static GEN
F2xq_elltrace_Harley(GEN a6, GEN T2)
{
pari_sp ltop = avma;
pari_timer ti;
GEN T, sqx;
GEN x, x2, t;
long n = F2x_degree(T2), N = ((n + 1)>>1) + 2;
long ispcyc;
if (n==1) return gen_m1;
if (n==2) return F2x_degree(a6) ? gen_1 : stoi(-3);
if (n==3) return F2x_degree(a6) ? (F2xq_trace(a6,T2) ? stoi(-3): gen_1)
: stoi(5);
timer_start(&ti);
sqx = mkvec2(F2xq_sqrt(polx_F2x(T2[1]),T2), T2);
if (DEBUGLEVEL>1) timer_printf(&ti,"Sqrtx");
ispcyc = zx_is_pcyc(F2x_to_Flx(T2));
T = ispcyc? F2x_to_ZX(T2): F2x_canonlift(T2, N-2);
if (DEBUGLEVEL>1) timer_printf(&ti,"Teich");
T = FpX_get_red(T, int2n(N));
if (DEBUGLEVEL>1) timer_printf(&ti,"Barrett");
x = solve_AGM_eqn(F2x_to_ZX(a6), N-2, T, sqx);
if (DEBUGLEVEL>1) timer_printf(&ti,"Lift");
x2 = ZX_Z_add_shallow(ZX_shifti(x,2), gen_1);
t = (ispcyc? Z2XQ_invnorm_pcyc: Z2XQ_invnorm)(x2, T, N);
if (DEBUGLEVEL>1) timer_printf(&ti,"Norm");
if (cmpii(sqri(t), int2n(n + 2)) > 0)
t = subii(t, int2n(N));
return gerepileuptoint(ltop, t);
}
/* Finds a random non-singular point on E */
GEN
random_F2xqE(GEN a, GEN a6, GEN T)
{
pari_sp ltop = avma;
GEN x, y, rhs, u;
do
{
avma= ltop;
x = random_F2x(F2x_degree(T),T[1]);
if (typ(a) == t_VECSMALL)
{
GEN a2 = a, x2;
if (!lgpol(x))
{ avma=ltop; retmkvec2(pol0_Flx(T[1]), F2xq_sqrt(a6,T)); }
u = x; x2 = F2xq_sqr(x, T);
rhs = F2x_add(F2xq_mul(x2,F2x_add(x,a2),T),a6);
rhs = F2xq_div(rhs,x2,T);
}
else
{
GEN a3 = gel(a,1), a4 = gel(a,2), a3i = gel(a,3), u2i;
u = a3; u2i = F2xq_sqr(a3i,T);
rhs = F2x_add(F2xq_mul(x,F2x_add(F2xq_sqr(x,T),a4),T),a6);
rhs = F2xq_mul(rhs,u2i,T);
}
} while (F2xq_trace(rhs,T));
y = F2xq_mul(F2xq_Artin_Schreier(rhs, T), u, T);
return gerepilecopy(ltop, mkvec2(x, y));
}
static GEN
_can_invd(void *E, GEN V, GEN v, GEN q, long M)
{
GEN F;
(void)E; (void)q;
F = mkvec2(ZX_shifti(gel(v,2),1), ZX_shifti(RgX_shift_shallow(gel(v,3),1),1));
return gen_Z2X_Dixon(F, V, M, NULL, _can_lin, _can_lins, _can_invls);
}
static GEN
F2xqE_neg_i(GEN P, GEN a)
{
GEN LHS;
if (ell_is_inf(P)) return P;
LHS = typ(a)==t_VECSMALL ? gel(P,1): gel(a,1);
return mkvec2(gel(P,1), F2x_add(LHS, gel(P,2)));
}
GEN
F2xqE_neg(GEN P, GEN a, GEN T)
{
GEN LHS;
(void) T;
if (ell_is_inf(P)) return ellinf();
LHS = typ(a)==t_VECSMALL ? gel(P,1): gel(a,1);
return mkvec2(gcopy(gel(P,1)), F2x_add(LHS, gel(P,2)));
}
GEN
nfrootsall_and_pr(GEN nf, GEN pol)
{
GEN J1,J2, pr, T;
pari_sp av = avma;
GEN z = gerepileupto(av, nfsqff(checknf(nf), pol, 2));
if (lg(z) == 1) return NULL;
(void)nf_pick_prime(1, nf, unifpol(nf, pol, t_COL), 2,
&J1, &J2, &pr, &T);
return mkvec2(z, pr);
}
GEN
shallowconcat(GEN x, GEN y)
{
long tx=typ(x),ty=typ(y),lx=lg(x),ly=lg(y),i;
GEN z,p1;
if (tx==t_STR || ty==t_STR) return strconcat(x,y);
if (tx==t_LIST || ty==t_LIST) return listconcat(x,y);
if (tx==t_MAT && lx==1)
{
if (ty!=t_VEC) return gtomat(y);
if (ly==1) return cgetg(1, t_MAT);
err_cat(x,y);
}
if (ty==t_MAT && ly==1)
{
if (tx!=t_VEC) return gtomat(x);
if (lx==1) return cgetg(1, t_MAT);
err_cat(x,y);
}
if (tx == ty)
{
if (tx == t_MAT)
{ if (lgcols(x) != lgcols(y)) err_cat(x,y); }
else
if (!is_matvec_t(tx) && tx != t_VECSMALL) return mkvec2(x, y);
z=cgetg(lx+ly-1,tx);
for (i=1; i<lx; i++) z[i] = x[i];
for (i=1; i<ly; i++) z[lx+i-1]= y[i];
return z;
}
if (! is_matvec_t(tx))
{
if (! is_matvec_t(ty)) return mkvec2(x, y);
z=cgetg(ly+1,ty);
if (ty != t_MAT) p1 = x;
else
{
if (lgcols(y)!=2) err_cat(x,y);
p1 = mkcol(x);
}
for (i=2; i<=ly; i++) z[i] = y[i-1];
gel(z, 1) = p1; return z;
}
if (! is_matvec_t(ty))
{
z=cgetg(lx+1,tx);
if (tx != t_MAT) p1 = y;
else
{
if (lgcols(x)!=2) err_cat(x,y);
p1 = mkcol(y);
}
for (i=1; i<lx; i++) z[i]=x[i];
gel(z, lx) = p1; return z;
}
switch(tx)
{
case t_VEC:
switch(ty)
{
case t_COL:
if (lx<=2) return (lx==1)? y: shallowconcat(gel(x,1),y);
if (ly>=3) break;
return (ly==1)? x: shallowconcat(x,gel(y,1));
case t_MAT:
z=cgetg(ly,t_MAT); if (lx != ly) break;
for (i=1; i<ly; i++) gel(z,i) = shallowconcat(gel(x,i),gel(y,i));
return z;
}
break;
case t_COL:
switch(ty)
{
case t_VEC:
if (lx<=2) return (lx==1)? y: shallowconcat(gel(x,1), y);
if (ly>=3) break;
return (ly==1)? x: shallowconcat(x, gel(y,1));
case t_MAT:
if (lx != lgcols(y)) break;
z=cgetg(ly+1,t_MAT); gel(z,1) = x;
for (i=2; i<=ly; i++) gel(z,i) = gel(y,i-1);
return z;
}
break;
case t_MAT:
switch(ty)
{
case t_VEC:
z=cgetg(lx, t_MAT); if (ly != lx) break;
for (i=1; i<lx; i++) gel(z,i) = shallowconcat(gel(x,i), gel(y,i));
return z;
case t_COL:
if (ly != lgcols(x)) break;
z=cgetg(lx+1,t_MAT); gel(z,lx) = y;
for (i=1; i<lx; i++) z[i]=x[i];
return z;
}
break;
}
err_cat(x,y);
return NULL; /* not reached */
}
static GEN
bnflog_i(GEN bnf, GEN ell)
{
long prec0, prec;
GEN nf, US, vdegS, S, T, M, CLp, CLt, Ftilde, vtG, ellk;
GEN D, Ap, cycAp, bnfS;
long i, j, lS, lvAp;
checkbnf(bnf);
nf = checknf(bnf);
S = idealprimedec(nf, ell);
bnfS = bnfsunit0(bnf, S, nf_GENMAT, LOWDEFAULTPREC); /* S-units */
US = leafcopy(gel(bnfS,1));
prec0 = maxss(30, vtilde_prec(nf, US, ell));
US = shallowconcat(bnf_get_fu(bnf), US);
settyp(US, t_COL);
T = padicfact(nf, S, prec0);
lS = lg(S); Ftilde = cgetg(lS, t_VECSMALL);
for (j = 1; j < lS; j++) Ftilde[j] = ftilde(nf, gel(S,j), gel(T,j));
CLp = CL_prime(bnf, ell, S);
cycAp = gel(CLp,1);
Ap = gel(CLp,2);
for(;;)
{
CLt = CL_tilde(nf, US, ell, T, Ftilde, &vtG, prec0);
if (CLt) break;
prec0 <<= 1;
T = padicfact(nf, S, prec0);
}
prec = ellexpo(cycAp, ell) + ellexpo(CLt,ell) + 1;
if (prec == 1) return mkvec3(cgetg(1,t_VEC), cgetg(1,t_VEC), cgetg(1,t_VEC));
vdegS = get_vdegS(Ftilde, ell, prec0);
ellk = powiu(ell, prec);
lvAp = lg(Ap);
if (lvAp > 1)
{
GEN Kcyc = bnf_get_cyc(bnf);
GEN C = zeromatcopy(lvAp-1, lS-1);
GEN Rell = gel(CLp,3), Uell = gel(CLp,4), ordS = gel(CLp,5);
for (i = 1; i < lvAp; i++)
{
GEN a, b, bi, A = gel(Ap,i), d = gel(cycAp,i);
bi = isprincipal(bnf, A);
a = vecmodii(ZC_Z_mul(bi,d), Kcyc);
/* a in subgroup generated by S = Rell; hence b integral */
b = hnf_invimage(Rell, a);
b = vecmodii(ZM_ZC_mul(Uell, ZC_neg(b)), ordS);
A = mkvec2(A, cgetg(1,t_MAT));
A = idealpowred(nf, A, d);
/* find a principal representative of A_i^cycA_i up to elements of S */
a = isprincipalfact(bnf,gel(A,1),S,b,nf_GENMAT|nf_FORCE);
if (!gequal0(gel(a,1))) pari_err_BUG("bnflog");
a = famat_mul_shallow(gel(A,2), gel(a,2)); /* principal part */
if (lg(a) == 1) continue;
for (j = 1; j < lS; j++)
gcoeff(C,i,j) = vtilde(nf, a, gel(T,j), gel(vdegS,j), ell, prec0);
}
C = gmod(gneg(C),ellk);
C = shallowtrans(C);
M = mkmat2(mkcol2(diagonal_shallow(cycAp), C), mkcol2(gen_0, vtG));
M = shallowmatconcat(M); /* relation matrix */
}
else
M = vtG;
M = ZM_hnfmodid(M, ellk);
D = matsnf0(M, 4);
if (lg(D) == 1 || !dvdii(gel(D,1), ellk))
pari_err_BUG("bnflog [missing Z_l component]");
D = vecslice(D,2,lg(D)-1);
return mkvec3(D, CLt, ellsylow(cycAp, ell));
}
/* g = polynomial, h = embedding. Return [[g,h]] */
static GEN
_subfield(GEN g, GEN h) { return mkvec(mkvec2(g,h)); }
