    1251           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"find_trace_Elkies_power");
    1252           0 :       gerepileall(btop, 6, &Eba4, &Eba6, &Eca4, &Eca6, &kpoly, &Ib);
    1253             :     }
    1254        1288 :     if (DEBUGLEVEL>1) err_printf(" [%ld ms]", timer_delay(ti));
    1255             :   }
    1256        4998 :   set_avma(ltop);
    1257        4998 :   ellk = upowuu(ell, k);
    1258        4998 :   pellk = umodiu(q, ellk);
    1259        4998 :   *pt_k = k;
    1260        4998 :   return mkvecsmall(Fl_add(lambda, Fl_div(pellk, lambda, ellk), ellk));
    1261             : }
    1262             : 
    1263             : /*Returns the possible values of the trace when ell is an Atkin prime, */
    1264             : /*given r the splitting degree of the modular equation at J = E.j */
    1265             : static GEN
    1266        4571 : find_trace_Atkin(ulong ell, long r, GEN q)
    1267             : {
    1268        4571 :   pari_sp ltop = avma;
    1269        4571 :   long nval = 0;
    1270        4571 :   ulong teta, pell = umodiu(q, ell), invp = Fl_inv(pell, ell);
    1271        4571 :   GEN val_pos = cgetg(1+ell, t_VECSMALL), P = gel(factoru(r), 1);
    1272        4571 :   GEN S = mkvecsmall4(0, pell, 0, 1);
    1273        4571 :   GEN U = mkvecsmall3(0, ell-1, 0);
    1274        4571 :   pari_sp btop = avma;
    1275        4571 :   if (r==2 && krouu(ell-pell, ell) < 0)
    1276         812 :     val_pos[++nval] = 0;
    1277       87073 :   for (teta = 1; teta < ell; teta++, set_avma(btop))
    1278             :   {
    1279       82502 :     ulong disc = Fl_sub(Fl_sqr(teta,ell), Fl_mul(4UL,pell,ell), ell);
    1280             :     GEN a;
    1281       82502 :     if (krouu(disc, ell) >= 0) continue;
    1282       40628 :     S[3] = Fl_neg(teta, ell);
    1283       40628 :     U[3] = Fl_mul(invp, teta, ell);
    1284       40628 :     a = Flxq_powu(U, r/P[1], S, ell);
    1285       40628 :     if (!Flx_equal1(a) && Flx_equal1(Flxq_powu(a, P[1], S, ell)))
    1286             :     {
    1287       26586 :       pari_sp av = avma;
    1288       26586 :       long i, l=lg(P);
    1289       45290 :       for (i = 2; i < l; i++, set_avma(av))
    1290       23856 :         if (Flx_equal1(Flxq_powu(U, r/P[i], S, ell))) break;
    1291       26586 :       if (i==l) val_pos[++nval] = teta;
    1292             :     }
    1293             :   }
    1294        4571 :   return gerepileupto(ltop, vecsmall_shorten(val_pos, nval));
    1295             : }
    1296             : 
    1297             : /*Returns the possible traces when there is only one root */
    1298             : static GEN
    1299         518 : find_trace_one_root(ulong ell, GEN q)
    1300             : {
    1301         518 :   ulong a = Fl_double(Fl_sqrt(umodiu(q,ell), ell), ell);
    1302         518 :   return mkvecsmall2(a, ell - a);
    1303             : }
    1304             : 
    1305             : static GEN
    1306          70 : find_trace_lp1_roots(long ell, GEN q)
    1307             : {
    1308          70 :   ulong ell2 = ell * ell, pell = umodiu(q, ell2);
    1309          70 :   ulong a  = Fl_sqrt(pell%ell, ell);
    1310          70 :   ulong pa = Fl_add(Fl_div(pell, a, ell2), a, ell2);
    1311          70 :   return mkvecsmall2(pa, ell2 - pa);
    1312             : }
    1313             : 
    1314             : /*trace modulo ell^k: [], [t] or [t1,...,td] */
    1315             : static GEN
    1316        9723 : find_trace(GEN a4, GEN a6, GEN j, ulong ell, GEN q, GEN T, GEN p, long *ptr_kt,
    1317             :   long smallfact, long vx, long vy)
    1318             : {
    1319        9723 :   pari_sp ltop = avma;
    1320             :   GEN g, meqnj, tr, tr2;
    1321             :   long kt, r;
    1322             :   enum mod_type mt;
    1323             :   struct meqn MEQN;
    1324             :   pari_timer ti;
    1325             : 
    1326        9723 :   kt = maxss((long)(log(expi(q)*M_LN2)/log((double)ell)), 1);
    1327        9723 :   if (DEBUGLEVEL)
    1328           0 :   { err_printf("SEA: Prime %5ld ", ell); timer_start(&ti); }
    1329        9723 :   get_modular_eqn(&MEQN, ell, vx, vy);
    1330        9723 :   meqnj = meqn_j(&MEQN, j, ell, T, p);
    1331        9723 :   g = study_modular_eqn(ell, meqnj, T, p, &mt, &r);
    1332             :   /* If l is an Elkies prime, search for a factor of the l-division polynomial.
    1333             :   * Then deduce the trace by looking for eigenvalues of the Frobenius by
    1334             :   * computing modulo this factor */
    1335        9723 :   switch (mt)
    1336             :   {
    1337             :   case MTone_root:
    1338         518 :     tr2 = find_trace_one_root(ell, q);
    1339         518 :     tr = find_trace_Elkies_power(a4,a6,ell, &kt, &MEQN, g, tr2, q, T, p, smallfact, &ti);
    1340         518 :     if (!tr) { tr = tr2; kt = 1; }
    1341         518 :     break;
    1342             :   case MTElkies:
    1343             :     /* Contrary to MTone_root, may look mod higher powers of ell */
    1344        4557 :     if (abscmpiu(p, 2*ell+3) <= 0)
    1345          14 :       kt = 1; /* Not implemented in this case */
    1346        4557 :     tr = find_trace_Elkies_power(a4,a6,ell, &kt, &MEQN, g, NULL, q, T, p, smallfact, &ti);
    1347        4557 :     if (!tr)
    1348             :     {
    1349          14 :       if (DEBUGLEVEL) err_printf("[fail]\n");
    1350          14 :       return gc_NULL(ltop);
    1351             :     }
    1352        4543 :     break;
    1353             :   case MTroots:
    1354          70 :     tr = find_trace_lp1_roots(ell, q);
    1355          70 :     kt = 2;
    1356          70 :     break;
    1357             :   case MTAtkin:
    1358        4571 :     tr = find_trace_Atkin(ell, r, q);
    1359        4571 :     if (lg(tr)==1) pari_err_PRIME("ellap",p);
    1360        4571 :     kt = 1;
    1361        4571 :     break;
    1362             :   default: /* case MTpathological: */
    1363           7 :     return gc_NULL(ltop);
    1364             :   }
    1365        9702 :   if (DEBUGLEVEL) {
    1366           0 :     long n = lg(tr)-1;
    1367           0 :     if (n > 1 || mt == MTAtkin)
    1368             :     {
    1369           0 :       err_printf("%3ld trace(s)",n);
    1370           0 :       if (DEBUGLEVEL>1) err_printf(" [%ld ms]", timer_delay(&ti));
    1371             :     }
    1372           0 :     if (n > 1) err_printf("\n");
    1373             :   }
    1374        9702 :   *ptr_kt = kt;
    1375        9702 :   return gerepileupto(ltop, tr);
    1376             : }
    1377             : 
    1378             : /* A partition of compile_atkin in baby and giant is represented as the binary
    1379             :    developpement of an integer; if the i-th bit is 1, the i-th prime in
    1380             :    compile-atkin is a baby. The optimum is obtained when the ratio between
    1381             :    the number of possibilities for traces modulo giants (p_g) and babies (p_b)
    1382             :    is near 3/4. */
    1383             : static long
    1384         889 : separation(GEN cnt)
    1385             : {
    1386             :   pari_sp btop;
    1387         889 :   long k = lg(cnt)-1, l = (1L<<k)-1, best_i, i, j;
    1388             :   GEN best_r, P, P3, r;
    1389             : 
    1390         889 :   P = gen_1;
    1391         889 :   for (j = 1; j <= k; ++j) P = mulis(P, cnt[j]);
    1392             :   /* p_b * p_g = P is constant */
    1393         889 :   P3 = mulsi(3, P);
    1394         889 :   btop = avma;
    1395         889 :   best_i = 0;
    1396         889 :   best_r = P3;
    1397       32564 :   for (i = 1; i < l; i++)
    1398             :   {
    1399             :     /* scan all possibilities */
    1400       31759 :     GEN p_b = gen_1;
    1401      272447 :     for (j = 0; j < k; j++)
    1402      240688 :       if (i & (1L<<j)) p_b = mulis(p_b, cnt[1+j]);
    1403       31759 :     r = subii(shifti(sqri(p_b), 2), P3); /* (p_b/p_g - 3/4)*4*P */
    1404       31759 :     if (!signe(r)) { best_i = i; break; }
    1405       31675 :     if (abscmpii(r, best_r) < 0) { best_i = i; best_r = r; }
    1406       31675 :     if (gc_needed(btop, 1))
    1407           0 :       best_r = gerepileuptoint(btop, best_r);
    1408             :   }
    1409         889 :   return best_i;
    1410             : }
    1411             : 
    1412             : /* x VEC defined modulo P (= *P), y VECSMALL modulo q, (q,P) = 1. */
    1413             : /* Update in place:
    1414             :  *   x to vector mod q P congruent to x mod P (resp. y mod q). */
    1415             : /*   P ( <-- qP ) */
    1416             : static void
    1417        1757 : multiple_crt(GEN x, GEN y, GEN q, GEN P)
    1418             : {
    1419        1757 :   pari_sp ltop = avma, av;
    1420        1757 :   long i, j, k, lx = lg(x)-1, ly = lg(y)-1;
    1421             :   GEN  a1, a2, u, v, A2X;
    1422        1757 :   (void)bezout(P,q,&u,&v);
    1423        1757 :   a1 = mulii(P,u);
    1424        1757 :   a2 = mulii(q,v); A2X = ZC_Z_mul(x, a2);
    1425        1757 :   av = avma; affii(mulii(P,q), P);
    1426       61733 :   for (i = 1, k = 1; i <= lx; i++, set_avma(av))
    1427             :   {
    1428       59976 :     GEN a2x = gel(A2X,i);
    1429     1016960 :     for (j = 1; j <= ly; ++j)
    1430             :     {
    1431      956984 :       GEN t = Fp_add(Fp_mulu(a1, y[j], P), a2x, P);
    1432      956984 :       affii(t, gel(x, k++));
    1433             :     }
    1434             :   }
    1435        1757 :   setlg(x, k); set_avma(ltop);
    1436        1757 : }
    1437             : 
    1438             : /****************************************************************************/
    1439             : /*                              MATCH AND SORT                              */
    1440             : /****************************************************************************/
    1441             : 
    1442             : static GEN
    1443        1778 : possible_traces(GEN compile, GEN mask, GEN *P, int larger)
    1444             : {
    1445        1778 :   GEN V, Pfinal = gen_1, C = shallowextract(compile, mask);
    1446        1778 :   long i, lfinal = 1, lC = lg(C), lP;
    1447        1778 :   pari_sp av = avma;
    1448             : 
    1449        5313 :   for (i = 1; i < lC; i++)
    1450             :   {
    1451        3535 :     GEN c = gel(C,i), t;
    1452        3535 :     Pfinal = mulii(Pfinal, gel(c,1));
    1453        3535 :     t = muluu(lfinal, lg(gel(c,2))-1);
    1454        3535 :     lfinal = itou(t);
    1455             :   }
    1456        1778 :   Pfinal = gerepileuptoint(av, Pfinal);
    1457        1778 :   if (larger)
    1458         889 :     lP = lgefint(shifti(Pfinal,1));
    1459             :   else
    1460         889 :     lP = lgefint(Pfinal);
    1461        1778 :   lfinal++;
    1462             :   /* allocate room for final result */
    1463        1778 :   V = cgetg(lfinal, t_VEC);
    1464        1778 :   for (i = 1; i < lfinal; i++) gel(V,i) = cgeti(lP);
    1465             : 
    1466             :   {
    1467        1778 :     GEN c = gel(C,1), v = gel(c,2);
    1468        1778 :     long l = lg(v);
    1469        1778 :     for (i = 1; i < l; i++) affsi(v[i], gel(V,i));
    1470        1778 :     setlg(V, l); affii(gel(c,1), Pfinal); /* reset Pfinal */
    1471             :   }
    1472        3535 :   for (i = 2; i < lC; i++)
    1473             :   {
    1474        1757 :     GEN c = gel(C,i);
    1475        1757 :     multiple_crt(V, gel(c,2), gel(c,1), Pfinal); /* Pfinal updated! */
    1476             :   }
    1477        1778 :   *P = Pfinal; return V;
    1478             : }
    1479             : 
    1480             : static GEN
    1481      189399 : cost(long mask, GEN cost_vec)
    1482             : {
    1483      189399 :   pari_sp ltop = avma;
    1484             :   long i;
    1485      189399 :   GEN c = gen_1;
    1486     2007950 :   for (i = 1; i < lg(cost_vec); i++)
    1487     1818551 :     if (mask&(1L<<(i-1)))
    1488      788732 :       c = mulis(c, cost_vec[i]);
    1489      189399 :   return gerepileuptoint(ltop, c);
    1490             : }
    1491             : 
    1492             : static GEN
    1493      152166 : value(long mask, GEN atkin, long k)
    1494             : {
    1495      152166 :   pari_sp ltop = avma;
    1496             :   long i;
    1497      152166 :   GEN c = gen_1;
    1498     1613626 :   for (i = 1; i <= k; i++)
    1499     1461460 :     if (mask&(1L<<(i-1)))
    1500      636538 :       c = mulii(c, gmael(atkin, i, 1));
    1501      152166 :   return gerepileuptoint(ltop, c);
    1502             : }
    1503             : 
    1504             : static void
    1505       74739 : set_cost(GEN B, long b, GEN cost_vec, long *pi)
    1506             : {
    1507       74739 :   pari_sp av = avma;
    1508       74739 :   GEN costb = cost(b, cost_vec);
    1509       74739 :   long i = *pi;
    1510       74739 :   while (cmpii(costb, cost(B[i], cost_vec)) < 0) --i;
    1511       74739 :   B[++i] = b;
    1512       74739 :   *pi = i; set_avma(av);
    1513       74739 : }
    1514             : 
    1515             : static GEN
    1516        1862 : get_lgatkin(GEN compile_atkin, long k)
    1517             : {
    1518        1862 :   GEN v = cgetg(k+1, t_VECSMALL);
    1519             :   long j;
    1520        1862 :   for (j = 1; j <= k; ++j) v[j] = lg(gmael(compile_atkin, j, 2))-1;
    1521        1862 :   return v;
    1522             : }
    1523             : 
    1524             : static GEN
    1525         973 : champion(GEN atkin, long k, GEN bound_champ)
    1526             : {
    1527         973 :   const long two_k = 1L<<k;
    1528         973 :   pari_sp ltop = avma;
    1529             :   long i, j, n, i1, i2;
    1530         973 :   GEN B, Bp, cost_vec, res = NULL;
    1531             : 
    1532         973 :   cost_vec = get_lgatkin(atkin, k);
    1533         973 :   if (k == 1) return mkvec2(gen_1, utoipos(cost_vec[1]));
    1534             : 
    1535         959 :   B  = zero_zv(two_k);
    1536         959 :   Bp = zero_zv(two_k);
    1537         959 :   Bp[2] = 1;
    1538        4144 :   for (n = 2, j = 2; j <= k; j++)
    1539             :   {
    1540             :     long b;
    1541        3185 :     i = 1;
    1542       72268 :     for (i1 = 2, i2 = 1; i1 <= n; )
    1543             :     {
    1544       65898 :       pari_sp av = avma;
    1545       65898 :       long b1 = Bp[i1], b2 = Bp[i2]|(1L<<(j-1));
    1546       65898 :       if (cmpii(value(b1, atkin, k), value(b2, atkin, k)) < 0)
    1547       35777 :         { b = b1; i1++; } else { b = b2; i2++; }
    1548       65898 :       set_avma(av);
    1549       65898 :       set_cost(B, b, cost_vec, &i);
    1550             :     }
    1551       12026 :     for ( ; i2 <= n; i2++)
    1552             :     {
    1553        8841 :       b = Bp[i2]|(1L<<(j-1));
    1554        8841 :       set_cost(B, b, cost_vec, &i);
    1555             :     }
    1556        3185 :     n = i;
    1557       57694 :     for (i = 1; i <= n; i++)
    1558       54509 :       Bp[i] = B[i];
    1559             :   }
    1560      232715 :   for (i = 1; i <= two_k; i++)
    1561      231756 :     if (B[i])
    1562             :     {
    1563       16506 :       GEN b = cost (B[i], cost_vec);
    1564       16506 :       GEN v = value(B[i], atkin, k);
    1565       16506 :       if (cmpii(v, bound_champ) <=0) continue;
    1566        1883 :       if (res && gcmp(b, gel(res, 2)) >=0) continue;
    1567         959 :       res = mkvec2(utoi(B[i]), b);
    1568             :     }
    1569         959 :   return gerepilecopy(ltop, res);
    1570             : }
    1571             : 
    1572             : static GEN
    1573        1778 : compute_diff(GEN v)
    1574             : {
    1575        1778 :   pari_sp av = avma;
    1576        1778 :   long i, l = lg(v) - 1;
    1577        1778 :   GEN diff = cgetg(l, t_VEC);
    1578        1778 :   for (i = 1; i < l; i++) gel(diff, i) = subii(gel(v, i+1), gel(v, i));
    1579        1778 :   return gerepileupto(av, ZV_sort_uniq(diff));
    1580             : }
    1581             : 
    1582             : static int
    1583       16198 : cmp_atkin(void*E, GEN a, GEN b)
    1584             : {
    1585       16198 :   long ta=typ(a)==t_INT, tb=typ(b)==t_INT, c;
    1586             :   (void) E;
    1587       16198 :   if (ta || tb) return ta-tb;
    1588        5173 :   c = lg(gel(a,2)) - lg(gel(b,2));
    1589        5173 :   if (c) return c;
    1590         728 :   return cmpii(gel(b,1), gel(a,1));
    1591             : }
    1592             : 
    1593             : static void
    1594        3864 : add_atkin(GEN atkin, GEN trace, long *nb)
    1595             : {
    1596        3864 :   long l = lg(atkin)-1;
    1597        3864 :   long i, k = gen_search(atkin, trace, 1, NULL, cmp_atkin);
    1598        3864 :   if (k==0 || k > l) return;
    1599       75705 :   for (i = l; i > k; i--)
    1600       71841 :     gel(atkin,i) = gel(atkin,i-1);
    1601        3864 :   if (typ(gel(atkin,l))==t_INT) (*nb)++;
    1602        3864 :   gel(atkin,k) = trace;
    1603             : }
    1604             : 
    1605             : /* V = baby / giant, P = Pb / Pg */
    1606             : static GEN
    1607        1778 : BSGS_pre(GEN *pdiff, GEN V, GEN P, void *E, const struct bb_group *grp)
    1608             : {
    1609        1778 :   GEN diff = compute_diff(V);
    1610        1778 :   GEN pre = cgetg(lg(diff), t_VEC);
    1611        1778 :   long i, l = lg(diff);
    1612        1778 :   gel(pre, 1) = grp->pow(E, P, gel(diff, 1));
    1613             :   /* what we'd _really_ want here is a hashtable diff[i] -> pre[i]  */
    1614       33957 :   for (i = 2; i < l; i++)
    1615             :   {
    1616       32179 :     pari_sp av = avma;
    1617       32179 :     GEN d = subii(gel(diff, i), gel(diff, i-1));
    1618       32179 :     GEN Q = grp->mul(E, gel(pre, i-1), grp->pow(E, P, d));
    1619       32179 :     gel(pre, i) = gerepilecopy(av, Q);
    1620             :   }
    1621        1778 :   *pdiff = diff; return pre;
    1622             : }
    1623             : 
    1624             : /* u = trace_elkies, Mu = prod_elkies. Let caller collect garbage */
    1625             : /* Match & sort: variant from Lercier's thesis, section 11.2.3 */
    1626             : /* baby/giant/table updated in place: this routines uses
    1627             :  *   size(baby)+size(giant)+size(table)+size(table_ind) + O(log p)
    1628             :  * bits of stack */
    1629             : static GEN
    1630         945 : match_and_sort(GEN compile_atkin, GEN Mu, GEN u, GEN q, void *E, const struct bb_group *grp)
    1631             : {
    1632             :   pari_sp av1, av2;
    1633         945 :   GEN baby, giant, SgMb, Mb, Mg, den, Sg, dec_inf, div, pp1 = addiu(q,1);
    1634             :   GEN P, Pb, Pg, point, diff, pre, table, table_ind;
    1635         945 :   long best_i, i, lbaby, lgiant, k = lg(compile_atkin)-1;
    1636         945 :   GEN bound = sqrti(shifti(q, 2)), card;
    1637         945 :   const long lcard = 100;
    1638         945 :   long lq = lgefint(q), nbcard;
    1639             :   pari_timer ti;
    1640             : 
    1641         945 :   if (k == 1)
    1642             :   { /*only one Atkin prime, check the cardinality with random points */
    1643          56 :     GEN r = gel(compile_atkin, 1), r1 = gel(r,1), r2 = gel(r,2);
    1644          56 :     long l = lg(r2), j;
    1645          56 :     GEN card = cgetg(l, t_VEC), Cs2, C, U;
    1646          56 :     Z_chinese_pre(Mu, r1, &C,&U, NULL);
    1647          56 :     Cs2 = shifti(C, -1);
    1648         378 :     for (j = 1, i = 1; i < l; i++)
    1649             :     {
    1650         322 :       GEN t = Z_chinese_post(u, stoi(r2[i]), C, U, NULL);
    1651         322 :       t = Fp_center_i(t, C, Cs2);
    1652         322 :       if (abscmpii(t, bound) <= 0) gel(card, j++) = subii(pp1, t);
    1653             :     }
    1654          56 :     setlg(card, j);
    1655          56 :     return gen_select_order(card, E, grp);
    1656             :   }
    1657         889 :   if (DEBUGLEVEL>=2) timer_start(&ti);
    1658         889 :   av1 = avma;
    1659         889 :   best_i = separation( get_lgatkin(compile_atkin, k) );
    1660         889 :   set_avma(av1);
    1661             : 
    1662         889 :   baby  = possible_traces(compile_atkin, utoi(best_i), &Mb, 1);
    1663         889 :   giant = possible_traces(compile_atkin, subiu(int2n(k), best_i+1), &Mg, 0);
    1664         889 :   lbaby = lg(baby);
    1665         889 :   lgiant = lg(giant);
    1666         889 :   den = Fp_inv(Fp_mul(Mu, Mb, Mg), Mg);
    1667         889 :   av2 = avma;
    1668      527037 :   for (i = 1; i < lgiant; i++, set_avma(av2))
    1669      526148 :     affii(Fp_mul(gel(giant,i), den, Mg), gel(giant,i));
    1670         889 :   ZV_sort_inplace(giant);
    1671         889 :   Sg = Fp_mul(negi(u), den, Mg);
    1672         889 :   den = Fp_inv(Fp_mul(Mu, Mg, Mb), Mb);
    1673         889 :   dec_inf = divii(mulii(Mb,addii(Mg,shifti(Sg,1))), shifti(Mg,1));
    1674         889 :   togglesign(dec_inf); /* now, dec_inf = ceil(- (Mb/2 + Sg Mb/Mg) ) */
    1675         889 :   div = mulii(truedivii(dec_inf, Mb), Mb);
    1676         889 :   av2 = avma;
    1677      378749 :   for (i = 1; i < lbaby; i++, set_avma(av2))
    1678             :   {
    1679      377860 :     GEN b = addii(Fp_mul(Fp_sub(gel(baby,i), u, Mb), den, Mb), div);
    1680      377860 :     if (cmpii(b, dec_inf) < 0) b = addii(b, Mb);
    1681      377860 :     affii(b, gel(baby,i));
    1682             :   }
    1683         889 :   ZV_sort_inplace(baby);
    1684             : 
    1685         889 :   SgMb = mulii(Sg, Mb);
    1686         889 :   card = cgetg(lcard+1,t_VEC);
    1687         889 :   for (i = 1; i <= lcard; i++) gel(card,i) = cgetipos(lq+1);
    1688             : 
    1689         889 :   av2 = avma;
    1690             : MATCH_RESTART:
    1691         889 :   set_avma(av2);
    1692         889 :   nbcard = 0;
    1693         889 :   P = grp->rand(E);
    1694         889 :   point = grp->pow(E,P, Mu);
    1695         889 :   Pb = grp->pow(E,point, Mg);
    1696         889 :   Pg = grp->pow(E,point, Mb);
    1697             :   /* Precomputation for babies */
    1698         889 :   pre = BSGS_pre(&diff, baby, Pb, E, grp);
    1699             : 
    1700             :   /*Now we compute the table of babies, this table contains only the */
    1701             :   /*lifted x-coordinate of the points in order to use less memory */
    1702         889 :   table = cgetg(lbaby, t_VECSMALL);
    1703         889 :   av1 = avma;
    1704             :   /* (p+1 - u - Mu*Mb*Sg) P - (baby[1]) Pb */
    1705         889 :   point = grp->pow(E,P, subii(subii(pp1, u), mulii(Mu, addii(SgMb, mulii(Mg, gel(baby,1))))));
    1706         889 :   table[1] = grp->hash(gel(point,1));
    1707      377860 :   for (i = 2; i < lbaby; i++)
    1708             :   {
    1709      376971 :     GEN d = subii(gel(baby, i), gel(baby, i-1));
    1710      376971 :     point =  grp->mul(E, point, grp->pow(E, gel(pre, ZV_search(diff, d)), gen_m1));
    1711      376971 :     table[i] = grp->hash(gel(point,1));
    1712      376971 :     if (gc_needed(av1,3))
    1713             :     {
    1714           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"match_and_sort, baby = %ld", i);
    1715           0 :       point = gerepileupto(av1, point);
    1716             :     }
    1717             :   }
    1718         889 :   set_avma(av1);
    1719             :   /* Precomputations for giants */
    1720         889 :   pre = BSGS_pre(&diff, giant, Pg, E, grp);
    1721             : 
    1722             :   /* Look for a collision among the x-coordinates */
    1723         889 :   table_ind = vecsmall_indexsort(table);
    1724         889 :   table = perm_mul(table,table_ind);
    1725             : 
    1726         889 :   av1 = avma;
    1727         889 :   point = grp->pow(E, Pg, gel(giant, 1));
    1728      526148 :   for (i = 1; ; i++)
    1729      525259 :   {
    1730             :     GEN d;
    1731      526148 :     long h = grp->hash(gel(point, 1));
    1732      526148 :     long s = zv_search(table, h);
    1733      526148 :     if (s) {
    1734         889 :       while (table[s] == h && s) s--;
    1735        1778 :       for (s++; s < lbaby && table[s] == h; s++)
    1736             :       {
    1737         889 :         GEN B = gel(baby,table_ind[s]), G = gel(giant,i);
    1738         889 :         GEN GMb = mulii(G, Mb), BMg = mulii(B, Mg);
    1739         889 :         GEN Be = subii(subii(pp1, u), mulii(Mu, addii(SgMb, BMg)));
    1740         889 :         GEN Bp = grp->pow(E,P, Be);
    1741             :         /* p+1 - u - Mu (Sg Mb + GIANT Mb + BABY Mg) */
    1742         889 :         if (gequal(gel(Bp,1),gel(point,1)))
    1743             :         {
    1744         889 :           GEN card1 = subii(Be, mulii(Mu, GMb));
    1745         889 :           GEN card2 = addii(card1, mulii(mulsi(2,Mu), GMb));
    1746         889 :           if (abscmpii(subii(pp1, card1), bound) <= 0)
    1747         777 :             affii(card1, gel(card, ++nbcard));
    1748         889 :           if (nbcard >= lcard) goto MATCH_RESTART;
    1749         889 :           if (abscmpii(subii(pp1, card2), bound) <= 0)
    1750         476 :             affii(card2, gel(card, ++nbcard));
    1751         889 :           if (nbcard >= lcard) goto MATCH_RESTART;
    1752             :         }
    1753             :       }
    1754             :     }
    1755      526148 :     if (i==lgiant-1) break;
    1756      525259 :     d = subii(gel(giant, i+1), gel(giant, i));
    1757      525259 :     point = grp->mul(E,point, gel(pre, ZV_search(diff, d)));
    1758      525259 :     if (gc_needed(av1,3))
    1759             :     {
    1760           0 :       if(DEBUGMEM>1) pari_warn(warnmem,"match_and_sort, giant = %ld", i);
    1761           0 :       point = gerepileupto(av1, point);
    1762             :     }
    1763             :   }
    1764         889 :   setlg(card, nbcard+1);
    1765         889 :   if (DEBUGLEVEL>=2) timer_printf(&ti,"match_and_sort");
    1766         889 :   return gen_select_order(card, E, grp);
    1767             : }
    1768             : 
    1769             : static GEN
    1770         994 : get_bound_bsgs(long lp)
    1771             : {
    1772             :   GEN B;
    1773         994 :   if (lp <= 160)
    1774         966 :     B = divru(powru(dbltor(1.048), lp), 9);
    1775          28 :   else if (lp <= 192)
    1776          21 :     B = divrr(powru(dbltor(1.052), lp), dbltor(16.65));
    1777             :   else
    1778           7 :     B = mulrr(powru(dbltor(1.035), minss(lp,307)), dbltor(1.35));
    1779         994 :   return mulru(B, 1000000);
    1780             : }
    1781             : 
    1782             : /*FIXME: the name of the function does not quite match what it does*/
    1783             : static const struct bb_group *
    1784         945 : get_FqE_group(void ** pt_E, GEN a4, GEN a6, GEN T, GEN p)
    1785             : {
    1786         945 :   if (!T) return get_FpE_group(pt_E,a4,a6,p);
    1787          42 :   else if (lgefint(p)==3)
    1788             :   {
    1789          34 :     ulong pp = uel(p,2);
    1790          34 :     GEN Tp = ZXT_to_FlxT(T,pp);
    1791          34 :     return get_FlxqE_group(pt_E, Fq_to_Flx(a4, Tp, pp), Fq_to_Flx(a6, Tp, pp),
    1792             :                            Tp, pp);
    1793             :   }
    1794           8 :   return get_FpXQE_group(pt_E,a4,a6,T,p);
    1795             : }
    1796             : 
    1797             : /* E is an elliptic curve defined over Z or over Fp in ellinit format, defined
    1798             :  * by the equation E: y^2 + a1*x*y + a2*y = x^3 + a2*x^2 + a4*x + a6
    1799             :  * p is a prime number
    1800             :  * set smallfact to stop whenever a small factor of the order, not dividing smallfact,
    1801             :  * is detected. Useful when searching for a good curve for cryptographic
    1802             :  * applications */
    1803             : GEN
    1804        1022 : Fq_ellcard_SEA(GEN a4, GEN a6, GEN q, GEN T, GEN p, long smallfact)
    1805             : {
    1806        1022 :   const long MAX_ATKIN = 21;
    1807        1022 :   pari_sp ltop = avma, btop;
    1808             :   long ell, i, nb_atkin, vx,vy;
    1809             :   GEN TR, TR_mod, compile_atkin, bound, bound_bsgs, champ;
    1810        1022 :   GEN prod_atkin = gen_1, max_traces = gen_0;
    1811             :   GEN j;
    1812        1022 :   double bound_gr = 1.;
    1813        1022 :   const double growth_factor = 1.26;
    1814             :   forprime_t TT;
    1815             : 
    1816        1022 :   j = Fq_ellj(a4, a6, T, p);
    1817        1022 :   if (signe(j) == 0 || signe(Fq_sub(j, utoi(1728), T, p)) == 0)
    1818           0 :     return T ? FpXQ_ellcard(Fq_to_FpXQ(a4, T, p), Fq_to_FpXQ(a6, T, p), T, p)
    1819          14 :              : Fp_ellcard(a4, a6, p);
    1820             :   /*First compute the trace modulo 2 */
    1821        1008 :   switch(FqX_nbroots(rhs(a4, a6, 0), T, p))
    1822             :   {
    1823             :   case 3: /* bonus time: 4 | #E(Fq) = q+1 - t */
    1824          77 :     i = mod4(q)+1; if (i > 2) i -= 4;
    1825          77 :     TR_mod = utoipos(4);
    1826          77 :     TR = stoi(i); break;
    1827             :   case 1:
    1828         490 :     TR_mod = gen_2;
    1829         490 :     TR = gen_0; break;
    1830             :   default : /* 0 */
    1831         441 :     TR_mod = gen_2;
    1832         441 :     TR = gen_1; break;
    1833             :   }
    1834        1008 :   if (odd(smallfact) && !mpodd(TR))
    1835             :   {
    1836          14 :     if (DEBUGLEVEL) err_printf("Aborting: #E(Fq) divisible by 2\n");
    1837          14 :     set_avma(ltop); return gen_0;
    1838             :   }
    1839         994 :   vy = fetch_var();
    1840         994 :   vx = fetch_var_higher();
    1841             : 
    1842             :   /* compile_atkin is a vector containing informations about Atkin primes,
    1843             :    * informations about Elkies primes lie in Mod(TR, TR_mod). */
    1844         994 :   u_forprime_init(&TT, 3, ULONG_MAX);
    1845         994 :   bound = sqrti(shifti(q, 4));
    1846         994 :   bound_bsgs = get_bound_bsgs(expi(q));
    1847         994 :   compile_atkin = zerovec(MAX_ATKIN); nb_atkin = 0;
    1848         994 :   btop = avma;
    1849       10724 :   while ( (ell = u_forprime_next(&TT)) )
    1850             :   {
    1851        9730 :     long ellkt, kt = 1, nbtrace;
    1852             :     GEN trace_mod;
    1853        9758 :     if (absequalui(ell, p)) continue;
    1854        9723 :     trace_mod = find_trace(a4, a6, j, ell, q, T, p, &kt, smallfact, vx,vy);
    1855        9723 :     if (!trace_mod) continue;
    1856             : 
    1857        9702 :     nbtrace = lg(trace_mod) - 1;
    1858        9702 :     ellkt = (long)upowuu(ell, kt);
    1859        9702 :     if (nbtrace == 1)
    1860             :     {
    1861        5838 :       long t_mod_ellkt = trace_mod[1];
    1862        5838 :       if (smallfact && smallfact%ell!=0)
    1863             :       { /* does ell divide q + 1 - t ? */
    1864         385 :         long q_mod_ell_plus_one = umodiu(q,ell) + 1;
    1865         385 :         ulong  card_mod_ell = umodsu(q_mod_ell_plus_one - t_mod_ellkt, ell);
    1866         385 :         ulong tcard_mod_ell = 1;
    1867         385 :         if (card_mod_ell && smallfact < 0)
    1868         133 :           tcard_mod_ell = umodsu(q_mod_ell_plus_one + t_mod_ellkt, ell);
    1869         385 :         if (!card_mod_ell || !tcard_mod_ell)
    1870             :         {
    1871          28 :           if (DEBUGLEVEL)
    1872           0 :             err_printf("\nAborting: #E%s(Fq) divisible by %ld\n",
    1873             :                        tcard_mod_ell ? "" : "_twist", ell);
    1874          28 :           delete_var();
    1875          28 :           delete_var();
    1876        1022 :           set_avma(ltop); return gen_0;
    1877             :         }
    1878             :       }
    1879        5810 :       (void)Z_incremental_CRT(&TR, t_mod_ellkt, &TR_mod, ellkt);
    1880        5810 :       if (DEBUGLEVEL)
    1881           0 :         err_printf(", missing %ld bits\n",expi(bound)-expi(TR_mod));
    1882             :     }
    1883             :     else
    1884             :     {
    1885        3864 :       add_atkin(compile_atkin, mkvec2(utoipos(ellkt), trace_mod), &nb_atkin);
    1886        3864 :       prod_atkin = value(-1, compile_atkin, nb_atkin);
    1887             :     }
    1888        9674 :     if (cmpii(mulii(TR_mod, prod_atkin), bound) > 0)
    1889             :     {
    1890             :       GEN bound_tr;
    1891        1008 :       if (!nb_atkin)
    1892             :       {
    1893          21 :         delete_var();
    1894          21 :         delete_var();
    1895          21 :         return gerepileuptoint(ltop, subii(addiu(q, 1), TR));
    1896             :       }
    1897         987 :       bound_tr = mulrr(bound_bsgs, dbltor(bound_gr));
    1898         987 :       bound_gr *= growth_factor;
    1899         987 :       if (signe(max_traces))
    1900             :       {
    1901          42 :         max_traces = divis(muliu(max_traces,nbtrace), ellkt);
    1902          42 :         if (DEBUGLEVEL>=3)
    1903           0 :           err_printf("At least %Ps remaining possibilities.\n",max_traces);
    1904             :       }
    1905         987 :       if (cmpir(max_traces, bound_tr) < 0)
    1906             :       {
    1907         973 :         GEN bound_atkin = truedivii(bound, TR_mod);
    1908         973 :         champ = champion(compile_atkin, nb_atkin, bound_atkin);
    1909         973 :         max_traces = gel(champ,2);
    1910         973 :         if (DEBUGLEVEL>=2)
    1911           0 :           err_printf("%Ps remaining possibilities.\n", max_traces);
    1912         973 :         if (cmpir(max_traces, bound_tr) < 0)
    1913             :         {
    1914         945 :           GEN res, cat = shallowextract(compile_atkin, gel(champ,1));
    1915             :           const struct bb_group *grp;
    1916             :           void *E;
    1917         945 :           if (DEBUGLEVEL)
    1918           0 :             err_printf("Match and sort for %Ps possibilities.\n", max_traces);
    1919         945 :           delete_var();
    1920         945 :           delete_var();
    1921         945 :           grp = get_FqE_group(&E,a4,a6,T,p);
    1922         945 :           res = match_and_sort(cat, TR_mod, TR, q, E, grp);
    1923         945 :           return gerepileuptoint(ltop, res);
    1924             :         }
    1925             :       }
    1926             :     }
    1927        8708 :     if (gc_needed(btop, 1))
    1928           0 :       gerepileall(btop,5, &TR,&TR_mod, &compile_atkin, &max_traces, &prod_atkin);
    1929             :   }
    1930             :   return NULL;/*LCOV_EXCL_LINE*/
    1931             : }
    1932             : 
    1933             : GEN
    1934         973 : Fp_ellcard_SEA(GEN a4, GEN a6, GEN p, long smallfact)
    1935         973 : { return Fq_ellcard_SEA(a4, a6, p, NULL, p, smallfact); }

Generated by: LCOV version 1.13