Line data Source code
1 : /* Copyright (C) 2000, 2012 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 : #include "pari.h"
15 : #include "paripriv.h"
16 :
17 : #define DEBUGLEVEL DEBUGLEVEL_subcyclo
18 :
19 : /* written by Takashi Fukuda */
20 :
21 : #define onevec(x) const_vec(x,gen_1)
22 : #define nullvec() cgetg(1, t_VEC)
23 : #define order_f_x(f, x) Fl_order(x%f, eulerphiu(f), f)
24 :
25 : #define USE_MLL (1L<<0)
26 : #define NO_PLUS_PART (1L<<1)
27 : #define NO_MINUS_PART (1L<<2)
28 : #define SKIP_PROPER (1L<<3)
29 : #define SAVE_MEMORY (1L<<4)
30 : #define USE_FULL_EL (1L<<5)
31 : #define USE_BASIS (1L<<6)
32 : #define USE_FACTOR (1L<<7)
33 : #define USE_GALOIS_POL (1L<<8)
34 : #define USE_F (1L<<9)
35 :
36 : static ulong
37 178002 : _get_d(GEN H) { return umael(H, 2, 1);}
38 : static ulong
39 119820 : _get_f(GEN H) { return umael(H, 2, 2);}
40 : static ulong
41 30843 : _get_h(GEN H) { return umael(H, 2, 3);}
42 : static long
43 28392 : _get_s(GEN H) { return umael(H, 2, 4);}
44 : static long
45 53957 : _get_g(GEN H) { return umael(H, 2, 5);}
46 : static GEN
47 30787 : _get_H(GEN H) { return gel(H, 3);}
48 : static ulong
49 112755 : K_get_d(GEN K) { return _get_d(gel(K,1)); }
50 : static ulong
51 82965 : K_get_f(GEN K) { return _get_f(gel(K,1)); }
52 : static ulong
53 17424 : K_get_h(GEN K) { return _get_h(gel(K,1)); }
54 : static long
55 0 : K_get_s(GEN K) { return _get_s(gel(K,1)); }
56 : static ulong
57 17102 : K_get_g(GEN K) { return _get_g(gel(K,1)); }
58 : static GEN
59 17368 : K_get_H(GEN K) { return _get_H(gel(K,1)); }
60 : static ulong
61 59314 : K_get_dchi(GEN K) { return gel(K,6)[1]; }
62 : static ulong
63 23660 : K_get_nconj(GEN K) { return gel(K,6)[2]; }
64 :
65 : /* G=<s> is a cyclic group of order n and t=s^(-1).
66 : * convert sum_i a_i*s^i to sum_i b_i*t^i */
67 : static GEN
68 14 : Flx_recip1_inplace(GEN x, long pn)
69 : {
70 14 : long i, lx = lg(x);
71 14 : if(lx-2 != pn) /* This case scarcely occurs */
72 : {
73 0 : long ly = pn+2;
74 0 : GEN y = const_vecsmall(ly, 0);
75 0 : y[1] = x[1];y[2] = x[2];
76 0 : for(i=3;i<lx;i++) y[ly+2-i] = x[i];
77 0 : return Flx_renormalize(y, ly);
78 : }
79 : else /* almost all cases */
80 : {
81 14 : long t, mid = (lx+1)>>1;
82 7168 : for(i=3;i<=mid;i++)
83 : {
84 7154 : t = x[i];x[i] = x[lx+2-i];x[lx+2-i] = t;
85 : }
86 14 : return Flx_renormalize(x, lx);
87 : }
88 : }
89 :
90 : /* Return h^degpol(P) P(x / h) */
91 : static GEN
92 14 : Flx_rescale_inplace(GEN P, ulong h, ulong p)
93 : {
94 14 : long i, l = lg(P);
95 14 : ulong hi = h;
96 14322 : for (i=l-2; i>=2; i--)
97 : {
98 14322 : P[i] = Fl_mul(P[i], hi, p);
99 14322 : if (i == 2) break;
100 14308 : hi = Fl_mul(hi,h, p);
101 : }
102 14 : return P;
103 : }
104 :
105 : static GEN
106 14 : zx_to_Flx_inplace(GEN x, ulong p)
107 : {
108 14 : long i, lx = lg(x);
109 14350 : for (i=2; i<lx; i++) uel(x,i) = umodsu(x[i], p);
110 14 : return Flx_renormalize(x, lx);
111 : }
112 :
113 : /* zero pol of n components (i.e. deg=n-1). need to pass to ZX_renormalize */
114 : INLINE GEN
115 53263 : pol_zero(long n)
116 : {
117 : long i;
118 53263 : GEN p = cgetg(n+2, t_POL);
119 53263 : p[1] = evalsigne(1) | evalvarn(0);
120 1799812 : for (i = 2; i < n+2; i++) gel(p, i) = gen_0;
121 53263 : return p;
122 : }
123 :
124 : /* e[i+1] = L*i + K for i >= n; determine K,L and reduce n if possible */
125 : static GEN
126 21 : vecsmall2vec2(GEN e, long n)
127 : {
128 21 : long L = e[n+1] - e[n], K = e[n+1] - L*n;
129 42 : n--; while (n >= 0 && e[n+1] - L*n == K) n--;
130 21 : if (n < 0) e = nullvec(); else { setlg(e, n+2); e = zv_to_ZV(e); }
131 21 : return mkvec3(utoi(L), stoi(K), e); /* L >= 0 */
132 : }
133 :
134 : /* z=zeta_{p^n}; return k s.t. (z-1)^k || f(z) assuming deg(f)<phi(p^n) */
135 : static long
136 42 : zx_p_val(GEN f, ulong p, ulong n)
137 : {
138 42 : pari_sp av = avma;
139 42 : ulong x = zx_lval(f, p);
140 42 : if (x) { f = zx_z_divexact(f, upowuu(p, x)); x *= (p-1)*upowuu(p, n-1); }
141 42 : x += Flx_val(Flx_translate1(zx_to_Flx(f, p), p));
142 42 : return gc_long(av, x);
143 : }
144 :
145 : static long
146 315 : ZX_p_val(GEN f, ulong p, ulong n)
147 : {
148 315 : pari_sp av = avma;
149 315 : ulong x = ZX_lval(f, p);
150 315 : if (x) { f = ZX_Z_divexact(f, powuu(p, x)); x *= (p-1)*upowuu(p, n-1); }
151 315 : x += Flx_val(Flx_translate1(ZX_to_Flx(f, p), p));
152 315 : return gc_long(av, x);
153 : }
154 :
155 : static GEN
156 35 : set_A(GEN B, int *chi)
157 : {
158 35 : long a, i, j, B1 = B[1], l = lg(B);
159 35 : GEN A = cgetg(l, t_VECSMALL);
160 1687014 : for (a = 0, j = 1; j < B1; j++) a += chi[j];
161 35 : A[1] = a;
162 714 : for (i = 2; i < l; i++)
163 : {
164 679 : long Bi = B[i];
165 28159243 : for (a = A[i-1], j = B[i-1]; j < Bi; j++) a += chi[j];
166 679 : A[i] = a;
167 : }
168 35 : return A;
169 : }
170 :
171 : /* g_n(a)=g_n(b) <==> a^2=b^2 mod 2^(n+2) <==> a=b,-b mod 2^(n+2)
172 : * g_n(a)=g_n(1+q0)^k <==> a=x(1+q0)^k x=1,-1
173 : * gam[1+a]=k, k<0 ==> g_n(a)=0
174 : * k>=0 ==> g_n(a)^(-1)=gamma^k, gamma=g_n(1+q0) */
175 : static GEN
176 14 : set_gam2(long q01, long n)
177 : {
178 : long i, x, x1, pn, pn2;
179 : GEN gam;
180 14 : pn = (1L<<n);
181 14 : pn2 = (pn<<2);
182 14 : gam = const_vecsmall(pn2, -1);
183 14 : x=Fl_inv(q01, pn2); x1=1;
184 14350 : for (i=0; i<pn; i++)
185 : {
186 14336 : gam[1+x1] = gam[1+Fl_neg(x1, pn2)] = i;
187 14336 : x1 = Fl_mul(x1, x, pn2);
188 : }
189 14 : return gam;
190 : }
191 :
192 : /* g_n(a)=g_n(b) <==> a^(p-1)=b^(p-1) mod p^(n+1) <==> a=xb x=<g^(p^n)>
193 : * g_n(a)=g_n(1+q0)^k <==> a=x(1+q0)^k x=<g^(p^n)>
194 : * gam[1+a]=k, k<0 ==> g_n(a)=0
195 : * k>=0 ==> g_n(a)^(-1)=gamma^k, gamma=g_n(1+q0) */
196 : static GEN
197 476 : set_gam(long q01, long p, long n)
198 : {
199 : long i, j, g, g1, x, x1, p1, pn, pn1;
200 : GEN A, gam;
201 476 : p1 = p-1; pn = upowuu(p, n); pn1 = p*pn;
202 476 : gam = const_vecsmall(pn1, -1);
203 476 : g = pgener_Zl(p); g1 = Fl_powu(g, pn, pn1);
204 476 : A = Fl_powers(g1, p1-1, pn1); /* A[1+i]=g^(i*p^n) mod p^(n+1), 0<=i<=p-2 */
205 476 : x = Fl_inv(q01, pn1); x1 = 1;
206 568694 : for (i=0; i<pn; i++)
207 : {
208 2086126 : for (j=1; j<=p1; j++) gam[1+Fl_mul(x1, A[j], pn1)] = i;
209 568218 : x1 = Fl_mul(x1, x, pn1);
210 : }
211 476 : return gam;
212 : }
213 :
214 : /* k=Q(sqrt(m)), A_n=p-class gr. of k_n, |A_n|=p^(e_n)
215 : * return e_n-e_(n-1)
216 : * essential assumption : m is not divisible by p
217 : * Gold, Acta Arith. XXVI (1974), p.22 formula (3) */
218 : static long
219 35 : ediff(ulong p, long m, ulong n, int *chi)
220 : {
221 35 : pari_sp av = avma;
222 : long j, lx, *px;
223 : ulong i, d, s, y, g, p1, pn, pn1, pn_1, phipn, phipn1;
224 : GEN A, B, x, gs, cs;
225 :
226 35 : d=((m-1)%4==0)?labs(m):4*labs(m);
227 35 : p1=p-1; pn_1=upowuu(p, n-1); pn=p*pn_1; pn1=p*pn; phipn=p1*pn_1; phipn1=p1*pn;
228 35 : lx=2*p1*phipn;
229 35 : y=Fl_inv(pn1%d, d); g=pgener_Zl(p); /* pn1 may > d */
230 35 : cs = cgetg(2+phipn, t_VECSMALL); cs[1] = evalvarn(0);
231 35 : x = cgetg(1+lx, t_VECSMALL);
232 35 : gs = Fl_powers(g, phipn1-1, pn1); /* gs[1+i]=g^i(mod p^(n+1)), 0<=i<p^(n+1) */
233 :
234 105 : for (px=x,i=0; i<p1; i++)
235 : {
236 70 : long ipn=i*pn+1,ipnpn=ipn+phipn;
237 546 : for (s=0; s<phipn; s++)
238 : {
239 476 : *++px = (y*gs[s+ipn])%d; /* gs[s+ipn] may > d */
240 476 : *++px = (y*gs[(s%pn_1)+ipnpn])%d;
241 : }
242 : }
243 35 : B = vecsmall_uniq(x);
244 35 : A = set_A(B, chi);
245 273 : for (s=0; s<phipn; s++)
246 : {
247 238 : long a=0, ipn=1, spn1=s%pn_1;
248 714 : for (i=0; i<p1; i++)
249 : {
250 476 : if ((j=zv_search(B, (y*gs[s+ipn])%d))<=0)
251 0 : pari_err_BUG("zv_search failed\n");
252 476 : a+=A[j];
253 476 : if ((j=zv_search(B, (y*gs[spn1+ipn+phipn])%d))<=0)
254 0 : pari_err_BUG("zv_search failed\n");
255 476 : a-=A[j];
256 476 : ipn+=pn;
257 : }
258 238 : cs[2+s] = a;
259 : }
260 35 : cs = zx_renormalize(cs, lg(cs));
261 35 : y = (lg(cs)==3) ? phipn*z_lval(cs[2], p) : zx_p_val(cs, p, n);
262 35 : return gc_long(av, y);
263 : }
264 :
265 : static GEN
266 0 : quadteichstk(GEN Chi, int *chi, GEN Gam, long p, long m, long n)
267 : {
268 0 : GEN Gam1 = Gam+1, xi;
269 : long i, j, j0, d, f0, pn, pn1, deg, pn1d;
270 :
271 0 : d = ((m&3)==1)?m:m<<2;
272 0 : f0 = ulcm(p, d)/p;
273 0 : pn = upowuu(p, n); pn1 = p*pn; pn1d = pn1%d;
274 0 : xi = cgetg(pn+2, t_POL); xi[1] = evalsigne(1) | evalvarn(0);
275 0 : for (i=0; i<pn; i++) gel(xi, 2+i) = const_vecsmall(p, 0);
276 0 : for (j=1; j<pn1; j++)
277 : {
278 : long jp, ipn1d, *xij0;
279 0 : if ((j0 = Gam1[j])<0) continue;
280 0 : jp = j%p; ipn1d = j%d; xij0 = gel(xi, 2+j0)+2;
281 0 : for (i=1; i<f0; i++)
282 : {
283 : int sgn;
284 0 : if ((ipn1d += pn1d) >= d) ipn1d -= d;
285 0 : if ((sgn = chi[ipn1d])==0) continue;
286 0 : deg = Chi[jp]; /* jp!=0 because j0>=0 */
287 0 : if (sgn>0) xij0[deg] += i;
288 0 : else xij0[deg] -= i;
289 : }
290 : }
291 0 : for (i=0; i<pn; i++) gel(xi, 2+i) = zx_renormalize(gel(xi, 2+i), p+1);
292 0 : return FlxX_renormalize(xi, pn+2); /* zxX_renormalize does not exist */
293 : }
294 :
295 : #ifdef DEBUG_QUADSTK
296 : /* return f0*xi_n */
297 : static GEN
298 : quadstkp_by_def(int *chi, GEN gam, long n, long p, long f, long f0)
299 : {
300 : long i, a, a1, pn, pn1, qn;
301 : GEN x, x2, gam1 = gam+1;
302 : pn = upowuu(p, n); pn1 = p*pn; qn = f0*pn1;
303 : x = const_vecsmall(pn+1, 0); x2 = x+2;
304 : for (a=1; a<qn; a++)
305 : {
306 : int sgn;
307 : if ((a1=gam1[a%pn1])<0 || (sgn=chi[a%f])==0) continue;
308 : if (sgn>0) x2[a1]+=a;
309 : else x2[a1]-=a;
310 : }
311 : for (i=0; i<pn; i++)
312 : {
313 : if (x2[i]%pn1) pari_err_BUG("stickel. ele. is not integral.\n");
314 : else x2[i]/=pn1;
315 : }
316 : return zx_renormalize(x, pn+2);
317 : }
318 : #endif
319 :
320 : /* f!=p
321 : * xi_n = f0^(-1)*
322 : * sum_{0<=j<pn1,(j,p)=1}(Q_n/Q,j)^(-1)*(sum_{0<=i<f0}i*chi^(-1)(pn1*i+j)) */
323 : static GEN
324 14 : quadstkp1(int *chi, GEN gam, long n, long p, long f, long f0)
325 : {
326 : long i, j, j0, pn, pn1, pn1f, den;
327 : GEN x, x2;
328 14 : pn = upowuu(p, n); pn1 = p*pn; pn1f = pn1%f;
329 14 : x = const_vecsmall(pn+1, 0); x2 = x+2;
330 14 : if (f==3) den = (chi[p%f]>0)?f0<<1:2;
331 14 : else if (f==4) den = (chi[p%f]>0)?f0<<1:f0;
332 14 : else den = f0<<1;
333 1653372 : for (j=1; j<pn1; j++)
334 : {
335 : long ipn1;
336 1653358 : if (j%p==0) continue;
337 1102248 : j0 = gam[1+j]; ipn1 = j%f;
338 263437272 : for (i=1; i<f0; i++)
339 : {
340 : int sgn;
341 262335024 : if ((ipn1+=pn1f)>=f) ipn1-=f;
342 262335024 : if ((sgn = chi[ipn1])>0) x2[j0]+=i;
343 131716319 : else if (sgn<0) x2[j0]-=i;
344 : }
345 : }
346 551138 : for (i=0; i<pn; i++)
347 : {
348 551124 : if (x2[i]%den) pari_err_BUG("stickel. ele. is not integral.\n");
349 551124 : else x2[i]/=den;
350 : }
351 14 : return zx_renormalize(x, pn+2);
352 : }
353 :
354 : /* f==p */
355 : static GEN
356 0 : quadstkp2(int *chi, GEN gam, long n, long p)
357 : {
358 : long a, a1, i, pn, pn1, amodp;
359 0 : GEN x, x2, gam1 = gam+1;
360 0 : pn = upowuu(p, n); pn1 = p*pn;
361 0 : x = const_vecsmall(pn+1, 0); x2 = x+2;
362 0 : for (a=1,amodp=0; a<pn1; a++)
363 : {
364 : int sgn;
365 0 : if (++amodp==p) {amodp = 0; continue; }
366 0 : if ((sgn = chi[amodp])==0) continue;
367 0 : a1=gam1[a];
368 0 : if (sgn>0) x2[a1]+=a;
369 0 : else x2[a1]-=a;
370 : }
371 0 : for (i=0; i<pn; i++)
372 : {
373 0 : if (x2[i]%pn1) pari_err_BUG("stickel. ele. is not integral.\n");
374 0 : else x2[i]/=pn1;
375 : }
376 0 : return zx_renormalize(x, pn+2);
377 : }
378 :
379 : /* p>=3
380 : * f = conductor of Q(sqrt(m))
381 : * q0 = lcm(f,p) = f0*p
382 : * qn = q0*p^n = f0*p^(n+1)
383 : * xi_n = qn^(-1)*sum_{1<=a<=qn,(a,qn)=1} a*chi(a)^(-1)*(Q_n/Q,a)^(-1) */
384 : static GEN
385 14 : quadstkp(long p, long m, long n, int *chi)
386 : {
387 : long f, f0, pn, pn1, q0;
388 : GEN gam;
389 14 : f = ((m-1)%4==0)?labs(m):4*labs(m);
390 14 : pn = upowuu(p, n); pn1 = p*pn;
391 14 : if (f % p) { q0 = f * p; f0 = f; } else { q0 = f; f0 = f / p; }
392 14 : gam = set_gam((1+q0)%pn1, p, n);
393 : #ifdef DEBUG_QUADSTK
394 : return quadstkp_by_def(chi, gam, n, p, f, f0);
395 : #else
396 14 : return (f0!=1)?quadstkp1(chi, gam, n, p, f, f0):quadstkp2(chi, gam, n, p);
397 : #endif
398 : }
399 :
400 : /* p=2 */
401 : static GEN
402 14 : quadstk2(long m, long n, int *chi)
403 : {
404 : long i, j, j0, f, f0, pn, pn1, pn2, pn2f, q0;
405 : GEN x, x2, gam;
406 14 : f = ((m-1)%4==0)?labs(m):4*labs(m);
407 14 : pn = 1L<<n; pn1 = pn<<1; pn2 = pn1<<1; pn2f = pn2%f;
408 14 : q0 = (f&1)?f*4:f;
409 14 : f0 = (f&1)?f:f/4;
410 14 : x = const_vecsmall(pn+1, 0); x2 = x+2;
411 14 : gam = set_gam2((1+q0)%pn2, n);
412 57344 : for (j=1; j<pn2; j++)
413 : {
414 : long ipn2;
415 57330 : if (!(j&1)) continue;
416 28672 : j0 = gam[1+j];
417 28672 : ipn2 = j%f;
418 : /* for (i=1; i<f0; i++) x2[j0]+=i*chi[(i*pn2+j)%f]; */
419 1691648 : for (i=1; i<f0; i++)
420 : {
421 : int sgn;
422 1662976 : if ((ipn2+=pn2f)>=f) ipn2-=f;
423 1662976 : if ((sgn=chi[ipn2])>0) x2[j0]+=i;
424 845390 : else if (sgn<0) x2[j0]-=i;
425 : }
426 : }
427 14350 : for (f0<<=1, i=0; i<pn; i++)
428 : {
429 14336 : if (x2[i]%f0) pari_err_BUG("stickel. ele. is not integral.\n");
430 14336 : else x2[i]/=f0;
431 : }
432 14 : return zx_renormalize(x, pn+2);
433 : }
434 :
435 : /* Chin is a generator of the group of the characters of G(Q_n/Q).
436 : * chin[1+a]=k, k<0 ==> Chin(a)=0
437 : * k>=0 ==> Chin(a)=zeta_{p^n}^k */
438 : static GEN
439 21 : set_chin(long p, long n)
440 : {
441 21 : long i, j, x = 1, g, gpn, pn, pn1;
442 : GEN chin, chin1;
443 21 : pn = upowuu(p, n); pn1 = p*pn;
444 21 : chin = const_vecsmall(pn1, -1); chin1 = chin+1;
445 21 : g = pgener_Zl(p); gpn = Fl_powu(g, pn, pn1);
446 294 : for (i=0; i<pn; i++)
447 : {
448 273 : long y = x;
449 819 : for (j=1; j<p; j++)
450 : {
451 546 : chin1[y] = i;
452 546 : y = Fl_mul(y, gpn, pn1);
453 : }
454 273 : x = Fl_mul(x, g, pn1);
455 : }
456 21 : return chin;
457 : }
458 :
459 : /* k=Q(sqrt(m)), A_n=p-class gr. of k_n, |A_n|=p^(e_n), p|m
460 : * return e_n-e_(n-1).
461 : * There is an another method using the Stickelberger element based on
462 : * Coates-Lichtenbaum, Ann. Math. vol.98 No.3 (1973), 498-550, Lemma 2.15.
463 : * If kro(m,p)!=1, then orders of two groups coincide.
464 : * ediff_ber is faster than the Stickelberger element. */
465 : static long
466 21 : ediff_ber(ulong p, long m, ulong n, int *chi)
467 : {
468 21 : pari_sp av = avma;
469 : long a, d, e, x, y, pn, pn1, qn1;
470 21 : GEN B, B2, chin = set_chin(p, n)+1;
471 :
472 21 : d = ((m-1)%4==0)?labs(m):4*labs(m);
473 21 : pn = upowuu(p, n); pn1 = p*pn; qn1 = (d*pn)>>1;
474 21 : B = const_vecsmall(pn+1, 0); B2 = B+2;
475 522105969 : for (a=x=y=1; a <= qn1; a++) /* x=a%d, y=a%pn1 */
476 : {
477 522105948 : int sgn = chi[x];
478 522105948 : if (sgn)
479 : {
480 172937856 : long k = chin[y];
481 172937856 : if (k >= 0) { if (sgn > 0) B2[k]++; else B2[k]--; }
482 : }
483 522105948 : if (++x == d) x = 0;
484 522105948 : if (++y == pn1) y = 0;
485 : }
486 21 : B = zx_renormalize(B, pn+2);
487 7 : e = (n==1)? zx_p_val(B, p, n)
488 21 : : ZX_p_val(ZX_rem(zx_to_ZX(B), polcyclo(pn, 0)), p, n);
489 21 : if (p==3 && chi[2] < 0) e--; /* 2 is a primitive root of 3^n (n>=1) */
490 21 : return gc_long(av, e);
491 : }
492 :
493 : #ifdef DEBUG
494 : /* slow */
495 : static int*
496 : set_quad_chi_1(long m)
497 : {
498 : long a, d, f;
499 : int *chi;
500 : d=((m-1)%4==0)?m:4*m; f=labs(d);
501 : chi= (int*)stack_calloc(sizeof(int)*f);
502 : for (a=1; a<f; a++) chi[a]=kross(d, a);
503 : return chi;
504 : }
505 : #endif
506 :
507 : /* chi[a]=kross(d, a) 0<=a<=f-1
508 : * d=discriminant of Q(sqrt(m)), f=abs(d)
509 : *
510 : * Algorithm: m=-p1*p2*...*pr ==> kross(d,gi)=-1 (1<=i<=r), gi=proot(pi)
511 : * set_quad_chi_1(m)=set_quad_chi_2(m) for all square-free m s.t. |m|<10^5. */
512 : static int*
513 49 : set_quad_chi_2(long m)
514 : {
515 49 : long d = (m-1) % 4? 4*m: m, f = labs(d);
516 49 : GEN fa = factoru(f), P = gel(fa, 1), E = gel(fa,2), u, v;
517 49 : long i, j, np, nm, l = lg(P);
518 49 : int *chi = (int*)stack_calloc(sizeof(int)*f);
519 49 : pari_sp av = avma;
520 49 : int *plus = (int*)stack_calloc(sizeof(int)*f), *p0 = plus;
521 49 : int *minus = (int*)stack_calloc(sizeof(int)*f), *p1 = minus;
522 :
523 49 : u = cgetg(32, t_VECSMALL);
524 49 : v = cgetg(32, t_VECSMALL);
525 147 : for (i = 1; i < l; i++)
526 : {
527 98 : ulong p = upowuu(P[i], E[i]);
528 98 : u[i] = p * Fl_inv(p, f / p);
529 98 : v[i] = Fl_sub(1, u[i], f);
530 : }
531 49 : if (E[1]==2) /* f=4*(-m) */
532 : {
533 14 : *p0++ = Fl_add(v[1], u[1], f);
534 14 : *p1++ = Fl_add(Fl_mul(3, v[1], f), u[1], f);
535 14 : i = 2;
536 : }
537 35 : else if (E[1]==3) /* f=8*(-m) */
538 : {
539 : ulong a;
540 7 : *p0++ = Fl_add(v[1], u[1], f);
541 7 : a = Fl_add(Fl_mul(3, v[1], f), u[1], f);
542 7 : if (kross(d, a) > 0) *p0++ = a; else *p1++ = a;
543 7 : a = Fl_add(Fl_mul(5, v[1], f), u[1], f);
544 7 : if (kross(d, a) > 0) *p0++ = a; else *p1++ = a;
545 7 : a = Fl_add(Fl_mul(7, v[1], f), u[1], f);
546 7 : if (kross(d, a) > 0) *p0++ = a; else *p1++ = a;
547 7 : i = 2;
548 : }
549 : else /* f=-m */
550 28 : {*p0++ = 1; i = 1; }
551 126 : for (; i < l; i++)
552 : {
553 77 : ulong gn, g = pgener_Fl(P[i]);
554 77 : gn = g = Fl_add(Fl_mul(g, v[i], f), u[i], f);
555 77 : np = p0-plus;
556 77 : nm = p1-minus;
557 : for (;;)
558 : {
559 4802406 : for (j = 0; j < np; j++) *p1++ = Fl_mul(plus[j], gn, f);
560 4799655 : for (j = 0; j < nm; j++) *p0++ = Fl_mul(minus[j], gn, f);
561 7679 : gn = Fl_mul(gn, g, f); if (gn == 1) break;
562 4763745 : for (j= 0; j < np; j++) *p0++ = Fl_mul(plus[j], gn, f);
563 4761022 : for (j = 0; j < nm; j++) *p1++ = Fl_mul(minus[j], gn, f);
564 7602 : gn = Fl_mul(gn, g, f); if (gn == 1) break;
565 : }
566 : }
567 49 : np = p0-plus;
568 49 : nm = p1-minus;
569 9548224 : for (i = 0; i < np; i++) chi[plus[i]] = 1;
570 9548224 : for (i = 0; i < nm; i++) chi[minus[i]] = -1;
571 49 : set_avma(av); return chi;
572 : }
573 :
574 : static long
575 8995 : srh_x(GEN T, long n, long x)
576 : {
577 30086 : for (; x<n; x++) if (!T[x]) return x;
578 623 : return -1;
579 : }
580 :
581 : /* G is a cyclic group of order d. hat(G)=<chi>.
582 : * chi, chi^p, ... , chi^(p^(d_chi-1)) are conjugate.
583 : * {chi^j | j in C} are repre. of Q_p-congacy classes of inj. chars.
584 : *
585 : * C is a set of representatives of H/<p>, where H=(Z/dZ)^* */
586 : static GEN
587 1134 : set_C(long p, long d, long d_chi, long n_conj)
588 : {
589 1134 : long i, j, x, y, pmodd = p%d;
590 1134 : GEN T = const_vecsmall(d, 0)+1;
591 1134 : GEN C = cgetg(1+n_conj, t_VECSMALL);
592 1134 : if (n_conj==1) { C[1] = 1; return C; }
593 9618 : for (i=0, x=1; x >= 0; x = srh_x(T, d, x))
594 : {
595 8995 : if (cgcd(x, d)==1) C[++i] = x;
596 40929 : for (j=0, y=x; j<d_chi; j++) T[y = Fl_mul(y, pmodd, d)] = 1;
597 : }
598 623 : return C;
599 : }
600 :
601 : static GEN
602 343 : FpX_one_cyclo(long n, GEN p)
603 : {
604 343 : if (lgefint(p)==3)
605 301 : return Flx_to_ZX(Flx_factcyclo(n, p[2], 1));
606 : else
607 42 : return FpX_factcyclo(n, p, 1);
608 : }
609 :
610 : static void
611 17094 : Flx_red_inplace(GEN x, ulong p)
612 : {
613 17094 : long i, l = lg(x);
614 274540 : for (i=2; i<l; i++) x[i] = uel(x, i)%p;
615 17094 : Flx_renormalize(x, l);
616 17094 : }
617 :
618 : /* x[i], T[i] < pn */
619 : static GEN
620 39046 : Flxq_xi_conj(GEN x, GEN T, long j, long d, long pn)
621 : {
622 39046 : long i, deg = degpol(x);
623 39046 : GEN z = const_vecsmall(d+1, 0);
624 1032304 : for (i=0; i<=deg; i++) z[2+Fl_mul(i, j, d)] = x[2+i];
625 39046 : return Flx_rem(Flx_renormalize(z, d+2), T, pn);
626 : }
627 :
628 : static GEN
629 966 : FlxqX_xi_conj(GEN x, GEN T, long j, long d, long pn)
630 : {
631 966 : long i, l = lg(x);
632 : GEN z;
633 966 : z = cgetg(l, t_POL); z[1] = evalsigne(1) | evalvarn(0);
634 40012 : for (i=2; i<l; i++) gel(z, i) = Flxq_xi_conj(gel(x, i), T, j, d, pn);
635 966 : return z;
636 : }
637 :
638 : static GEN
639 0 : FlxqX_xi_norm(GEN x, GEN T, long p, long d, long pn)
640 : {
641 0 : long i, d_chi = degpol(T);
642 0 : GEN z = x, z1 = x;
643 0 : for (i=1; i<d_chi; i++)
644 : {
645 0 : z1 = FlxqX_xi_conj(z1, T, p, d, pn);
646 0 : z = FlxqX_mul(z, z1, T, pn);
647 : }
648 0 : return z;
649 : }
650 :
651 : /* assume 0 <= x[i], y[j] <= m-1 */
652 : static GEN
653 15 : FpV_shift_add(GEN x, GEN y, GEN m, long start, long end)
654 : {
655 : long i, j;
656 222300 : for (i=start, j=1; i<=end; i++, j++)
657 : {
658 222285 : pari_sp av = avma;
659 222285 : GEN z = addii(gel(x, i), gel(y, j));
660 222285 : gel(x, i) = (cmpii(z, m) >= 0)? gerepileuptoint(av, subii(z, m)): z;
661 : }
662 15 : return x;
663 : }
664 :
665 : /* assume 0 <= x[i], y[j] <= m-1 */
666 : static GEN
667 10 : FpV_shift_sub(GEN x, GEN y, GEN m, long start, long end)
668 : {
669 : long i, j;
670 112430 : for (i=start, j=1; i<=end; i++, j++)
671 : {
672 112420 : pari_sp av = avma;
673 112420 : GEN z = subii(gel(x, i), gel(y, j));
674 112420 : gel(x, i) = (signe(z) < 0)? gerepileuptoint(av, addii(z, m)): z;
675 : }
676 10 : return x;
677 : }
678 :
679 : /* assume 0 <= x[i], y[j] <= m-1 */
680 : static GEN
681 173 : Flv_shift_add(GEN x, GEN y, ulong m, long start, long end)
682 : {
683 : long i, j;
684 2320113 : for (i=start, j=1; i<=end; i++, j++)
685 : {
686 2319940 : ulong xi = x[i], yj = y[j];
687 2319940 : x[i] = Fl_add(xi, yj, m);
688 : }
689 173 : return x;
690 : }
691 :
692 : /* assume 0 <= x[i], y[j] <= m-1 */
693 : static GEN
694 102 : Flv_shift_sub(GEN x, GEN y, ulong m, long start, long end)
695 : {
696 : long i, j;
697 1165182 : for (i=start, j=1; i<=end; i++, j++)
698 : {
699 1165080 : ulong xi = x[i], yj = y[j];
700 1165080 : x[i] = Fl_sub(xi, yj, m);
701 : }
702 102 : return x;
703 : }
704 :
705 : /* return 0 if p|x. else return 1 */
706 : INLINE long
707 896 : Flx_divcheck(GEN x, ulong p)
708 : {
709 896 : long i, l = lg(x);
710 910 : for (i=2; i<l; i++) if (uel(x, i)%p) return 1;
711 448 : return 0;
712 : }
713 :
714 : static long
715 448 : FlxX_weier_deg(GEN pol, long p)
716 : {
717 448 : long i, l = lg(pol);
718 896 : for (i=2; i<l && Flx_divcheck(gel(pol, i), p)==0; i++);
719 448 : return (i<l)?i-2:-1;
720 : }
721 :
722 : static long
723 1582 : Flx_weier_deg(GEN pol, long p)
724 : {
725 1582 : long i, l = lg(pol);
726 3997 : for (i=2; i<l && pol[i]%p==0; i++);
727 1582 : return (i<l)?i-2:-1;
728 : }
729 :
730 : static GEN
731 308 : Flxn_shift_mul(GEN g, long n, GEN p, long d, long m)
732 : {
733 308 : return Flx_shift(Flxn_mul(g, p, d, m), n);
734 : }
735 :
736 : INLINE long
737 1057 : deg_trunc(long lam, long p, long n, long pn)
738 : {
739 : long r, x, d;
740 1260 : for (r=1,x=p; x<lam; r++) x *= p; /* r is min int s.t. lam<=p^r */
741 1057 : if ((d = (n-r+2)*lam+1)>=pn) d = pn;
742 1057 : return d;
743 : }
744 :
745 : /* Flx_translate1_basecase(g, pn) becomes slow when degpol(g)>1000.
746 : * So I wrote Flxn_translate1().
747 : * I need lambda to truncate pol.
748 : * But I need to translate T --> 1+T to know lambda.
749 : * Though the code has a little overhead, it is still fast. */
750 : static GEN
751 756 : Flxn_translate1(GEN g, long p, long n)
752 : {
753 : long i, j, d, lam, pn, start;
754 756 : if (n==1) start = 3;
755 70 : else if (n==2) start = 9;
756 70 : else start = 10;
757 756 : pn = upowuu(p, n);
758 756 : for (lam=start; lam; lam<<=1) /* least upper bound is 3 */
759 : {
760 : GEN z;
761 756 : d = deg_trunc(lam, p, n, pn);
762 756 : z = const_vecsmall(d+1, 0); /* z[2],...,z[d+1] <--> a_0,...,a_{d-1} */
763 44954 : for (i=degpol(g); i>=0; i--)
764 : {
765 1683486 : for (j=d+1; j>2; j--) z[j] = Fl_add(z[j], z[j-1], pn); /* z = z*(1+T) */
766 44198 : z[2] = Fl_add(z[2], g[2+i], pn);
767 : }
768 756 : z = Flx_renormalize(z, d+2);
769 756 : if (Flx_weier_deg(z, p) <= lam) return z;
770 : }
771 : return NULL; /*LCOV_EXCL_LINE*/
772 : }
773 :
774 : static GEN
775 224 : FlxXn_translate1(GEN g, long p, long n)
776 : {
777 : long i, j, d, lam, pn, start;
778 : GEN z;
779 224 : if (n==1) start = 3;
780 0 : else if (n==2) start = 9;
781 0 : else start = 10;
782 224 : pn = upowuu(p, n);
783 224 : for (lam=start; lam; lam<<=1) /* least upper bound is 3 */
784 : {
785 224 : d = deg_trunc(lam, p, n, pn);
786 224 : z = const_vec(d+1, pol0_Flx(0)); /* z[2],...,z[d+1] <--> a_0,...,a_{d-1} */
787 224 : settyp(z, t_POL); z[1] = evalsigne(1) | evalvarn(0);
788 9408 : for (i=degpol(g); i>=0; i--)
789 : {
790 64288 : for (j=d+1; j>2; j--) gel(z, j) = Flx_add(gel(z, j), gel(z, j-1), pn);
791 9184 : gel(z, 2) = Flx_add(gel(z, 2), gel(g, 2+i), pn);
792 : }
793 224 : z = FlxX_renormalize(z, d+2);
794 224 : if (FlxX_weier_deg(z, p) <= lam) return z;
795 : }
796 : return NULL; /*LCOV_EXCL_LINE*/
797 : }
798 :
799 : /* lam < 0 => error (lambda can't be determined)
800 : * lam = 0 => return 1
801 : * lam > 0 => return dist. poly. of degree lam. */
802 : static GEN
803 84 : Flxn_Weierstrass_prep(GEN g, long p, long n, long d_chi)
804 : {
805 84 : long i, r0, d, dg = degpol(g), lam, pn, t;
806 : ulong lam0;
807 : GEN U, UINV, P, PU, g0, g1, gp, gU;
808 84 : if ((lam = Flx_weier_deg(g, p))==0) return(pol1_Flx(0));
809 77 : else if (lam<0)
810 0 : pari_err(e_MISC,"Flxn_Weierstrass_prep: precision too low. Increase n!");
811 77 : lam0 = lam/d_chi;
812 77 : pn = upowuu(p, n);
813 77 : d = deg_trunc(lam, p, n, pn);
814 77 : if (d>dg) d = dg;
815 77 : if (d<=lam) d=1+lam;
816 140 : for (r0=1; upowuu(p, r0)<lam0; r0++);
817 77 : g = Flxn_red(g, d);
818 77 : t = Fl_inv(g[2+lam], pn);
819 77 : g = Flx_Fl_mul(g, t, pn); /* normalized so as g[2+lam]=1 */
820 77 : U = Flx_shift(g, -lam);
821 77 : UINV = Flxn_inv(U, d, pn);
822 77 : P = zx_z_divexact(Flxn_red(g, lam), p); /* assume g[i] <= LONG_MAX */
823 77 : PU = Flxn_mul(P, UINV, d, pn);
824 77 : gU = Flxn_mul(g, UINV, d, pn);
825 77 : g0 = pol1_Flx(0);
826 77 : g1 = pol1_Flx(0);
827 385 : for (i=1; i<n; i++)
828 : {
829 308 : g1 = Flxn_shift_mul(g1, -lam, PU, d, pn);
830 308 : gp = Flx_Fl_mul(g1, upowuu(p, i), pn);
831 308 : g0 = (i&1)?Flx_sub(g0, gp, pn):Flx_add(g0, gp, pn);
832 : }
833 77 : g0 = Flxn_mul(g0, gU, lam+1, pn);
834 77 : g0 = Flx_red(g0, upowuu(p, (p==2)?n-r0:n+1-r0));
835 77 : return g0;
836 : }
837 :
838 : /* xi_n and Iwasawa pol. for Q(sqrt(m)) and p
839 : *
840 : * (flag&1)!=0 ==> output xi_n
841 : * (flag&2)!=0 ==> output power series
842 : * (flag&4)!=0 ==> output Iwasawa polynomial */
843 : static GEN
844 14 : imagquadstkpol(long p, long m, long n)
845 : {
846 14 : long pn = upowuu(p, n);
847 : GEN pol, stk, stk2;
848 : int *chi;
849 14 : if (p==2 && (m==-1 || m==-2 || m==-3 || m==-6)) return nullvec();
850 14 : if (p==3 && m==-3) return nullvec();
851 14 : if (p==2 && m%2==0) m /= 2;
852 14 : chi = set_quad_chi_2(m);
853 14 : stk = (p==2)? quadstk2(m, n, chi): quadstkp(p, m, n, chi);
854 14 : stk2 = zx_to_Flx(stk, pn);
855 14 : pol = Flxn_Weierstrass_prep(zlx_translate1(stk2, p, n), p, n, 1);
856 14 : return degpol(pol)? mkvec(Flx_to_ZX(pol)): nullvec();
857 : }
858 :
859 : /* a mod p == g^i mod p ==> omega(a)=zeta_(p-1)^(-i)
860 : * Chi[g^i mod p]=i (0 <= i <= p-2) */
861 : static GEN
862 0 : get_teich(long p, long g)
863 : {
864 0 : long i, gi = 1, p1 = p-1;
865 0 : GEN Chi = cgetg(p, t_VECSMALL);
866 0 : for (i=0; i<p1; i++) { Chi[gi] = i; gi = Fl_mul(gi, g, p); }
867 0 : return Chi;
868 : }
869 :
870 : /* Ichimura-Sumida criterion for Greenberg conjecture for real quadratic field.
871 : * chi: character of Q(sqrt(m)), omega: Teichmuller character mod p or 4.
872 : * Get Stickelberger element from chi^* = omega*chi^(-1) and convert it to
873 : * power series by the correspondence (Q_n/Q,1+q0)^(-1) <-> (1+T)(1+q0)^(-1) */
874 : static GEN
875 14 : realquadstkpol(long p, long m, long n)
876 : {
877 : int *chi;
878 14 : long pnm1 = upowuu(p, n-1),pn = p*pnm1, pn1 = p*pn, d, q0;
879 : GEN stk, ser, pol;
880 14 : if (m==1) pari_err_DOMAIN("quadstkpol", "m", "=", gen_1, gen_1);
881 14 : if (p==2 && (m&1)==0) m>>=1;
882 14 : d = ((m&3)==1)?m:m<<2;
883 14 : q0 = ulcm((p==2)?4:p, d);
884 14 : if (p==2)
885 : {
886 14 : chi = set_quad_chi_2(-m);
887 14 : stk = quadstk2(-m, n, chi);
888 14 : stk = zx_to_Flx_inplace(stk, pn);
889 : }
890 0 : else if (p==3 && m%3==0 && kross(-m/3,3)==1)
891 0 : {
892 0 : long m3 = m/3;
893 0 : chi = set_quad_chi_2(-m3);
894 0 : stk = quadstkp(3, -m3, n, chi);
895 0 : stk = zx_to_Flx_inplace(stk, pn);
896 : }
897 : else
898 : {
899 0 : long g = pgener_Zl(p);
900 0 : long x = Fl_powu(Fl_inv(g, p), pnm1, pn);
901 0 : GEN Chi = get_teich(p, g);
902 0 : GEN Gam = set_gam((1+q0)%pn1, p, n);
903 0 : chi = set_quad_chi_2(m);
904 0 : stk = quadteichstk(Chi, chi, Gam, p, m, n); /* exact */
905 0 : stk = zxX_to_FlxX(stk, pn); /* approx. */
906 0 : stk = FlxY_evalx(stk, x, pn);
907 : }
908 14 : stk = Flx_rescale_inplace(Flx_recip1_inplace(stk, pn), (1+q0)%pn, pn);
909 14 : ser = Flxn_translate1(stk, p, n);
910 14 : pol = Flxn_Weierstrass_prep(ser, p, n, 1);
911 14 : return degpol(pol)? mkvec(Flx_to_ZX(pol)): nullvec();
912 : }
913 :
914 : /* m > 0 square-free. lambda_2(Q(sqrt(-m)))
915 : * Kida, Tohoku Math. J. vol.31 (1979), 91-96, Theorem 1. */
916 : static GEN
917 0 : quadlambda2(long m)
918 : {
919 : long i, l, L;
920 : GEN P;
921 0 : if ((m&1)==0) m >>= 1; /* lam_2(Q(sqrt(-m)))=lam_2(Q(sqrt(-2*m))) */
922 0 : if (m <= 3) return mkvecs(0);
923 0 : P = gel(factoru(m), 1); l = lg(P);
924 0 : for (L = -1,i = 1; i < l; i++) L += 1L << (-3 + vals(P[i]-1) + vals(P[i]+1));
925 0 : return mkvecs(L);
926 : }
927 :
928 : /* Iwasawa lambda invariant of Q(sqrt(m)) (m<0) for p
929 : * |A_n|=p^(e[n])
930 : * kross(m,p)!=1 : e[n]-e[n-1]<eulerphi(p^n) ==> lambda=e[n]-e[n-1]
931 : * kross(m,p)==1 : e[n]-e[n-1]<=eulerphi(p^n) ==> lambda=e[n]-e[n-1]
932 : * Gold, Acta Arith. XXVI (1974), p.25, Cor. 3
933 : * Gold, Acta Arith. XXVI (1975), p.237, Cor. */
934 : static GEN
935 21 : quadlambda(long p, long m)
936 : {
937 : long flag, n, phipn;
938 21 : GEN e = cgetg(31, t_VECSMALL);
939 : int *chi;
940 21 : if (m>0) pari_err_IMPL("plus part of lambda invariant in quadlambda()");
941 21 : if (p==2) return quadlambda2(-m);
942 21 : if (p==3 && m==-3) return mkvec3(gen_0, gen_0, nullvec());
943 21 : flag = kross(m, p);
944 21 : e[1] = Z_lval(quadclassno(quaddisc(stoi(m))), p);
945 21 : if (flag!=1 && e[1]==0) return mkvec3(gen_0, gen_0, nullvec());
946 21 : chi = set_quad_chi_2(m);
947 21 : phipn = p-1; /* phipn=phi(p^n) */
948 56 : for (n=1; n; n++, phipn *= p)
949 : {
950 56 : long L = flag? ediff(p, m, n, chi): ediff_ber(p, m, n, chi);
951 56 : e[n+1] = e[n] + L;
952 56 : if ((flag!=1 && (L < phipn))|| (flag==1 && (L <= phipn))) break;
953 : }
954 21 : return vecsmall2vec2(e, n);
955 : }
956 :
957 : /* factor n-th cyclotomic polynomial mod p^r and return a minimal
958 : * polynomial of zeta_n over Q_p.
959 : * phi(n)=deg*n_conj, n_conj == 1 <=> polcyclo(n) is irred mod p. */
960 : static GEN
961 945 : set_minpol(ulong n, GEN p, ulong r, long n_conj)
962 : {
963 : GEN z, v, pol, pr;
964 : pari_timer ti;
965 945 : if (umodiu(p, n)==1) /* zeta_n in Z_p, faster than polcyclo() */
966 : {
967 420 : GEN prm1 = powiu(p, r-1), pr = mulii(prm1, p); /* pr=p^r */
968 420 : GEN prn = diviuexact(subii(pr, prm1), n); /* prn=phi(p^r)/n */
969 420 : z = Fp_pow(pgener_Fp(p), prn, pr);
970 420 : return deg1pol_shallow(gen_1, Fp_neg(z, pr), 0);
971 : }
972 525 : pr = powiu(p, r);
973 525 : pol = polcyclo(n, 0);
974 525 : if (n_conj==1) return FpX_red(pol, pr);
975 343 : if (DEBUGLEVEL>3) timer_start(&ti);
976 343 : z = FpX_one_cyclo(n, p);
977 343 : if (DEBUGLEVEL>3) timer_printf(&ti, "FpX_one_cyclo:n=%ld ", n);
978 343 : v = ZpX_liftfact(pol, mkvec2(z, FpX_div(pol, z, p)), pr, p, r);
979 343 : return gel(v, 1);
980 : }
981 :
982 : static GEN
983 91 : set_minpol_teich(ulong g_K, GEN p, ulong r)
984 : {
985 91 : GEN prm1 = powiu(p, r-1), pr = mulii(prm1, p), z;
986 91 : z = Fp_pow(Fp_inv(utoi(g_K), p), prm1, pr);
987 91 : return deg1pol_shallow(gen_1, Fp_neg(z, pr), 0);
988 : }
989 :
990 : static long
991 18963 : srh_1(GEN H)
992 : {
993 18963 : GEN bits = gel(H, 3);
994 18963 : ulong f = bits[1];
995 18963 : return F2v_coeff(bits, f-1);
996 : }
997 :
998 : /* (1/f)sum_{1<=a<=f}a*chi^{-1}(a) = -(1/(2-chi(a)))sum_{1<=a<=f/2} chi^{-1}(a)
999 : * does not overflow */
1000 : static GEN
1001 13146 : zx_ber_num(GEN Chi, long f, long d)
1002 : {
1003 13146 : long i, f2 = f>>1;
1004 13146 : GEN x = const_vecsmall(d+1, 0), x2 = x+2;
1005 51965081 : for (i = 1; i <= f2; i++)
1006 51951935 : if (Chi[i] >= 0) x2[Chi[i]] ++;
1007 13146 : return zx_renormalize(x, d+2);
1008 : }
1009 :
1010 : /* x a zx
1011 : * zx_ber_num is O(f). ZX[FpX,Flx]_ber_conj is O(d). Sometimes d<<f. */
1012 : static GEN
1013 26257 : ZX_ber_conj(GEN x, long j, long d)
1014 : {
1015 26257 : long i, deg = degpol(x);
1016 26257 : GEN y = pol_zero(d), x2 = x+2, y2 = y+2;
1017 818202 : for (i=0; i<=deg; i++) gel(y2, Fl_mul(i, j, d)) = stoi(x2[i]);
1018 26257 : return ZX_renormalize(y, d+2);
1019 : }
1020 :
1021 : /* x a zx */
1022 : static GEN
1023 252 : FpX_ber_conj(GEN x, long j, long d, GEN p)
1024 : {
1025 252 : long i, deg = degpol(x);
1026 252 : GEN y = pol_zero(d), x2 = x+2, y2 = y+2;
1027 756 : for (i=0; i<=deg; i++) gel(y2, Fl_mul(i, j, d)) = modsi(x2[i], p);
1028 252 : return FpX_renormalize(y, d+2);
1029 : }
1030 :
1031 : /* x a zx */
1032 : static GEN
1033 21756 : Flx_ber_conj(GEN x, long j, long d, ulong p)
1034 : {
1035 21756 : long i, deg = degpol(x);
1036 21756 : GEN y = const_vecsmall(d+1, 0), x2 = x+2, y2 = y+2;
1037 1076565 : for (i=0; i<=deg; i++) y2[Fl_mul(i, j, d)] = umodsu(x2[i], p);
1038 21756 : return Flx_renormalize(y, d+2);
1039 : }
1040 :
1041 : static GEN
1042 26257 : ZX_ber_den(GEN Chi, long j, long d)
1043 : {
1044 26257 : GEN x = pol_zero(d), x2 = x+2;
1045 26257 : if (Chi[2]>=0) gel(x2, Fl_neg(Fl_mul(Chi[2], j, d), d)) = gen_1;
1046 26257 : gel(x2, 0) = subiu(gel(x2, 0), 2);
1047 26257 : return ZX_renormalize(x, d+2);
1048 : }
1049 :
1050 : static GEN
1051 14490 : Flx_ber_den(GEN Chi, long j, long d, ulong p)
1052 : {
1053 14490 : GEN x = const_vecsmall(d+1, 0), x2 = x+2;
1054 14490 : if (Chi[2]>=0) x2[Fl_neg(Fl_mul(Chi[2], j, d), d)] = 1;
1055 14490 : x2[0] = Fl_sub(x2[0], 2, p);
1056 14490 : return Flx_renormalize(x, d+2);
1057 : }
1058 :
1059 : /* x is ZX of deg <= d-1 */
1060 : static GEN
1061 196 : ber_conj(GEN x, long k, long d)
1062 : {
1063 196 : long i, deg = degpol(x);
1064 196 : GEN z = pol_zero(d);
1065 196 : if (k==1)
1066 0 : for (i=0; i<=deg; i++) gel(z, 2+i) = gel(x, 2+i);
1067 : else
1068 122206 : for (i=0; i<=deg; i++) gel(z, 2+Fl_mul(i, k, d)) = gel(x, 2+i);
1069 196 : return ZX_renormalize(z, d+2);
1070 : }
1071 :
1072 : /* The computation is fast when p^n and el=1+k*f*p^n are less than 2^64
1073 : * for m <= n <= M
1074 : * We believe M>=3 is enough when f%p=0 and M>=2 is enough for other case
1075 : * because we expect that p^2 does not divide |A_{K,psi}| for a large p.
1076 : * FIXME: M should be set according to p and f. */
1077 : static void
1078 196 : set_p_f(GEN pp, ulong f, long *pm, long *pM)
1079 : {
1080 196 : ulong p = itou_or_0(pp);
1081 196 : if (!p || p >= 2000000) { *pm=2; *pM = dvdui(f, pp)? 3: 2; }
1082 189 : else if (p == 3) { *pm=5; *pM=20; }
1083 119 : else if (p == 5) { *pm=5; *pM=13; }
1084 56 : else if (p == 7) { *pm=5; *pM=11; }
1085 42 : else if (p == 11) { *pm=5; *pM=9; }
1086 35 : else if (p == 13) { *pm=5; *pM=8; }
1087 28 : else if (p < 400) { *pm=5; *pM=7; }
1088 0 : else if (p < 5000) { *pm=3; *pM=5; }
1089 0 : else if (p < 50000) { *pm=2; *pM=4; }
1090 0 : else { *pm=2; *pM=3; }
1091 196 : }
1092 :
1093 : static GEN
1094 18795 : subgp2ary(GEN H, long n)
1095 : {
1096 18795 : GEN v = gel(H, 3), w = cgetg(n+1, t_VECSMALL);
1097 18795 : long i, j, f = v[1];
1098 399982464 : for (i = 1, j = 0; i <= f; i++)
1099 399963669 : if (F2v_coeff(v,i)) w[++j] = i;
1100 18795 : return w;
1101 : }
1102 :
1103 : static GEN
1104 19124 : Flv_FlvV_factorback(GEN g, GEN x, ulong q)
1105 90216 : { pari_APPLY_ulong(Flv_factorback(g, gel(x,i), q)) }
1106 :
1107 : /* lift chi character on G/H to character on G */
1108 : static GEN
1109 18795 : zncharlift(GEN chi, GEN ncycGH, GEN U, GEN cycG)
1110 : {
1111 18795 : GEN nchi = char_normalize(chi, ncycGH);
1112 18795 : GEN c = ZV_ZM_mul(gel(nchi, 2), U), d = gel(nchi, 1);
1113 18795 : return char_denormalize(cycG, d, c);
1114 : }
1115 :
1116 : /* 0 <= c[i] < d, i=1..r; (c[1],...,c[r], d) = 1; find e[i] such that
1117 : * sum e[i]*c[i] = 1 mod d */
1118 : static GEN
1119 18795 : Flv_extgcd(GEN c, ulong d)
1120 : {
1121 18795 : long i, j, u, f, l = lg(c);
1122 18795 : GEN e = zero_zv(l-1);
1123 18795 : if (l == 1) return e;
1124 46053 : for (f = d, i = 1; f != 1 && i < l; i++)
1125 : {
1126 27258 : f = cbezout(f, itou(gel(c,i)), &u, &e[i]);
1127 27258 : if (!e[i]) continue;
1128 25004 : e[i] = umodsu(e[i], d);
1129 25004 : u = umodsu(u, d);
1130 32998 : if (u != 1) for (j = 1; j < i; j++) e[j] = Fl_mul(e[j], u, d);
1131 : }
1132 18795 : return e;
1133 : }
1134 :
1135 : /* f!=p; return exact xi. */
1136 : static GEN
1137 462 : get_xi_1(GEN Chi, GEN Gam, long p, long f, long n, long d, ulong pm)
1138 : {
1139 462 : GEN Gam1 = Gam+1, xi;
1140 : long i, j, j0, f0, pn, pn1, deg, pn1f;
1141 :
1142 462 : f0 = (f%p)?f:f/p;
1143 462 : pn = upowuu(p, n); pn1 = p*pn; pn1f = pn1%f;
1144 462 : xi = cgetg(pn+2, t_POL); xi[1] = evalsigne(1) | evalvarn(0);
1145 17556 : for (i=0; i<pn; i++) gel(xi, 2+i) = const_vecsmall(d+1, 0);
1146 432754 : for (j=1; j<pn1; j++)
1147 : {
1148 : long ipn1,*xij0;
1149 432292 : if ((j0 = Gam1[j])<0) continue;
1150 415660 : ipn1 = j%f; xij0 = gel(xi, 2+j0)+2;
1151 4027401588 : for (i=1; i<f0; i++)
1152 : {
1153 4026985928 : if ((ipn1 += pn1f) >= f) ipn1 -= f;
1154 4026985928 : if (ipn1==0 || (deg = Chi[ipn1])<0) continue;
1155 2203029612 : xij0[deg] += i;
1156 : }
1157 : }
1158 17556 : for (i=0; i<pn; i++) Flx_red_inplace(gel(xi, 2+i), pm);
1159 462 : return FlxX_renormalize(xi, pn+2);
1160 : }
1161 :
1162 : /* f=p; return p^(n+1)*xi mod pm. */
1163 : static GEN
1164 0 : get_xi_2(GEN Chi, GEN Gam, long p, long f, long n, long d, ulong pm)
1165 : {
1166 : long a, amodf, i, j0, pn, pn1, deg;
1167 0 : GEN Gam1 = Gam+1, xi;
1168 :
1169 0 : pn = upowuu(p, n); pn1 = p*pn;
1170 0 : xi = cgetg(pn+2, t_POL); xi[1] = evalsigne(1) | evalvarn(0);
1171 0 : for (i=0; i<pn; i++) gel(xi, 2+i) = const_vecsmall(d+1, 0);
1172 0 : for (a=1,amodf=0; a<pn1; a++) /* xi is exact */
1173 : {
1174 0 : if (++amodf==f) amodf = 0;
1175 0 : if ((j0=Gam1[a])<0 || amodf==0 || (deg=Chi[amodf])<0) continue;
1176 0 : mael(xi, 2+j0, 2+deg) += a;
1177 : }
1178 0 : for (i=0; i<pn; i++) Flx_red_inplace(gel(xi, 2+i), pm);
1179 0 : return FlxX_renormalize(xi, pn+2);
1180 : }
1181 :
1182 : static GEN
1183 56 : pol_chi_xi(GEN K, long p, long j, long n)
1184 : {
1185 56 : pari_sp av = avma;
1186 56 : GEN MinPol2 = gel(K, 7), xi = gel(K, 8);
1187 56 : long d = K_get_d(K), f = K_get_f(K), d_chi = K_get_dchi(K);
1188 56 : long wd, minpolpow = (f==p)?2*n+1:n, pm = upowuu(p, minpolpow);
1189 : GEN ser, pol, xi_conj;
1190 : pari_timer ti;
1191 :
1192 : /* xi is FlxX mod p^m, MinPol2 is Flx mod p^m, xi_conj is FlxqX. */
1193 56 : xi_conj = FlxqX_xi_conj(xi, MinPol2, j, d, pm);
1194 56 : if (d_chi==1) /* d_chi==1 if f==p */
1195 : {
1196 56 : xi_conj = FlxX_to_Flx(xi_conj);
1197 56 : if (f==p) xi_conj = zx_z_divexact(xi_conj, upowuu(p, n+1));
1198 : }
1199 : /* Now xi_conj is mod p^n */
1200 56 : if (DEBUGLEVEL>1) timer_start(&ti);
1201 56 : ser = (d_chi==1) ? Flxn_translate1(xi_conj, p, n)
1202 56 : : FlxXn_translate1(xi_conj, p, n);
1203 56 : if (DEBUGLEVEL>1) timer_printf(&ti, "Flx%sn_translate1",(d_chi==1)?"":"X");
1204 56 : wd = (d_chi==1)?Flx_weier_deg(ser, p):FlxX_weier_deg(ser, p);
1205 56 : if (wd<0) pari_err(e_MISC,"pol_chi_xi: precision too low. Increase n!\n");
1206 56 : else if (wd==0) return pol_1(0);
1207 : /* wd>0, convert to dist. poly. */
1208 56 : if (d_chi>1) /* f!=p. minpolpow==n */
1209 : {
1210 0 : ser = FlxqX_xi_norm(ser, MinPol2, p, d, upowuu(p, n));
1211 0 : ser = FlxX_to_Flx(ser);
1212 : }
1213 56 : pol = Flx_to_ZX(Flxn_Weierstrass_prep(ser, p, n, d_chi));
1214 56 : setvarn(pol, fetch_user_var("T"));
1215 : #ifdef DEBUG
1216 : if (wd>0 && d_chi>1)
1217 : err_printf("(wd,d_chi,p,f,d,j,H)=(%ld,%ld,%ld,%ld,%ld,%ld,%Ps)\n",
1218 : wd,d_chi,p,f,d,j,gmael3(K, 1, 1, 1));
1219 : #endif
1220 56 : return gerepilecopy(av, pol);
1221 : }
1222 :
1223 : /* return 0 if lam_psi (psi=chi^j) is determined to be zero.
1224 : * else return -1.
1225 : * If psi(p)!=1, then N_{Q(zeta_d)/Q}(1-psi(p))!=0 (mod p) */
1226 : static long
1227 14504 : lam_chi_ber(GEN K, long p, long j)
1228 : {
1229 14504 : pari_sp av = avma;
1230 14504 : GEN B1, B2, Chi = gel(K, 2), MinPol2 = gel(K, 7), B_num = gel(K, 8);
1231 14504 : long x, p2 = p*p, d = K_get_d(K), f = K_get_f(K);
1232 :
1233 14504 : if (f == d+1 && p == f && j == 1) return 0; /* Teichmuller */
1234 :
1235 14490 : B1 = Flx_rem(Flx_ber_conj(B_num, j, d, p2), MinPol2, p2);
1236 14490 : B2 = Flx_rem(Flx_ber_den(Chi, j, d, p2), MinPol2, p2);
1237 14490 : if (degpol(B1)<0 || degpol(B2)<0)
1238 49 : return gc_long(av, -1); /* 0 mod p^2 */
1239 14441 : x = zx_lval(B1, p) - zx_lval(B2, p);
1240 14441 : if (x<0) pari_err_BUG("subcycloiwasawa [Bernoulli number]");
1241 14441 : return gc_long(av, x==0 ? 0: -1);
1242 : }
1243 :
1244 : static long
1245 910 : lam_chi_xi(GEN K, long p, long j, long n)
1246 : {
1247 910 : pari_sp av = avma;
1248 910 : GEN xi_conj, z, MinPol2 = gel(K, 7), xi = gel(K, 8);
1249 910 : long d = K_get_d(K), f = K_get_f(K), d_chi = K_get_dchi(K);
1250 910 : long wd, minpolpow = (f==p)?n+2:1, pm = upowuu(p, minpolpow);
1251 :
1252 : /* xi is FlxX mod p^m, MinPol2 is Flx mod p^m, xi_conj is FlxqX. */
1253 910 : xi_conj = FlxqX_xi_conj(xi, MinPol2, j, d, pm);
1254 910 : if (d_chi==1) /* d_chi==1 if f==p */
1255 : {
1256 686 : xi_conj = FlxX_to_Flx(xi_conj);
1257 686 : if (f==p) xi_conj = zx_z_divexact(xi_conj, upowuu(p, n+1));
1258 : }
1259 : /* Now xi_conj is mod p^n */
1260 686 : z = (d_chi==1) ? Flxn_translate1(xi_conj, p, n)
1261 910 : : FlxXn_translate1(xi_conj, p, n);
1262 910 : wd = (d_chi==1)?Flx_weier_deg(z, p):FlxX_weier_deg(z, p);
1263 : #ifdef DEBUG
1264 : if (wd>0 && d_chi>1)
1265 : err_printf("(wd,d_chi,p,f,d,j,H)=(%ld,%ld,%ld,%ld,%ld,%ld,%Ps)\n",
1266 : wd,d_chi,p,f,d,j,gmael3(K, 1, 1, 1));
1267 : #endif
1268 910 : return gc_long(av, wd<0 ? -1 : wd*d_chi);
1269 : }
1270 :
1271 : /* K = [H1, Chi, Minpol, C, [d_chi, n_conj]] */
1272 : static GEN
1273 56 : imag_cyc_pol(GEN K, long p, long n)
1274 : {
1275 56 : pari_sp av = avma;
1276 56 : GEN Chi = gel(K, 2), MinPol = gel(K, 3), C = gel(K, 4), MinPol2;
1277 56 : long d_K = K_get_d(K), f = K_get_f(K), n_conj = K_get_nconj(K);
1278 56 : long i, q0, pn1, pM, pmodf = p%f, n_done = 0;
1279 56 : GEN z = nullvec(), Gam, xi, Lam, K2;
1280 :
1281 56 : Lam = const_vecsmall(n_conj, -1);
1282 56 : if (pmodf==0 || Chi[pmodf]) /* mark trivial chi-part using Bernoulli number */
1283 : {
1284 14 : MinPol2 = ZX_to_Flx(MinPol, p*p); /* p^2 for B_{1,chi} */
1285 14 : K2 = shallowconcat(K, mkvec2(MinPol2, zx_ber_num(Chi, f, d_K)));
1286 42 : for (i=1; i<=n_conj; i++)
1287 28 : if ((Lam[i] = lam_chi_ber(K2, p, C[i])) == 0) n_done++;
1288 14 : if (n_conj==n_done) return gerepilecopy(av, z); /* all chi-parts trivial */
1289 : }
1290 49 : q0 = (f%p)? f*p: f;
1291 49 : pn1 = upowuu(p, n+1);
1292 49 : Gam = set_gam((1+q0)%pn1, p, n);
1293 49 : pM = upowuu(p, (f==p)? 2*n+1: n);
1294 49 : MinPol2 = ZX_to_Flx(MinPol, pM);
1295 0 : xi = (f==p)? get_xi_2(Chi, Gam, p, f, n, d_K, pM)
1296 49 : : get_xi_1(Chi, Gam, p, f, n, d_K, pM);
1297 49 : K2 = shallowconcat(K, mkvec2(MinPol2, xi));
1298 105 : for (i=1; i<=n_conj; i++)
1299 : {
1300 : GEN z1;
1301 56 : if (Lam[i]>=0) continue;
1302 56 : z1 = pol_chi_xi(K2, p, C[i], n);
1303 56 : if (degpol(z1)) z = vec_append(z, z1); /* degpol(z1) may be zero */
1304 : }
1305 49 : return gerepilecopy(av, z);
1306 : }
1307 :
1308 : /* K is an imaginary cyclic extension of degree d contained in Q(zeta_f)
1309 : * H is the subgr of G=(Z/fZ)^* corresponding to K
1310 : * h=|H|, d*h=phi(f)
1311 : * G/H=<g> i.e. g^d \in H
1312 : * d_chi=[Q_p(zeta_d):Q_p], i.e. p^d_chi=1 (mod d)
1313 : * An inj. char. of G(K/Q) is automatically imaginary.
1314 : *
1315 : * G(K/Q)=G/H=<g>, chi:G(K/Q) -> overline{Q_p} s.t. chi(g)=zeta_d^(-1)
1316 : * Chi[a]=k, k<0 => chi(a)=0
1317 : * k>=0 => chi(a)=zeta_d^(-k)
1318 : * psi=chi^j, j in C : repre. of inj. odd char.
1319 : * psi(p)==1 <=> chi(p)^j==0 <=> j*Chi[p]=0 (mod d) <=> Chi[p]==0
1320 : *
1321 : * K = [H1, Chi, Minpol, C, [d_chi, n_conj]] */
1322 : static long
1323 3262 : imag_cyc_lam(GEN K, long p)
1324 : {
1325 3262 : pari_sp av = avma;
1326 3262 : GEN Chi = gel(K, 2), MinPol = gel(K, 3), C = gel(K, 4), MinPol2;
1327 3262 : long d_K = K_get_d(K), f = K_get_f(K), n_conj = K_get_nconj(K);
1328 3262 : long i, q0, n, pmodf = p%f, n_done = 0;
1329 : ulong pn1, pM;
1330 3262 : GEN p0 = utoi(p), Gam, Lam, xi, K2;
1331 :
1332 3262 : q0 = (f%p)? f*p: f;
1333 3262 : Lam = const_vecsmall(n_conj, -1);
1334 3262 : if (pmodf==0 || Chi[pmodf]) /* 1st trial is Bernoulli number */
1335 : {
1336 3052 : MinPol2 = ZX_to_Flx(MinPol, p*p); /* p^2 for B_{1,chi} */
1337 3052 : K2 = shallowconcat(K, mkvec2(MinPol2, zx_ber_num(Chi, f, d_K)));
1338 17528 : for (i=1; i<=n_conj; i++)
1339 14476 : if ((Lam[i] = lam_chi_ber(K2, p, C[i])) == 0) n_done++;
1340 3052 : if (n_conj==n_done) return gc_long(av, 0); /* all chi-parts trivial */
1341 : }
1342 413 : pM = pn1 = p;
1343 413 : for (n=1; n>=0; n++) /* 2nd trial is Stickelberger element */
1344 : {
1345 413 : pn1 *= p; /* p^(n+1) */
1346 413 : if (f == p)
1347 : { /* do not use set_minpol: it returns a new pol for each call */
1348 0 : GEN fac, cofac, v, pol = polcyclo(d_K, 0);
1349 0 : pM = pn1 * p; /* p^(n+2) */
1350 0 : fac = FpX_red(MinPol, p0); cofac = FpX_div(pol, fac, p0);
1351 0 : v = ZpX_liftfact(pol, mkvec2(fac, cofac), utoipos(pM), p0, n+2);
1352 0 : MinPol2 = gel(v, 1);
1353 : }
1354 413 : Gam = set_gam((1+q0)%pn1, p, n);
1355 413 : MinPol2 = ZX_to_Flx(MinPol, pM);
1356 0 : xi = (f==p)? get_xi_2(Chi, Gam, p, f, n, d_K, pM)
1357 413 : : get_xi_1(Chi, Gam, p, f, n, d_K, pM);
1358 413 : K2 = shallowconcat(K, mkvec2(MinPol2, xi));
1359 2205 : for (i=1; i<=n_conj; i++)
1360 1792 : if (Lam[i]<0 && (Lam[i] = lam_chi_xi(K2, p, C[i], n)) >= 0) n_done++;
1361 413 : if (n_conj==n_done) break;
1362 : }
1363 413 : return gc_long(av, zv_sum(Lam));
1364 : }
1365 : static GEN
1366 329 : GHinit(long f, GEN HH, GEN *pcycGH)
1367 : {
1368 329 : GEN G = znstar0(utoipos(f), 1);
1369 329 : GEN U, Ui, cycG, cycGH, ncycGH, gG, gGH, vChar, vH1, P, gH = gel(HH, 1);
1370 329 : long i, expG, n_f, lgH = lg(gH); /* gens. of H */
1371 329 : P = cgetg(lgH, t_MAT);
1372 805 : for (i = 1; i < lgH; i++) gel(P,i) = Zideallog(G, utoi(gH[i]));
1373 :
1374 : /* group structure of G/H */
1375 329 : cycG = znstar_get_cyc(G);
1376 329 : expG = itou(gel(cycG, 1));
1377 : /* gG generators of G, gGH generators of G/H: gGH = g.Ui, g = gGH.U */
1378 329 : cycGH = ZM_snf_group(hnfmodid(P, cycG), &U, &Ui);
1379 329 : ncycGH = cyc_normalize(cycGH);
1380 329 : gG = ZV_to_Flv(znstar_get_gen(G), f); /* gens. of G */
1381 : /* generators of G/H */
1382 329 : gGH = Flv_FlvV_factorback(gG, ZM_to_Flm(Ui, expG), f);
1383 329 : vChar = chargalois(cycGH, NULL); n_f = lg(vChar)-2;
1384 329 : vH1 = cgetg(n_f+1, t_VEC);
1385 19124 : for (i = 1; i <= n_f; i++)
1386 : { /* skip trivial character */
1387 18795 : GEN chi = gel(vChar,i+1), nchi = char_normalize(chi, ncycGH);
1388 18795 : GEN chiG, E, H1, C = gel(nchi, 2);
1389 18795 : long e, he, gen, d = itou(gel(nchi, 1));
1390 : /* chi(prod g[i]^e[i]) = e(sum e[i]*C[i] / d), chi has order d = #(G/H1)*/
1391 18795 : E = Flv_extgcd(C, d); /* \sum C[i]*E[i] = 1 in Z/dZ */
1392 :
1393 18795 : chiG = zncharlift(chi, ncycGH, U, cycG);
1394 18795 : H1 = charker(cycG, chiG); /* H1 < G with G/H1 cyclic */
1395 18795 : e = itou( zncharconductor(G, chiG) ); /* cond H1 = cond chi */
1396 18795 : H1 = Flv_FlvV_factorback(zv_to_Flv(gG, e), ZM_to_Flm(H1, expG), e);
1397 18795 : gen = Flv_factorback(zv_to_Flv(gGH, e), E, e);
1398 18795 : H1 = znstar_generate(e, H1); /* G/H1 = <gen>, chi(gen) = e(1/d) */
1399 18795 : he = eulerphiu(e) / d;
1400 : /* G/H1 = <gen> cyclic of index d, e = cond(H1) */
1401 18795 : gel(vH1,i) = mkvec3(H1, mkvecsmall5(d,e,he,srh_1(H1), gen),
1402 : subgp2ary(H1, he));
1403 : }
1404 329 : if (pcycGH) *pcycGH = cycGH;
1405 329 : return vH1;
1406 : }
1407 :
1408 : /* aH=g^iH ==> chi(a)=zeta_n^(-i); Chi[g^iH]=i; Chi[0] is never accessed */
1409 : static GEN
1410 13419 : get_chi(GEN H1)
1411 : {
1412 13419 : GEN H = _get_H(H1);
1413 13419 : long i, j, gi, d = _get_d(H1), f = _get_f(H1), h = _get_h(H1), g = _get_g(H1);
1414 13419 : GEN Chi = const_vecsmall(f-1, -1);
1415 :
1416 5584159 : for (j=1; j<=h; j++) Chi[H[j]] = 0; /* i = 0 */
1417 396592 : for (i = 1, gi = g; i < d; i++)
1418 : {
1419 40492081 : for (j=1; j<=h; j++) Chi[Fl_mul(gi, H[j], f)] = i;
1420 383173 : gi = Fl_mul(gi, g, f);
1421 : }
1422 13419 : return Chi;
1423 : }
1424 :
1425 : static void
1426 14 : errpdiv(const char *f, GEN p, long d)
1427 : {
1428 14 : pari_err_DOMAIN(f, "p", "divides",
1429 14 : strtoGENstr(stack_sprintf("[F:Q] = %ld", d)), p);
1430 0 : }
1431 : /* p odd doesn't divide degF; return lambda invariant if n==0 and
1432 : * iwasawa polynomials if n>=1 */
1433 : static GEN
1434 49 : abeliwasawa(long p, long f, GEN HH, long degF, long n)
1435 : {
1436 49 : long lam = 0, i, n_f;
1437 49 : GEN vH1, vData, z = nullvec(), p0 = utoi(p) ;
1438 :
1439 49 : vH1 = GHinit(f, HH, NULL); n_f = lg(vH1)-1;
1440 49 : vData = const_vec(degF, NULL);
1441 6608 : for (i=1; i<=n_f; i++) /* prescan. set Teichmuller */
1442 : {
1443 6573 : GEN H1 = gel(vH1,i);
1444 6573 : long d_K = _get_d(H1), f_K = _get_f(H1), g_K = _get_g(H1);
1445 :
1446 6573 : if (f_K == d_K+1 && p == f_K) /* found K=Q(zeta_p) */
1447 : {
1448 14 : long d_chi = 1, n_conj = eulerphiu(d_K);
1449 14 : GEN C = set_C(p, d_K, d_chi, n_conj);
1450 14 : long minpow = n? 2*n+1: 2;
1451 14 : GEN MinPol = set_minpol_teich(g_K, p0, minpow);
1452 14 : gel(vData, d_K) = mkvec4(MinPol, C, NULL, mkvecsmall2(d_chi, n_conj));
1453 14 : break;
1454 : }
1455 : }
1456 :
1457 6664 : for (i=1; i<=n_f; i++)
1458 : {
1459 6615 : GEN H1 = gel(vH1,i), z1, Chi, K;
1460 6615 : long d_K = _get_d(H1), s = _get_s(H1);
1461 :
1462 6615 : if (s) continue; /* F is real */
1463 : #ifdef DEBUG
1464 : err_printf(" handling %s cyclic subfield K, deg(K)=%ld, cond(K)=%ld\n",
1465 : s? "a real": "an imaginary", d_K, _get_f(H1));
1466 : #endif
1467 3318 : if (!gel(vData, d_K))
1468 : {
1469 126 : long d_chi = order_f_x(d_K, p), n_conj = eulerphiu(d_K)/d_chi;
1470 126 : GEN C = set_C(p, d_K, d_chi, n_conj);
1471 126 : long minpow = n? n+1: 2;
1472 126 : GEN MinPol = set_minpol(d_K, p0, minpow, n_conj);
1473 126 : gel(vData, d_K) = mkvec4(MinPol, C, NULL, mkvecsmall2(d_chi, n_conj));
1474 : }
1475 3318 : Chi = get_chi(H1);
1476 3318 : K = shallowconcat(mkvec2(H1, Chi), gel(vData, d_K));
1477 3318 : if (n==0) lam += imag_cyc_lam(K, p);
1478 56 : else if (lg(z1 = imag_cyc_pol(K, p, n)) > 1) z = shallowconcat(z, z1);
1479 : }
1480 49 : return n? z: mkvecs(lam);
1481 : }
1482 :
1483 : static GEN
1484 77 : ary2mat(GEN x, long n)
1485 : {
1486 : long i, j;
1487 77 : GEN z = cgetg(n+1,t_MAT);
1488 182 : for (i=1; i<=n; i++)
1489 : {
1490 105 : gel(z,i) = cgetg(n+1,t_COL);
1491 280 : for (j=1; j<=n; j++) gmael(z, i, j) = utoi(x[(i-1)*n+j-1]);
1492 : }
1493 77 : return z;
1494 : }
1495 :
1496 : static long
1497 0 : is_cyclic(GEN x)
1498 : {
1499 0 : GEN y = gel(x, 2);
1500 0 : long i, l = lg(y), n = 0;
1501 0 : for (i = 1; i < l; i++) if (signe(gel(y,i))) n++;
1502 0 : return n <= 1;
1503 : }
1504 :
1505 : static GEN
1506 77 : make_p_part(GEN y, ulong p, long d_pow)
1507 : {
1508 77 : long i, l = lg(y);
1509 77 : GEN z = cgetg(l, t_VECSMALL);
1510 182 : for (i = 1; i < l; i++) z[i] = signe(gel(y,i))? Z_lval(gel(y,i), p): d_pow;
1511 77 : return z;
1512 : }
1513 :
1514 : static GEN
1515 77 : structure_MLL(GEN y, long d_pow)
1516 : {
1517 77 : long y0, i, l = lg(y);
1518 77 : GEN x = gen_0, E = cgetg(l, t_VEC);
1519 182 : for (i = 1; i < l; i++)
1520 : {
1521 105 : if ((y0 = d_pow-y[i]) < 0) y0 = 0;
1522 105 : x = addiu(x, y0);
1523 105 : gel(E, l-i) = utoi(y0);
1524 : }
1525 77 : return mkvec2(x, E);
1526 : }
1527 :
1528 : static long
1529 14 : find_del_el(GEN *oldgr, GEN newgr, long n, long n_el, long d_chi)
1530 : {
1531 14 : if (n_el==1) return 1;
1532 14 : if (equalis(gmael(newgr, 2, 1), n_el)) return n;
1533 14 : if (cmpii(gel(*oldgr, 1), gel(newgr, 1)) >= 0) return n;
1534 14 : if (n > 1 && is_cyclic(newgr)) { *oldgr = newgr; return n-1; }
1535 14 : if (n == n_el) return n;
1536 14 : if (cmpis(gel(newgr, 1), n*d_chi) < 0) return n;
1537 14 : return 0;
1538 : }
1539 :
1540 : static GEN
1541 7 : subgr2vecsmall(GEN H, long h, long f)
1542 : {
1543 : long i;
1544 7 : GEN z = const_vecsmall(f-1, 0); /* f=lg(z) */
1545 2023 : for (i=1; i<=h; i++) z[H[i]] = 1; /* H[i]!=0 */
1546 7 : return z;
1547 : }
1548 :
1549 : /* K is the subfield of Q(zeta_f) with degree d corresponding to the subgroup
1550 : * H in (Z/fZ)^*; for a divisor e of f, zeta_e \in K <=> H \subset He. */
1551 : static long
1552 119 : root_of_1(long f, GEN H)
1553 : {
1554 119 : GEN g = gel(H, 1); /* generators */
1555 119 : long e = f, i, l = lg(g);
1556 119 : for (i = 1; i < l; i++)
1557 : {
1558 98 : e = cgcd(e, g[i] - 1);
1559 98 : if (e <= 2) return 2;
1560 : }
1561 21 : return odd(e)? (e<<1): e;
1562 : }
1563 :
1564 : static long
1565 259 : find_ele(GEN H)
1566 : {
1567 259 : long i, f=lg(H);
1568 369852 : for (i=1; i<f; i++) if (H[i]) return i;
1569 7 : return 0;
1570 : }
1571 :
1572 : static void
1573 252 : delete_ele(GEN H, long j, long el)
1574 : {
1575 252 : long f = lg(H), x = 1;
1576 2016 : do H[Fl_mul(j,x,f)] = 0;
1577 2016 : while ((x=Fl_mul(x,el,f))!=1);
1578 252 : }
1579 :
1580 : static GEN
1581 7 : get_coset(GEN H, long h, long f, long el)
1582 : {
1583 7 : long i, j, k = h/order_f_x(f, el);
1584 7 : GEN H2, coset = const_vecsmall(k, 0);
1585 7 : H2 = subgr2vecsmall(H, h, f);
1586 259 : for (i=0; (j=find_ele(H2))>0; i++)
1587 : {
1588 252 : coset[1+i] = j;
1589 252 : delete_ele(H2, j, el);
1590 : }
1591 7 : if (i != k) pari_err_BUG("failed to find coset\n");
1592 7 : return coset;
1593 : }
1594 :
1595 : static long
1596 3024 : srh_pol(GEN xpows, GEN vn, GEN pols, long el, long k, long f)
1597 : {
1598 3024 : pari_sp av = avma;
1599 3024 : long i, j, l = lg(pols), d = degpol(gel(pols, 1));
1600 3024 : GEN pol = gel(pols, 1);
1601 :
1602 654696 : for (i=1; i<l; i++)
1603 : {
1604 : GEN x, y, z;
1605 654696 : if (vn[i]==0) continue;
1606 331170 : y = gel(pols, vn[i]);
1607 331170 : z = pol0_Flx(0);
1608 3311700 : for (j=0; j<=d; j++)
1609 2980530 : z = Flx_add(z, Flx_Fl_mul(gel(xpows, 1+Fl_mul(j, k, f)), y[2+j], el), el);
1610 331170 : x = Flx_rem(z, pol, el);
1611 331170 : if (lg(x)==2)
1612 3024 : {vn[i] = 0; return gc_long(av, i); } /* pols[i] is min pol of zeta_f^k */
1613 : }
1614 0 : pari_err_BUG("subcyclopclgp [minimal polinomial]");
1615 0 : return 0; /* to suppress warning */
1616 : }
1617 :
1618 : /* e_chi[i mod dK] = chi(i*j), i = 0..dK-1; beware: e_chi is translated ! */
1619 : static GEN
1620 35857 : get_e_chi(GEN K, ulong j, ulong d, ulong *pdK)
1621 : {
1622 35857 : ulong i, dK = K_get_d(K);
1623 35857 : GEN TR = gel(K,4) + 2, e_chi = cgetg(dK+1, t_VECSMALL) + 1;
1624 35857 : if (j == 1)
1625 286902 : for (i = 0; i < dK; i++) e_chi[i] = umodiu(gel(TR, i), d);
1626 : else
1627 983345 : for (i = 0; i < dK; i++) e_chi[i] = umodiu(gel(TR, Fl_mul(i, j, dK)), d);
1628 35857 : *pdK = dK; return e_chi;
1629 : }
1630 : static GEN
1631 5145 : get_e_chi_ll(GEN K, ulong j, GEN d)
1632 : {
1633 5145 : ulong i, dK = umael3(K, 1, 2, 1);
1634 5145 : GEN TR = gel(K,4) + 2, e_chi = cgetg(dK+1, t_VEC) + 1;
1635 246785 : for (i = 0; i < dK; i++) gel(e_chi,i) = modii(gel(TR, Fl_mul(i, j, dK)), d);
1636 5145 : return e_chi;
1637 : }
1638 :
1639 : /* el=1 (mod f) */
1640 : static long
1641 0 : chk_el_real_f(GEN K, ulong p, ulong d_pow, ulong el, ulong j0)
1642 : {
1643 0 : pari_sp av = avma;
1644 0 : GEN H = K_get_H(K);
1645 0 : ulong d_K, f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
1646 0 : ulong i, j, gi, d = upowuu(p, d_pow), dp = d*p;
1647 0 : ulong g_el, z_f, flag = 0, el1f = (el-1)/f, el1dp = (el-1)/dp;
1648 0 : GEN e_chi = get_e_chi(K, j0, dp, &d_K);
1649 0 : GEN vz_f, xi_el = cgetg(d_K+1, t_VECSMALL)+1;
1650 :
1651 0 : g_el = pgener_Fl(el);
1652 0 : z_f = Fl_powu(g_el, el1f, el);
1653 0 : vz_f = Fl_powers(z_f, f-1, el)+1;
1654 :
1655 0 : for (gi = 1, i = 0; i < d_K; i++)
1656 : {
1657 0 : ulong x = 1;
1658 0 : for (j = 1; j <= h; j++)
1659 : {
1660 0 : ulong y = Fl_mul(H[j], gi, f);
1661 0 : x = Fl_mul(x, vz_f[y]-1, el); /* vz_f[y] = z_f^y */
1662 : }
1663 0 : gi = Fl_mul(gi, g_K, f);
1664 0 : xi_el[i] = x; /* xi_el[i] = xi^{g_K^i} mod el */
1665 : }
1666 0 : for (i=0; i<d_K; i++)
1667 : {
1668 0 : ulong x = 1;
1669 0 : for (j=0; j<d_K; j++)
1670 0 : x = Fl_mul(x, Fl_powu(xi_el[j], e_chi[(i+j)%d_K], el), el);
1671 0 : if ((x = Fl_powu(x, el1dp, el))!=1) flag = 1;
1672 0 : if (Fl_powu(x, p, el)!=1) return gc_long(av,0);
1673 : }
1674 0 : return gc_long(av, flag?1:0);
1675 : }
1676 :
1677 : /* For a cyclic field K contained in Q(zeta_f),
1678 : * computes minimal polynomial T of theta=Tr_{Q(zeta_f)/K}(zeta_f) over Q
1679 : * and conjugates of theta */
1680 : static GEN
1681 14 : minpol_theta(GEN K)
1682 : {
1683 14 : GEN HH = gmael3(K,1,1,1);
1684 14 : return galoissubcyclo(utoi(K_get_f(K)), HH, 0, 0);
1685 : }
1686 :
1687 : /* xi[1+i] = i-th conj of xi = Tr_{Q(zeta_f)/K}(1-zeta_f).
1688 : * |1-(cos(x)+i*sin(x))|^2 = 2(1-cos(x)) */
1689 : static GEN
1690 0 : xi_approx(GEN K, long prec)
1691 : {
1692 0 : pari_sp av = avma;
1693 0 : GEN H = K_get_H(K);
1694 0 : long d_K = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
1695 0 : GEN xi = cgetg(d_K+1, t_COL), vz_f = grootsof1(f, prec);
1696 0 : long i, j, g = 1, h2 = h>>1;
1697 0 : for (i=1; i<=d_K; i++)
1698 : {
1699 0 : GEN y = real_1(prec);
1700 0 : for (j=1; j<=h2; j++)
1701 : {
1702 0 : GEN z = gmael(vz_f, 1+Fl_mul(H[j], g, f), 1);
1703 0 : y = mulrr(y, shiftr(subsr(1, z), 1));
1704 : }
1705 0 : gel(xi, i) = y;
1706 0 : g = Fl_mul(g, g_K, f);
1707 : }
1708 0 : return gerepilecopy(av, xi);
1709 : }
1710 :
1711 : static GEN
1712 47 : theta_xi_el(GEN K, ulong el)
1713 : {
1714 47 : pari_sp av = avma;
1715 47 : GEN H = K_get_H(K);
1716 47 : ulong d_K = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
1717 47 : GEN theta = cgetg(d_K+1, t_VECSMALL), xi = cgetg(d_K+1, t_VECSMALL), vz_f;
1718 47 : ulong i, j, g = 1, x, y, g_el, z_f;
1719 :
1720 47 : g_el = pgener_Fl(el);
1721 47 : z_f = Fl_powu(g_el, (el-1)/f, el);
1722 47 : vz_f = Fl_powers(z_f, f-1, el);
1723 1109 : for (i=1; i<=d_K; i++)
1724 : {
1725 1062 : x = 0; y = 1;
1726 28254 : for (j=1; j<=h; j++)
1727 : {
1728 27192 : ulong z = vz_f[1+Fl_mul(H[j], g, f)];
1729 27192 : x = Fl_add(x, z, el);
1730 27192 : y = Fl_mul(y, z-1, el);
1731 : }
1732 1062 : theta[i] = x;
1733 1062 : xi[i] = y;
1734 1062 : g = Fl_mul(g, g_K, f);
1735 : }
1736 47 : return gerepilecopy(av, mkvec2(theta, xi));
1737 : }
1738 :
1739 : static GEN
1740 47 : make_Xi(GEN xi, long d)
1741 : {
1742 : long i, j;
1743 47 : GEN Xi = cgetg(d+1, t_MAT);
1744 1109 : for (j=0; j<d; j++)
1745 : {
1746 1062 : GEN x = cgetg(d+1, t_VECSMALL);
1747 1062 : gel(Xi, 1+j) = x;
1748 27714 : for (i=0; i<d; i++) x[1+i] = xi[1+(i+j)%d];
1749 : }
1750 47 : return Xi;
1751 : }
1752 :
1753 : static GEN
1754 47 : make_Theta(GEN theta, ulong d, ulong el)
1755 : {
1756 : ulong i;
1757 47 : GEN Theta = cgetg(d+1, t_MAT);
1758 1109 : for (i=1; i<=d; i++) gel(Theta, i) = Fl_powers(theta[i], d-1, el);
1759 47 : return Flm_inv(Theta, el);
1760 : }
1761 :
1762 : static GEN
1763 47 : Xi_el(GEN K, GEN tInvA, ulong el)
1764 : {
1765 47 : pari_sp av = avma;
1766 47 : ulong d_K = K_get_d(K);
1767 47 : GEN tx = theta_xi_el(K, el), Theta, Xi, X;
1768 :
1769 47 : if ((Theta = make_Theta(gel(tx, 1), d_K, el))==NULL) return NULL;
1770 47 : Xi = make_Xi(gel(tx, 2), d_K);
1771 47 : X = Flm_mul(Flm_mul(Xi, Theta, el), ZM_to_Flm(tInvA, el), el);
1772 47 : return gerepilecopy(av, X);
1773 : }
1774 :
1775 : static GEN
1776 0 : pol_xi_el(GEN K, ulong el)
1777 : {
1778 0 : pari_sp av = avma;
1779 0 : ulong d_K = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
1780 0 : GEN H = K_get_H(K), xi = cgetg(d_K+1, t_VECSMALL), vz_f;
1781 0 : ulong i, j, g = 1, y, g_el, z_f;
1782 :
1783 0 : g_el = pgener_Fl(el);
1784 0 : z_f = Fl_powu(g_el, (el-1)/f, el);
1785 0 : vz_f = Fl_powers(z_f, f-1, el);
1786 0 : for (i=1; i<=d_K; i++)
1787 : {
1788 0 : y = 1;
1789 0 : for (j=1; j<=h; j++)
1790 : {
1791 0 : ulong z = vz_f[1+Fl_mul(H[j], g, f)];
1792 0 : y = Fl_mul(y, z-1, el);
1793 : }
1794 0 : xi[i] = y;
1795 0 : g = Fl_mul(g, g_K, f);
1796 : }
1797 0 : return gerepilecopy(av, Flv_roots_to_pol(xi, el, 0));
1798 : }
1799 :
1800 : /* theta[1+i] = i-th conj of theta; xi[1+i] = i-th conj of xi. */
1801 : static GEN
1802 14 : theta_xi_approx(GEN K, long prec)
1803 : {
1804 14 : pari_sp av = avma;
1805 14 : GEN H = K_get_H(K);
1806 14 : long d_K = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
1807 14 : GEN theta = cgetg(d_K+1, t_VEC), xi = cgetg(d_K+1, t_VEC);
1808 14 : GEN vz_f = grootsof1(f, prec);
1809 14 : long i, j, g = 1, h2 = h>>1;
1810 :
1811 238 : for (i=1; i<=d_K; i++)
1812 : {
1813 224 : GEN x = real_0(prec), y = real_1(prec);
1814 5068 : for (j=1; j<=h2; j++)
1815 : {
1816 4844 : GEN z = gmael(vz_f, 1+Fl_mul(H[j], g, f), 1);
1817 4844 : x = addrr(x, z);
1818 4844 : y = mulrr(y, shiftr(subsr(1, z), 1));
1819 : }
1820 224 : gel(theta, i) = shiftr(x, 1);
1821 224 : gel(xi, i) = y;
1822 224 : g = Fl_mul(g, g_K, f);
1823 : }
1824 14 : return gerepilecopy(av, mkvec2(theta, xi));
1825 : }
1826 :
1827 : static GEN
1828 14 : bound_coeff_xi(GEN K, GEN tInvA)
1829 : {
1830 14 : pari_sp av = avma;
1831 14 : long d_K = K_get_d(K), prec = MEDDEFAULTPREC, i;
1832 14 : GEN tInvV, R = cgetg(d_K+1, t_MAT), theta_xi = theta_xi_approx(K, prec);
1833 14 : GEN theta = gel(theta_xi, 1), xi = gel(theta_xi, 2), x1, x2, bound;
1834 :
1835 238 : for (i=1; i<=d_K; i++)
1836 : {
1837 224 : GEN z = gpowers(gel(theta, i), d_K-1);
1838 224 : settyp(z, t_COL);
1839 224 : gel(R, i) = z;
1840 : }
1841 14 : tInvV = RgM_mul(RgM_inv(R), tInvA);
1842 14 : x1 = gsupnorm(tInvV, prec); x2 = gsupnorm(xi, prec);
1843 14 : bound = mulrs(mulrr(x1, x2), 3*d_K);
1844 14 : return gerepilecopy(av, bound);
1845 : }
1846 :
1847 : static GEN
1848 14 : get_Xi(GEN K, GEN tInvA)
1849 : {
1850 14 : pari_sp av = avma;
1851 : GEN M0, XI, EL, Xi;
1852 14 : ulong f = K_get_f(K), el, e, n, i;
1853 : forprime_t T0;
1854 :
1855 14 : M0 = bound_coeff_xi(K, tInvA);
1856 14 : e = expo(M0)+1; n = e/(BITS_IN_LONG-1); n++;
1857 14 : EL = cgetg(1+n, t_VECSMALL);
1858 14 : XI = cgetg(1+n, t_VEC);
1859 14 : u_forprime_arith_init(&T0, LONG_MAX, ULONG_MAX, 1, f);
1860 :
1861 61 : for (i=1; i<=n; i++)
1862 : {
1863 47 : el = u_forprime_next(&T0);
1864 47 : while ((Xi=Xi_el(K, tInvA, el))==NULL) el = u_forprime_next(&T0);
1865 47 : gel(XI, i) = Xi;
1866 47 : EL[i] = el;
1867 : }
1868 14 : return gerepileupto(av, nmV_chinese_center(XI, EL, NULL));
1869 : }
1870 :
1871 : /* K is a cyclic field of conductor f with degree d=d_K
1872 : * xi = Norm_{Q(zeta_f)/K}(1-zeta_f)
1873 : * 1: T, min poly of a=Tr_{Q(zeta_f)/K}(zeta_f) over Q
1874 : * 2: B, power basis of K with respect to a
1875 : * 3: A, rational matrix s.t. t(v_1,...v_d) = A*t(1,a,...,a^{d-1})
1876 : * 4: Xi, integer matrix s.t. t(xi^(1),...,xi^(d)) = Xi*t(v_1,...,v_d) */
1877 : static GEN
1878 14 : xi_data_basis(GEN K)
1879 : {
1880 14 : pari_sp av = avma;
1881 14 : GEN T = minpol_theta(K);
1882 : GEN InvA, A, M, Xi, A_den;
1883 14 : GEN D, B = nfbasis(T, &D);
1884 : pari_timer ti;
1885 14 : if (DEBUGLEVEL>1) timer_start(&ti);
1886 14 : A = RgXV_to_RgM(B, lg(B)-1);
1887 14 : M = gmael(A, 1, 1);
1888 14 : if (!equali1(M)) A = RgM_Rg_div(A, M);
1889 14 : InvA = QM_inv(A);
1890 14 : A = Q_remove_denom(A, &A_den);
1891 14 : if (A_den==NULL) A_den = gen_1;
1892 14 : Xi = get_Xi(K, shallowtrans(InvA));
1893 14 : if (DEBUGLEVEL>1) timer_printf(&ti, "xi_data_basis");
1894 14 : return gerepilecopy(av, mkvec5(T, B, shallowtrans(A), Xi, A_den));
1895 : }
1896 :
1897 : /* When factorization of polcyclo mod el is difficult, one can try to
1898 : * check the condition of el using an integral basis of K.
1899 : * This is useful when d_K is small. */
1900 : static long
1901 14 : chk_el_real_basis(GEN K, long p, long d_pow, long el, long j0)
1902 : {
1903 14 : pari_sp av = avma;
1904 14 : GEN xi = gel(K, 7), T = gel(xi, 1), A = gel(xi, 3), Xi = gel(xi, 4);
1905 14 : GEN A_den = gel(xi, 5);
1906 14 : ulong i, j, x, found = 0;
1907 : GEN v_el, xi_el;
1908 : GEN e_chi, xi_e_chi;
1909 : ulong d_K, d, dp, el1dp;
1910 :
1911 14 : if (dvdiu(A_den, el)) return 0;
1912 :
1913 14 : d = upowuu(p, d_pow); dp = d*p; el1dp = (el-1)/dp;
1914 14 : e_chi = get_e_chi(K, j0, dp, &d_K);
1915 14 : xi_e_chi = cgetg(d_K+1, t_VECSMALL)+1;
1916 :
1917 14 : if (DEBUGLEVEL>1) err_printf("chk_el_real_basis: d_K=%ld el=%ld\n",d_K,el);
1918 14 : A = ZM_to_Flm(A, el);
1919 14 : A = Flm_Fl_mul(A, Fl_inv(umodiu(A_den, el), el), el);
1920 14 : x = Flx_oneroot_split(ZX_to_Flx(T, el), el);
1921 14 : v_el = Flm_Flc_mul(A, Fl_powers(x, d_K-1, el), el);
1922 14 : xi_el = Flm_Flc_mul(ZM_to_Flm(Xi, el), v_el, el);
1923 14 : if (DEBUGLEVEL>2) err_printf("el=%ld xi_el=%Ps\n", el, xi_el);
1924 238 : for (i=0; i<d_K; i++)
1925 : {
1926 224 : long z = 1;
1927 5208 : for (j=0; j<d_K; j++)
1928 4984 : z = Fl_mul(z, Fl_powu(xi_el[1+j], e_chi[(i+j)%d_K], el), el);
1929 224 : xi_e_chi[i] = z;
1930 : }
1931 14 : if (DEBUGLEVEL>2) err_printf("xi_e_chi=%Ps\n", xi_e_chi-1);
1932 238 : for (i=0; i<d_K; i++)
1933 : {
1934 224 : long x = Fl_powu(xi_e_chi[i], el1dp, el);
1935 224 : if (x!=1) found = 1;
1936 224 : if (Fl_powu(x, p, el)!=1) return gc_long(av, 0);
1937 : }
1938 14 : return gc_long(av, found);
1939 : }
1940 :
1941 : static GEN
1942 0 : bound_pol_xi(GEN K)
1943 : {
1944 0 : pari_sp av = avma;
1945 0 : GEN xi = xi_approx(K, MEDDEFAULTPREC);
1946 0 : GEN M = real_1(MEDDEFAULTPREC), one = rtor(dbltor(1.0001), MEDDEFAULTPREC);
1947 0 : long i, n = lg(xi);
1948 :
1949 0 : for (i=1; i<n; i++) M = mulrr(M, addrr(one, gel(xi, i)));
1950 0 : M = mulrs(M, 3);
1951 0 : return gerepilecopy(av, M);
1952 : }
1953 :
1954 : static GEN
1955 0 : minpol_xi(GEN K)
1956 : {
1957 0 : pari_sp av = avma;
1958 : GEN M0, POL, EL;
1959 0 : ulong f = K_get_f(K), el, e, n, i;
1960 : forprime_t T0;
1961 :
1962 0 : M0 = bound_pol_xi(K);
1963 0 : e = expo(M0)+1; n = e/(BITS_IN_LONG-1); n++;
1964 0 : EL = cgetg(1+n, t_VECSMALL);
1965 0 : POL = cgetg(1+n, t_VEC);
1966 0 : u_forprime_arith_init(&T0, LONG_MAX, ULONG_MAX, 1, f);
1967 0 : for (i=1; i<=n; i++)
1968 : {
1969 0 : el = u_forprime_next(&T0);
1970 0 : gel(POL, i) = pol_xi_el(K, el);
1971 0 : EL[i] = el;
1972 : }
1973 0 : return gerepileupto(av, nxV_chinese_center(POL, EL, NULL));
1974 : }
1975 :
1976 : static long
1977 0 : find_conj_el(GEN K, GEN pol, GEN Den)
1978 : {
1979 0 : pari_sp av = avma;
1980 0 : GEN H = K_get_H(K);
1981 0 : ulong d_K = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g = K_get_g(K);
1982 0 : ulong j, k, el, g_el, z_f, xi = 1, xi_g = 1;
1983 0 : GEN T = NULL, vz_f;
1984 :
1985 0 : for (el=f+1; el; el+=f)
1986 0 : if (uisprime(el) && dvdiu(Den, el)==0)
1987 : {
1988 0 : T = ZX_to_Flx(pol, el);
1989 0 : T = Flx_Fl_mul(T, Fl_inv(umodiu(Den, el), el), el);
1990 0 : break;
1991 : }
1992 0 : g_el = pgener_Fl(el);
1993 0 : z_f = Fl_powu(g_el, (el-1)/f, el);
1994 0 : vz_f = Fl_powers(z_f, f-1, el);
1995 0 : for (j=1; j<=h; j++)
1996 0 : xi = Fl_mul(xi, vz_f[1+H[j]]-1, el);
1997 0 : for (j=1; j<=h; j++)
1998 0 : xi_g = Fl_mul(xi_g, vz_f[1+Fl_mul(H[j], g, f)]-1, el);
1999 0 : for (k=1; k<=d_K; k++)
2000 : {
2001 0 : xi = Flx_eval(T, xi, el);
2002 0 : if (xi == xi_g) break;
2003 : }
2004 0 : if (xi != xi_g) pari_err_BUG("find_conj_el");
2005 0 : return gc_long(av, k);
2006 : }
2007 :
2008 : /* G = H_1*H_2*...*H_m is cyclic of order n, H_i=<perms[i]>
2009 : * G is not necessarily a direct product.
2010 : * If p^e || n, then p^e || |H_i| for some i.
2011 : * return a generator of G. */
2012 : static GEN
2013 0 : find_gen(GEN perms, long n)
2014 : {
2015 0 : GEN fa = factoru(n), P = gel(fa, 1), E = gel(fa, 2);
2016 0 : long i, j, l = lg(perms), r = lg(P);
2017 0 : GEN gen = cgetg(r, t_VEC), orders = cgetg(l, t_VECSMALL), perm;
2018 :
2019 0 : for (i=1; i<l; i++) orders[i] = perm_orderu(gel(perms, i));
2020 0 : for (i=1; i<r; i++)
2021 : {
2022 0 : long pe = upowuu(P[i], E[i]);
2023 0 : for (j=1; j<l; j++) if (orders[j]%pe==0) break;
2024 0 : gel(gen, i) = perm_powu(gel(perms, j), orders[j]/pe);
2025 : }
2026 0 : perm = gel(gen, 1);
2027 0 : for (i=2; i<l; i++) perm = perm_mul(perm, gel(gen, i));
2028 0 : return perm;
2029 : }
2030 :
2031 : /* R is the roots of T. R[1+i] = R[1]^(g_K^i), 0 <= i <= d_K-1
2032 : * 1: min poly T of xi over Q
2033 : * 2: F(x)\in Q[x] s.t. xi^(g_K)=F(xi) */
2034 : static GEN
2035 0 : xi_data_galois(GEN K)
2036 : {
2037 0 : pari_sp av = avma;
2038 : GEN T, G, perms, perm, pol, pol2, Den;
2039 0 : ulong k, d_K = K_get_d(K);
2040 : pari_timer ti;
2041 :
2042 0 : if (DEBUGLEVEL>1) timer_start(&ti);
2043 0 : T = minpol_xi(K);
2044 0 : if (DEBUGLEVEL>1) timer_printf(&ti, "minpol_xi");
2045 0 : G = galoisinit(T, NULL);
2046 0 : if (DEBUGLEVEL>1) timer_printf(&ti, "galoisinit");
2047 0 : perms = gal_get_gen(G);
2048 0 : perm = (lg(perms)==2)?gel(perms, 1):find_gen(perms, d_K);
2049 0 : if (DEBUGLEVEL>1) timer_start(&ti);
2050 0 : pol = galoispermtopol(G, perm);
2051 0 : if (DEBUGLEVEL>1) timer_printf(&ti, "galoispermtopol");
2052 0 : pol = Q_remove_denom(pol, &Den);
2053 0 : if (Den==NULL) Den = gen_1;
2054 0 : k = find_conj_el(K, pol, Den);
2055 0 : if (DEBUGLEVEL>1) timer_printf(&ti,"find_conj");
2056 0 : pol2 = galoispermtopol(G, perm_powu(perm, k));
2057 0 : pol2 = Q_remove_denom(pol2, &Den);
2058 0 : if (Den==NULL) Den = gen_1;
2059 0 : return gerepilecopy(av, mkvec3(T, pol2, Den));
2060 : }
2061 :
2062 : /* If g(X)\in Q[X] s.t. g(xi)=xi^{g_K} was found,
2063 : * then we fix an integer x_0 s.t. xi=x_0 (mod el) and construct x_i
2064 : * s.t. xi^{g_K^i}=x_i (mod el) using g(X). */
2065 : static long
2066 0 : chk_el_real_galois(GEN K, long p, long d_pow, long el, long j0)
2067 : {
2068 0 : pari_sp av = avma;
2069 0 : GEN xi = gel(K, 7), T = gel(xi, 1), F = gel(xi, 2), Den = gel(xi, 3);
2070 : GEN Fel, xi_el, xi_e_chi, e_chi;
2071 0 : ulong i, j, x, found = 0;
2072 : ulong d_K, d, dp, el1dp;
2073 :
2074 0 : if (dvdiu(Den, el)) return 0;
2075 :
2076 0 : d = upowuu(p, d_pow); dp = d*p; el1dp = (el-1)/dp;
2077 0 : e_chi = get_e_chi(K, j0, dp, &d_K);
2078 0 : xi_el = cgetg(d_K+1, t_VECSMALL)+1;
2079 0 : xi_e_chi = cgetg(d_K+1, t_VECSMALL)+1;
2080 :
2081 0 : if (DEBUGLEVEL>1) err_printf("chk_el_real_galois: d_K=%ld el=%ld\n",d_K,el);
2082 0 : Fel = ZX_to_Flx(F, el);
2083 0 : Fel = Flx_Fl_mul(Fel, Fl_inv(umodiu(Den, el), el), el);
2084 0 : x = Flx_oneroot_split(ZX_to_Flx(T, el), el);
2085 0 : for (i=0; i<d_K; i++) { xi_el[i] = x; x = Flx_eval(Fel, x, el); }
2086 0 : if (DEBUGLEVEL>2) err_printf("el=%ld xi_el=%Ps\n", el, xi_el-1);
2087 0 : for (i=0; i<d_K; i++)
2088 : {
2089 0 : long z = 1;
2090 0 : for (j=0; j<d_K; j++)
2091 0 : z = Fl_mul(z, Fl_powu(xi_el[j], e_chi[(i+j)%d_K], el), el);
2092 0 : xi_e_chi[i] = z;
2093 : }
2094 0 : if (DEBUGLEVEL>2) err_printf("xi_e_chi=%Ps\n", xi_e_chi-1);
2095 0 : for (i=0; i<d_K; i++)
2096 : {
2097 0 : long x = Fl_powu(xi_e_chi[i], el1dp, el);
2098 0 : if (x!=1) found = 1;
2099 0 : if (Fl_powu(x, p, el)!=1) return gc_long(av, 0);
2100 : }
2101 0 : return gc_long(av, found);
2102 : }
2103 :
2104 : /* checks the condition of el using the irreducible polynomial G_K(X) of zeta_f
2105 : * over K. G_K(X) mod el is enough for our purpose and it is obtained by
2106 : * factoring polcyclo(f) mod el */
2107 : static long
2108 7 : chk_el_real_factor(GEN K, long p, long d_pow, long el, long j0)
2109 : {
2110 7 : pari_sp av = avma;
2111 7 : GEN H = K_get_H(K);
2112 7 : ulong f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
2113 7 : ulong i, j, k, d_K, d = upowuu(p, d_pow), dp = d*p, found = 0;
2114 : GEN pols, coset, vn_g, polnum, xpows, G_K;
2115 7 : ulong el1dp = (el-1)/dp, n_coset, n_g, gi;
2116 7 : GEN e_chi = get_e_chi(K, j0, dp, &d_K);
2117 : pari_timer ti;
2118 :
2119 7 : if (DEBUGLEVEL>1) err_printf("chk_el_real_factor: f=%ld el=%ld\n",f,el);
2120 7 : coset = get_coset(H, h, f, el);
2121 7 : if (DEBUGLEVEL>1)
2122 : {
2123 0 : timer_start(&ti);
2124 0 : err_printf("factoring polyclo(d) (mod %ld)\n",f, el);
2125 : }
2126 7 : pols = Flx_factcyclo(f, el, 0);
2127 7 : if (DEBUGLEVEL>1) timer_printf(&ti,"Flx_factcyclo(%lu,%lu)",f,el);
2128 7 : n_coset = lg(coset)-1;
2129 7 : n_g = lg(pols)-1;
2130 7 : vn_g = identity_perm(n_g);
2131 :
2132 7 : polnum = const_vec(d_K, NULL);
2133 91 : for (i=1; i<=d_K; i++) gel(polnum, i) = const_vecsmall(n_coset, 0);
2134 7 : xpows = Flxq_powers(polx_Flx(0), f-1, gel(pols, 1), el);
2135 91 : for (gi=1,i=1; i<=d_K; i++)
2136 : {
2137 3108 : for (j=1; j<=n_coset; j++)
2138 : {
2139 3024 : long x, conj = Fl_mul(gi, coset[j], f);
2140 3024 : x = srh_pol(xpows, vn_g, pols, el, conj, f);
2141 3024 : gel(polnum, i)[j] = x;
2142 : }
2143 84 : gi = Fl_mul(gi, g_K, f);
2144 : }
2145 7 : G_K = const_vec(d_K, NULL);
2146 91 : for (i=1; i<=d_K; i++)
2147 : {
2148 84 : GEN z = pol1_Flx(0);
2149 3108 : for (j=1; j<=n_coset; j++) z = Flx_mul(z, gel(pols, gel(polnum,i)[j]), el);
2150 84 : gel(G_K, i) = z;
2151 : }
2152 7 : if (DEBUGLEVEL>2) err_printf("G_K(x)=%Ps\n",Flx_to_ZX(gel(G_K, 1)));
2153 91 : for (k=0; k<d_K; k++)
2154 : {
2155 84 : long x = 1;
2156 1092 : for (i = 0; i < d_K; i++)
2157 : {
2158 : long x0, t;
2159 1008 : x0 = Flx_eval(gel(G_K, 1+i), 1, el);
2160 1008 : t = Fl_powu(x0, e_chi[(i+k)%d_K], el);
2161 1008 : x = Fl_mul(x, t, el);
2162 : }
2163 84 : x = Fl_powu(x, el1dp, el);
2164 84 : if (x!=1) found = 1;
2165 84 : if (Fl_powu(x, p, el)!=1) return gc_long(av, 0);
2166 : }
2167 7 : return gc_long(av, found);
2168 : }
2169 :
2170 : static long
2171 21 : chk_el_real_chi(GEN K, ulong p, ulong d_pow, ulong el, ulong j0, long flag)
2172 : {
2173 21 : ulong f = K_get_f(K);
2174 :
2175 21 : if (el%f == 1)
2176 0 : return chk_el_real_f(K, p, d_pow, el, j0); /* must be faster */
2177 21 : if (flag&USE_BASIS)
2178 14 : return chk_el_real_basis(K, p, d_pow, el, j0);
2179 7 : if (flag&USE_GALOIS_POL)
2180 0 : return chk_el_real_galois(K, p, d_pow, el, j0);
2181 7 : return chk_el_real_factor(K, p, d_pow, el, j0);
2182 : }
2183 :
2184 : static long
2185 616 : chk_ell_real(GEN K, long d2, GEN ell, long j0)
2186 : {
2187 616 : pari_sp av = avma;
2188 616 : GEN H = K_get_H(K);
2189 616 : ulong f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
2190 : ulong d_K, i, j, gi;
2191 616 : GEN e_chi = get_e_chi(K, j0, d2, &d_K);
2192 616 : GEN g_ell, z_f, vz_f, xi_el = cgetg(d_K+1, t_VEC)+1;
2193 616 : GEN ell_1 = subiu(ell,1), ell1d2 = diviuexact(ell_1, d2);
2194 :
2195 616 : g_ell = pgener_Fp(ell);
2196 616 : z_f = Fp_pow(g_ell, diviuexact(subiu(ell, 1), f), ell);
2197 616 : vz_f = Fp_powers(z_f, f-1, ell)+1;
2198 12964 : for (gi=1, i=0; i<d_K; i++)
2199 : {
2200 12348 : GEN x = gen_1;
2201 1074948 : for (j = 1; j <= h; j++)
2202 : {
2203 1062600 : ulong y = Fl_mul(H[j], gi, f);
2204 1062600 : x = Fp_mul(x, subiu(gel(vz_f, y), 1), ell); /* vz_f[y] = z_f^y */
2205 : }
2206 12348 : gi = Fl_mul(gi, g_K, f);
2207 12348 : gel(xi_el, i) = x; /* xi_el[i]=xi^{g_K^i} */
2208 : }
2209 1421 : for (i=0; i<d_K; i++)
2210 : {
2211 1386 : GEN x = gen_1;
2212 31976 : for (j=0; j<d_K; j++)
2213 30590 : x = Fp_mul(x, Fp_powu(gel(xi_el, j), e_chi[(i+j)%d_K], ell), ell);
2214 1386 : x = Fp_pow(x, ell1d2, ell);
2215 1386 : if (!equali1(x)) return gc_long(av, 0);
2216 : }
2217 35 : return gc_long(av, 1);
2218 : }
2219 :
2220 : static GEN
2221 21 : next_el_real(GEN K, long p, long d_pow, GEN elg, long j0, long flag)
2222 : {
2223 21 : GEN Chi = gel(K, 2);
2224 21 : ulong f = K_get_f(K), h = K_get_h(K), d = upowuu(p, d_pow), d2 = d*d;
2225 21 : ulong D = (flag & USE_F)? d2*f: d2<<1, el = elg[1] + D;
2226 :
2227 : /* O(el*h) -> O(el*log(el)) by FFT */
2228 21 : if (1000 < h && el < h) { el = (h/D)*D+1; if (el < h) el += D; }
2229 21 : if (flag&USE_F) /* el=1 (mod f) */
2230 : {
2231 0 : for (;; el += D)
2232 0 : if (uisprime(el) && chk_el_real_f(K, p, d_pow, el, j0)) break;
2233 : }
2234 : else
2235 : {
2236 413 : for (;; el += D)
2237 455 : if (Chi[el%f]==0 && uisprime(el) &&
2238 42 : chk_el_real_chi(K, p, d_pow, el, j0, flag)) break;
2239 : }
2240 21 : return mkvecsmall2(el, pgener_Fl(el));
2241 : }
2242 :
2243 : static GEN
2244 35 : next_ell_real(GEN K, GEN ellg, long d2, GEN df0l, long j0)
2245 : {
2246 35 : GEN ell = gel(ellg, 1);
2247 7602 : for (ell = addii(ell, df0l);; ell = addii(ell, df0l))
2248 7602 : if (BPSW_psp(ell) && chk_ell_real(K, d2, ell, j0))
2249 35 : return mkvec2(ell, pgener_Fp(ell));
2250 : }
2251 :
2252 : /* #velg >= n */
2253 : static long
2254 0 : delete_el(GEN velg, long n)
2255 : {
2256 : long i, l;
2257 0 : if (DEBUGLEVEL>1) err_printf("deleting %ld ...\n", gmael(velg, n, 1));
2258 0 : for (l = lg(velg)-1; l >= 1; l--) if (gel(velg, l)) break;
2259 0 : for (i = n; i < l; i++) gel(velg, i) = gel(velg, i+1);
2260 0 : return l;
2261 : }
2262 :
2263 : /* velg has n components */
2264 : static GEN
2265 21 : set_ell_real(GEN K, GEN velg, long n, long d_chi, long d2, long f0, long j0)
2266 : {
2267 21 : long i, n_ell = n*d_chi;
2268 21 : GEN z = cgetg(n_ell + 1, t_VEC);
2269 21 : GEN df0l = muluu(d2, f0), ellg = mkvec2(gen_1, gen_1);
2270 42 : for (i=1; i<=n; i++) df0l = muliu(df0l, gel(velg, i)[1]);
2271 56 : for (i=1; i<=n_ell; i++) ellg = gel(z, i)= next_ell_real(K, ellg, d2, df0l, j0);
2272 21 : return z;
2273 : }
2274 :
2275 : static GEN
2276 182 : G_K_vell(GEN K, GEN vellg, ulong gk)
2277 : {
2278 182 : pari_sp av = avma;
2279 182 : GEN H = K_get_H(K);
2280 182 : ulong f = K_get_f(K), h = K_get_h(K);
2281 182 : GEN z_f, vz_f, A, P, M, z = cgetg(h+1, t_VEC);
2282 182 : ulong i, lv = lg(vellg);
2283 :
2284 182 : A=cgetg(lv, t_VEC);
2285 182 : P=cgetg(lv, t_VEC);
2286 728 : for (i=1; i<lv; i++)
2287 : {
2288 546 : GEN ell = gmael(vellg, i, 1), g_ell = gmael(vellg, i, 2);
2289 546 : gel(A, i) = Fp_pow(g_ell, diviuexact(subiu(ell, 1), f), ell);
2290 546 : gel(P, i) = ell;
2291 : }
2292 182 : z_f = ZV_chinese(A, P, &M);
2293 182 : vz_f = Fp_powers(z_f, f-1, M)+1;
2294 3822 : for (i=1; i<=h; i++) gel(z, i) = gel(vz_f, Fl_mul(H[i], gk, f));
2295 182 : return gerepilecopy(av, FpV_roots_to_pol(z, M, 0));
2296 : }
2297 :
2298 : /* f=cond(K), M=product of ell in vell, G(K/Q)=<g_K>
2299 : * G_K[1+i]=minimal polynomial of zeta_f^{g_k^i} over K mod M, 0 <= i < d_K */
2300 : static GEN
2301 7 : make_G_K(GEN K, GEN vellg)
2302 : {
2303 7 : ulong d_K = K_get_d(K), f = K_get_f(K), g_K = K_get_g(K);
2304 7 : GEN G_K = cgetg(d_K+1, t_VEC);
2305 7 : ulong i, g = 1;
2306 :
2307 189 : for (i=0; i<d_K; i++)
2308 : {
2309 182 : gel(G_K, 1+i) = G_K_vell(K, vellg, g);
2310 182 : g = Fl_mul(g, g_K, f);
2311 : }
2312 7 : return G_K;
2313 : }
2314 :
2315 : static GEN
2316 12 : G_K_p(GEN K, GEN ellg, ulong gk)
2317 : {
2318 12 : pari_sp av = avma;
2319 12 : ulong i, f = K_get_f(K), h = K_get_h(K);
2320 12 : GEN ell = gel(ellg, 1), g_ell = gel(ellg, 2);
2321 12 : GEN H = K_get_H(K), z_f, vz_f, z = cgetg(h+1, t_VEC);
2322 :
2323 12 : z_f = Fp_pow(g_ell, diviuexact(subiu(ell, 1), f), ell);
2324 12 : vz_f = Fp_powers(z_f, f-1, ell)+1;
2325 3468 : for (i=1; i<=h; i++) gel(z, i) = gel(vz_f, Fl_mul(H[i], gk, f));
2326 12 : return gerepilecopy(av, FpV_roots_to_pol(z, ell, 0));
2327 : }
2328 :
2329 : static GEN
2330 114 : G_K_l(GEN K, GEN ellg, ulong gk)
2331 : {
2332 114 : pari_sp av = avma;
2333 114 : ulong ell = itou(gel(ellg, 1)), g_ell = itou(gel(ellg, 2));
2334 114 : ulong f = K_get_f(K), h = K_get_h(K), i, z_f;
2335 114 : GEN H = K_get_H(K), vz_f, z = cgetg(h+1, t_VEC);
2336 :
2337 114 : z_f = Fl_powu(g_ell, (ell-1) / f, ell);
2338 114 : vz_f = Fl_powers(z_f, f-1, ell)+1;
2339 26898 : for (i=1; i<=h; i++) z[i] = vz_f[Fl_mul(H[i], gk, f)];
2340 114 : return gerepilecopy(av, Flv_roots_to_pol(z, ell, 0));
2341 : }
2342 :
2343 : static GEN
2344 6 : vz_vell(long d, GEN vellg, GEN *pM)
2345 : {
2346 6 : long i, l = lg(vellg);
2347 6 : GEN A = cgetg(l, t_VEC), P = cgetg(l, t_VEC), z;
2348 :
2349 18 : for (i = 1; i < l; i++)
2350 : {
2351 12 : GEN ell = gmael(vellg, i, 1), g_ell = gmael(vellg, i, 2);
2352 12 : gel(A, i) = Fp_pow(g_ell, diviuexact(subiu(ell, 1), d), ell);
2353 12 : gel(P, i) = ell;
2354 : }
2355 6 : z = ZV_chinese(A, P, pM); return Fp_powers(z, d-1, *pM);
2356 : }
2357 :
2358 : static GEN
2359 0 : D_xi_el_vell_FFT(GEN K, GEN elg, GEN vellg, ulong d, ulong j0, GEN vG_K)
2360 : {
2361 0 : pari_sp av = avma;
2362 0 : ulong d_K, h = K_get_h(K), d_chi = K_get_dchi(K);
2363 0 : ulong el = elg[1], g_el = elg[2], el_1 = el-1;
2364 : ulong i, j, i2, k, dwel;
2365 0 : GEN u = cgetg(el+2, t_POL) , v = cgetg(h+3, t_POL);
2366 0 : GEN w = cgetg(el+1, t_VEC), ww;
2367 0 : GEN M, vz_el, G_K, z = const_vec(d_chi, gen_1);
2368 0 : GEN e_chi = get_e_chi(K, j0, d, &d_K);
2369 :
2370 0 : vz_el = vz_vell(el, vellg, &M);
2371 0 : u[1] = evalsigne(1) | evalvarn(0);
2372 0 : v[1] = evalsigne(1) | evalvarn(0);
2373 :
2374 0 : for (i=i2=0; i<el; i++)
2375 : {
2376 0 : ulong j2 = i2?el-i2:i2; /* i2=(i*i)%el */
2377 0 : gel(u, 2+i) = gel(vz_el, 1+j2);
2378 0 : if ((i2+=i+i+1)>=el) i2%=el;
2379 : }
2380 0 : for (k=0; k<d_K; k++)
2381 : {
2382 0 : pari_sp av = avma;
2383 : pari_timer ti;
2384 : long gd, gi;
2385 0 : GEN x1 = gen_1;
2386 0 : G_K = gel(vG_K, 1+k);
2387 0 : for (i=i2=0; i<=h; i++)
2388 : {
2389 0 : gel(v, 2+i) = Fp_mul(gel(G_K, 2+i), gel(vz_el, 1+i2), M);
2390 0 : if ((i2+=i+i+1)>=el) i2%=el;
2391 : }
2392 0 : if (DEBUGLEVEL>2) timer_start(&ti);
2393 0 : ww = ZX_mul(u, v);
2394 0 : if (DEBUGLEVEL>2)
2395 0 : timer_printf(&ti, "ZX_mul:%ld/%ld h*el=%ld*%ld", k, d_K, h, el);
2396 0 : dwel = degpol(ww)-el;
2397 0 : for (i=0; i<=dwel; i++) gel(w, 1+i) = addii(gel(ww, 2+i), gel(ww, 2+i+el));
2398 0 : for (; i<el; i++) gel(w, 1+i) = gel(ww, 2+i);
2399 0 : for (i=i2=1; i<el; i++) /* w[i]=G_K(z_el^(2*i)) */
2400 : {
2401 0 : gel(w, i) = Fp_mul(gel(w, 1+i), gel(vz_el, 1+i2), M);
2402 0 : if ((i2+=i+i+1)>=el) i2%=el;
2403 : }
2404 0 : gd = Fl_powu(g_el, d, el); /* a bit faster */
2405 0 : gi = g_el;
2406 0 : for (i=1; i<d; i++)
2407 : {
2408 0 : GEN xi = gen_1;
2409 0 : long gdi = gi;
2410 0 : for (j=0; i+j<el_1; j+=d)
2411 : {
2412 0 : xi = Fp_mul(xi, gel(w, (gdi+gdi)%el), M);
2413 0 : gdi = Fl_mul(gdi, gd, el);
2414 : }
2415 0 : gi = Fl_mul(gi, g_el, el);
2416 0 : xi = Fp_powu(xi, i, M);
2417 0 : x1 = Fp_mul(x1, xi, M);
2418 : }
2419 0 : for (i=1; i<=d_chi; i++)
2420 : {
2421 0 : GEN x2 = Fp_powu(x1, e_chi[(k+i-1)%d_K], M);
2422 0 : gel(z, i) = Fp_mul(gel(z, i), x2, M);
2423 : }
2424 0 : z = gerepilecopy(av, z);
2425 : }
2426 0 : return gerepilecopy(av, z);
2427 : }
2428 :
2429 : static GEN
2430 0 : D_xi_el_vell(GEN K, GEN elg, GEN vellg, ulong d, ulong j0)
2431 : {
2432 0 : pari_sp av = avma;
2433 0 : GEN H = K_get_H(K);
2434 0 : ulong f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
2435 : GEN z_f, z_el, vz_f, vz_el;
2436 0 : ulong el = elg[1], g_el = elg[2], el_1 = el-1;
2437 0 : ulong i, j, k, d_K, lv = lg(vellg), d_chi = K_get_dchi(K);
2438 0 : GEN A, B, P, M, z = const_vec(d_chi, gen_1);
2439 0 : GEN e_chi = get_e_chi(K, j0, d, &d_K);
2440 :
2441 0 : A=cgetg(lv, t_VEC);
2442 0 : B=cgetg(lv, t_VEC);
2443 0 : P=cgetg(lv, t_VEC);
2444 0 : for (i = 1; i < lv; i++)
2445 : {
2446 0 : GEN ell = gmael(vellg, i, 1), g_ell = gmael(vellg, i, 2);
2447 0 : GEN ell_1 = subiu(ell, 1);
2448 0 : gel(A, i) = Fp_pow(g_ell, diviuexact(ell_1, f), ell);
2449 0 : gel(B, i) = Fp_pow(g_ell, diviuexact(ell_1, el), ell);
2450 0 : gel(P, i) = ell;
2451 : }
2452 0 : z_f = ZV_chinese(A, P, &M);
2453 0 : z_el = ZV_chinese(B, P, NULL);
2454 0 : vz_f = Fp_powers(z_f, f-1, M);
2455 0 : vz_el = Fp_powers(z_el, el-1, M);
2456 0 : for (k = 0; k < d_K; k++)
2457 : {
2458 0 : pari_sp av = avma;
2459 0 : GEN x0 = gen_1;
2460 0 : long gk = Fl_powu(g_K, k, f);
2461 0 : for (i=1; i<el_1; i++)
2462 : {
2463 0 : long gi = Fl_powu(g_el, i, el);
2464 0 : GEN x1 = gen_1;
2465 0 : GEN x2 = gel(vz_el, 1+gi);
2466 0 : for (j=1; j<=h; j++)
2467 : {
2468 0 : long y = Fl_mul(H[j], gk, f);
2469 0 : x1 = Fp_mul(x1, Fp_sub(x2, gel(vz_f, 1+y), M), M);
2470 : }
2471 0 : x1 = Fp_powu(x1, i%d, M);
2472 0 : x0 = Fp_mul(x0, x1, M);
2473 : }
2474 0 : for (i=1; i<=d_chi; i++)
2475 : {
2476 0 : GEN x2 = Fp_powu(x0, e_chi[(k+i-1)%d_K], M);
2477 0 : gel(z, i) = Fp_mul(gel(z, i), x2, M);
2478 : }
2479 0 : z = gerepilecopy(av, z);
2480 : }
2481 0 : return gerepilecopy(av, z);
2482 : }
2483 :
2484 : static GEN
2485 34 : D_xi_el_Flx_mul(GEN K, GEN elg, GEN ellg, GEN vG_K, ulong d, ulong j0)
2486 : {
2487 34 : pari_sp av = avma;
2488 34 : ulong d_K, f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
2489 34 : ulong el = elg[1], g_el = elg[2], el_1 = el-1, d_chi = K_get_dchi(K);
2490 34 : ulong ell = itou(gel(ellg, 1)), g_ell = itou(gel(ellg, 2)), z_el;
2491 34 : GEN u = cgetg(el+2, t_VECSMALL), v = cgetg(h+3, t_VECSMALL);
2492 34 : GEN w = cgetg(el+1, t_VECSMALL), ww;
2493 34 : GEN vz_el, G_K, z = const_vecsmall(d_chi, 1);
2494 34 : GEN e_chi = get_e_chi(K, j0, d, &d_K);
2495 : ulong i, j, i2, k, dwel;
2496 :
2497 34 : u[1] = evalvarn(0);
2498 34 : v[1] = evalvarn(0);
2499 34 : z_el = Fl_powu(g_ell, (ell - 1) / el, ell);
2500 34 : vz_el = Fl_powers(z_el, el_1, ell)+1;
2501 :
2502 700376 : for (i=i2=0; i<el; i++)
2503 : {
2504 700342 : ulong j2 = i2?el-i2:i2;
2505 700342 : u[2+i] = vz_el[j2];
2506 700342 : if ((i2+=i+i+1)>=el) i2%=el; /* i2=(i*i)%el */
2507 : }
2508 694 : for (k=0; k<d_K; k++)
2509 : {
2510 660 : pari_sp av = avma;
2511 : pari_timer ti;
2512 660 : ulong gk = Fl_powu(g_K, k, f);
2513 660 : long gd, gi, x1 = 1;
2514 660 : if (DEBUGLEVEL>2) timer_start(&ti);
2515 660 : G_K = (vG_K==NULL)?G_K_l(K, ellg, gk):ZX_to_Flx(gel(vG_K, 1+k), ell);
2516 660 : if (DEBUGLEVEL>2) timer_printf(&ti, "G_K_l");
2517 39024 : for (i=i2=0; i<=h; i++)
2518 : {
2519 38364 : v[2+i] = Fl_mul(G_K[2+i], vz_el[i2], ell);
2520 38364 : if ((i2+=i+i+1)>=el) i2%=el; /* i2=(i*i)%el */
2521 : }
2522 660 : if (DEBUGLEVEL>2) timer_start(&ti);
2523 660 : ww = Flx_mul(u, v, ell);
2524 660 : if (DEBUGLEVEL>2)
2525 0 : timer_printf(&ti, "Flx_mul:%ld/%ld h*el=%ld*%ld", k, d_K, h, el);
2526 660 : dwel=degpol(ww)-el; /* dwel=h-1 */
2527 38364 : for (i=0; i<=dwel; i++) w[1+i] = Fl_add(ww[2+i], ww[2+i+el], ell);
2528 8388480 : for (; i<el; i++) w[1+i] = ww[2+i];
2529 8425524 : for (i=i2=1; i<el; i++) /* w[i]=G_K(z_el^(2*i)) */
2530 : {
2531 8424864 : w[i] = Fl_mul(w[1+i], vz_el[i2], ell);
2532 8424864 : if ((i2+=i+i+1)>=el) i2%=el; /* i2=(i*i)%el */
2533 : }
2534 660 : gd = Fl_powu(g_el, d, el); /* a bit faster */
2535 660 : gi = g_el;
2536 4596 : for (i=1; i<d; i++)
2537 : {
2538 3936 : long xi = 1, gdi = gi;
2539 8163696 : for (j=0; i+j<el_1; j+=d)
2540 : {
2541 8159760 : xi = Fl_mul(xi, w[(gdi+gdi)%el], ell);
2542 8159760 : gdi = Fl_mul(gdi, gd, el);
2543 : }
2544 3936 : gi = Fl_mul(gi, g_el, el);
2545 3936 : xi = Fl_powu(xi, i, ell);
2546 3936 : x1 = Fl_mul(x1, xi, ell);
2547 : }
2548 2412 : for (i=1; i<=d_chi; i++)
2549 : {
2550 1752 : long x2 = Fl_powu(x1, e_chi[(k+i-1)%d_K], ell);
2551 1752 : z[i] = Fl_mul(z[i], x2, ell);
2552 : }
2553 660 : set_avma(av);
2554 : }
2555 34 : return gerepilecopy(av, Flv_to_ZV(z));
2556 : }
2557 :
2558 : static GEN
2559 35 : D_xi_el_ZX_mul(GEN K, GEN elg, GEN ellg, GEN vG_K, ulong d, ulong j0)
2560 : {
2561 35 : pari_sp av = avma;
2562 35 : GEN ell = gel(ellg,1), g_ell, u, v, w, ww, z_el, vz_el, G_K, z, e_chi;
2563 : ulong d_K, f, h, g_K, el, g_el, el_1, d_chi, i, j, i2, k, dwel;
2564 :
2565 35 : if (lgefint(ell) == 3) return D_xi_el_Flx_mul(K, elg, ellg, vG_K, d, j0);
2566 1 : f = K_get_f(K); h = K_get_h(K); g_K = K_get_g(K);
2567 1 : el = elg[1]; g_el = elg[2]; el_1 = el-1; d_chi = K_get_dchi(K);
2568 1 : g_ell = gel(ellg, 2);
2569 1 : z = const_vec(d_chi, gen_1);
2570 1 : e_chi = get_e_chi(K, j0, d, &d_K);
2571 :
2572 1 : u = cgetg(el+2,t_POL); u[1] = evalsigne(1) | evalvarn(0);
2573 1 : v = cgetg(h+3, t_POL); v[1] = evalsigne(1) | evalvarn(0);
2574 1 : w = cgetg(el+1, t_VEC);
2575 1 : z_el = Fp_pow(g_ell, diviuexact(subiu(ell, 1), el), ell);
2576 1 : vz_el = Fp_powers(z_el, el_1, ell)+1;
2577 :
2578 114998 : for (i=i2=0; i<el; i++)
2579 : {
2580 114997 : ulong j2 = i2?el-i2:i2; /* i2=(i*i)%el */
2581 114997 : gel(u, 2+i) = gel(vz_el, j2);
2582 114997 : if ((i2+=i+i+1)>=el) i2%=el;
2583 : }
2584 13 : for (k=0; k<d_K; k++)
2585 : {
2586 12 : pari_sp av = avma;
2587 : pari_timer ti;
2588 12 : long gd, gi, gk = Fl_powu(g_K, k, f);
2589 12 : GEN x1 = gen_1;
2590 12 : if (DEBUGLEVEL>2) timer_start(&ti);
2591 12 : G_K = (vG_K==NULL) ? G_K_p(K, ellg, gk):RgX_to_FpX(gel(vG_K, 1+k), ell);
2592 12 : if (DEBUGLEVEL>2) timer_printf(&ti, "G_K_p");
2593 3480 : for (i=i2=0; i<=h; i++)
2594 : {
2595 3468 : gel(v, 2+i) = Fp_mul(gel(G_K, 2+i), gel(vz_el, i2), ell);
2596 3468 : if ((i2+=i+i+1)>=el) i2%=el;
2597 : }
2598 12 : if (DEBUGLEVEL>2) timer_start(&ti);
2599 12 : ww = ZX_mul(u, v);
2600 12 : if (DEBUGLEVEL>2)
2601 0 : timer_printf(&ti, "ZX_mul:%ld/%ld h*el=%ld*%ld", k, d_K, h, el);
2602 12 : dwel = degpol(ww)-el;
2603 3468 : for (i=0; i<=dwel; i++) gel(w, 1+i) = addii(gel(ww, 2+i), gel(ww, 2+i+el));
2604 1376520 : for (; i<el; i++) gel(w, 1+i) = gel(ww, 2+i);
2605 1379964 : for (i=i2=1; i<el; i++) /* w[i]=G_K(z_el^(2*i)) */
2606 : {
2607 1379952 : gel(w, i) = Fp_mul(gel(w, 1+i), gel(vz_el, i2), ell);
2608 1379952 : if ((i2+=i+i+1)>=el) i2%=el;
2609 : }
2610 12 : gd = Fl_powu(g_el, d, el); /* a bit faster */
2611 12 : gi = g_el;
2612 444 : for (i=1; i<d; i++)
2613 : {
2614 432 : GEN xi = gen_1;
2615 432 : long gdi = gi;
2616 1343088 : for (j=0; i+j<el_1; j+=d)
2617 : {
2618 1342656 : xi = Fp_mul(xi, gel(w, (gdi+gdi)%el), ell);
2619 1342656 : gdi = Fl_mul(gdi, gd, el);
2620 : }
2621 432 : gi = Fl_mul(gi, g_el, el);
2622 432 : xi = Fp_powu(xi, i, ell);
2623 432 : x1 = Fp_mul(x1, xi, ell);
2624 : }
2625 24 : for (i=1; i<=d_chi; i++)
2626 : {
2627 12 : GEN x2 = Fp_powu(x1, e_chi[(k+i-1)%d_K], ell);
2628 12 : gel(z, i) = Fp_mul(gel(z, i), x2, ell);
2629 : }
2630 12 : z = gerepilecopy(av, z);
2631 : }
2632 1 : return gerepilecopy(av, z);
2633 : }
2634 :
2635 : static GEN
2636 0 : D_xi_el_ss(GEN K, GEN elg, GEN ellg, ulong d, ulong j0)
2637 : {
2638 0 : pari_sp av = avma;
2639 0 : GEN H = K_get_H(K);
2640 0 : ulong d_K, f = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
2641 0 : ulong el = elg[1], g_el = elg[2], el_1 = el-1;
2642 0 : ulong ell = itou(gel(ellg, 1)), g_ell = itou(gel(ellg, 2));
2643 0 : ulong i, j, k, gk, z_f, z_el, d_chi = K_get_dchi(K);
2644 0 : GEN vz_f, vz_el, z = const_vecsmall(d_chi, 1);
2645 0 : GEN e_chi = get_e_chi(K, j0, d, &d_K);
2646 :
2647 0 : z_f = Fl_powu(g_ell, (ell - 1) / f, ell);
2648 0 : z_el = Fl_powu(g_ell, (ell - 1) / el, ell);
2649 0 : vz_f = Fl_powers(z_f, f-1, ell)+1;
2650 0 : vz_el = Fl_powers(z_el, el-1, ell)+1;
2651 0 : gk = 1; /* g_K^k */
2652 0 : for (k = 0; k < d_K; k++)
2653 : {
2654 0 : ulong x0 = 1, gi = g_el; /* g_el^i */
2655 0 : for (i = 1; i < el_1; i++)
2656 : {
2657 0 : ulong x1 = 1, x2 = vz_el[gi];
2658 0 : for (j=1; j<=h; j++)
2659 : {
2660 0 : ulong y = Fl_mul(H[j], gk, f);
2661 0 : x1 = Fl_mul(x1, Fl_sub(x2, vz_f[y], ell), ell);
2662 : }
2663 0 : x1 = Fl_powu(x1, i%d, ell);
2664 0 : x0 = Fl_mul(x0, x1, ell);
2665 0 : gi = Fl_mul(gi, g_el, el);
2666 : }
2667 0 : for (i = 1; i <= d_chi; i++)
2668 : {
2669 0 : ulong x2 = Fl_powu(x0, e_chi[(k+i-1)%d_K], ell);
2670 0 : z[i] = Fl_mul(z[i], x2, ell);
2671 : }
2672 0 : gk = Fl_mul(gk, g_K, f);
2673 : }
2674 0 : return gerepileupto(av, Flv_to_ZV(z));
2675 : }
2676 :
2677 : static GEN
2678 0 : D_xi_el_sl(GEN K, GEN elg, GEN ellg, ulong d, ulong j0)
2679 : {
2680 0 : pari_sp av = avma;
2681 0 : GEN ell = gel(ellg, 1), H;
2682 : GEN g_ell, ell_1, z_f, z_el, vz_f, vz_el, z, e_chi;
2683 : ulong d_K, f, h, g_K, el, g_el, el_1, d_chi, i, j, k, gk;
2684 :
2685 0 : if (lgefint(ell) == 3) return D_xi_el_ss(K, elg, ellg, d, j0);
2686 0 : H = K_get_H(K);
2687 0 : f = K_get_f(K); h = K_get_h(K); g_K = K_get_g(K);
2688 0 : el = elg[1]; g_el = elg[2]; el_1 = el-1; d_chi = K_get_dchi(K);
2689 0 : g_ell = gel(ellg, 2); ell_1 = subiu(ell, 1);
2690 0 : z = const_vec(d_chi, gen_1);
2691 0 : e_chi = get_e_chi(K, j0, d, &d_K);
2692 :
2693 0 : z_f = Fp_pow(g_ell, diviuexact(ell_1, f), ell);
2694 0 : z_el = Fp_pow(g_ell, diviuexact(ell_1, el), ell);
2695 0 : vz_f = Fp_powers(z_f, f-1, ell) + 1;
2696 0 : vz_el = Fp_powers(z_el, el-1, ell) + 1;
2697 0 : gk = 1; /* g_K^k */
2698 0 : for (k = 0; k < d_K; k++)
2699 : {
2700 0 : pari_sp av2 = avma;
2701 0 : GEN x0 = gen_1;
2702 0 : ulong gi = g_el; /* g_el^i */
2703 0 : for (i = 1; i < el_1; i++)
2704 : {
2705 0 : pari_sp av3 = avma;
2706 0 : GEN x1 = gen_1, x2 = gel(vz_el, gi);
2707 0 : for (j = 1; j <= h; j++)
2708 : {
2709 0 : ulong y = Fl_neg(Fl_mul(H[j], gk, f), f);
2710 0 : x1 = Fp_mul(x1, Fp_sub(x2, gel(vz_f, y), ell), ell);
2711 : }
2712 0 : x1 = Fp_powu(x1, i%d, ell);
2713 0 : x0 = gerepileuptoint(av3, Fp_mul(x0, x1, ell));
2714 0 : gi = Fl_mul(gi, g_el, el);
2715 : }
2716 0 : for (i=1; i<=d_chi; i++)
2717 : {
2718 0 : GEN x2 = Fp_powu(x0, e_chi[(k+i-1)%d_K], ell);
2719 0 : gel(z, i) = Fp_mul(gel(z, i), x2, ell);
2720 : }
2721 0 : if (k == d_K-1) break;
2722 0 : z = gerepilecopy(av2, z);
2723 0 : gk = Fl_mul(gk, g_K, f);
2724 : }
2725 0 : return gerepilecopy(av, z);
2726 : }
2727 :
2728 : static long
2729 175 : get_y(GEN z, GEN ellg, long d)
2730 : {
2731 175 : GEN ell = gel(ellg, 1), g_ell = gel(ellg, 2);
2732 175 : GEN elld = diviuexact(subiu(ell, 1), d);
2733 175 : GEN g_elld = Fp_pow(g_ell, elld, ell);
2734 175 : GEN x = Fp_pow(modii(z, ell), elld, ell);
2735 : long k;
2736 211778 : for (k=0; k<d; k++)
2737 : {
2738 211778 : if (equali1(x)) break;
2739 211603 : x = Fp_mul(x, g_elld, ell);
2740 : }
2741 175 : if (k==0) k=d;
2742 161 : else if (d<=k) pari_err_BUG("subcyclopclgp [MLL]");
2743 175 : return k;
2744 : }
2745 :
2746 : static void
2747 0 : real_MLLn(long *y, GEN K, ulong p, ulong d_pow, ulong n,
2748 : GEN velg, GEN vellg, GEN vG_K, ulong j0)
2749 : {
2750 0 : pari_sp av = avma;
2751 0 : ulong i, j, k, d = upowuu(p, d_pow), h = gmael(K, 1, 2)[3];
2752 0 : ulong row = lg(vellg)-1;
2753 0 : for (i=1; i<=n; i++)
2754 : {
2755 0 : GEN elg = gel(velg, i), z;
2756 0 : ulong el = elg[1], nz;
2757 : pari_timer ti;
2758 0 : if (DEBUGLEVEL>1) timer_start(&ti);
2759 0 : z = (h<el) ? D_xi_el_vell_FFT(K, elg, vellg, d, j0, vG_K)
2760 0 : : D_xi_el_vell(K, elg, vellg, d, j0);
2761 0 : if (DEBUGLEVEL>1) timer_printf(&ti, "subcyclopclgp:[D_xi_el]");
2762 0 : if (DEBUGLEVEL>2) err_printf("z=%Ps\n", z);
2763 0 : nz = lg(z)-1;
2764 0 : for (k = 1; k <= nz; k++)
2765 0 : for (j=1; j<=row; j++)
2766 0 : y[(j-1)*row+(i-1)*nz+k-1] = get_y(gel(z, k), gel(vellg, j), d);
2767 0 : set_avma(av);
2768 : }
2769 0 : }
2770 :
2771 : static void
2772 14 : real_MLL1(long *y, GEN K, ulong p, ulong d_pow, GEN velg, GEN vellg, ulong j0)
2773 : {
2774 14 : ulong h = gmael(K, 1, 2)[3], d = upowuu(p, d_pow);
2775 14 : GEN elg = gel(velg, 1), ellg = gel(vellg, 1), z;
2776 14 : ulong el = elg[1];
2777 : pari_timer ti;
2778 :
2779 14 : if (DEBUGLEVEL>2) timer_start(&ti);
2780 14 : z = h < el? D_xi_el_ZX_mul(K, elg, ellg, NULL, d, j0)
2781 14 : : D_xi_el_sl(K, elg, ellg, d, j0);
2782 14 : if (DEBUGLEVEL>2) timer_printf(&ti, "subcyclopclgp:[D_xi_el]");
2783 14 : if (DEBUGLEVEL>2) err_printf("z=%Ps\n", z);
2784 14 : y[0] = get_y(gel(z, 1), ellg, d);
2785 14 : }
2786 :
2787 : static void
2788 7 : real_MLL(long *y, GEN K, ulong p, ulong d_pow, ulong n,
2789 : GEN velg, GEN vellg, GEN vG_K, ulong j0)
2790 : {
2791 7 : ulong i, j, k, d = upowuu(p, d_pow), h = gmael(K, 1, 2)[3];
2792 7 : ulong row = lg(vellg)-1;
2793 28 : for (j=1; j<=row; j++)
2794 : {
2795 21 : GEN ellg = gel(vellg, j);
2796 42 : for (i=1; i<=n; i++)
2797 : {
2798 21 : pari_sp av2 = avma;
2799 21 : GEN elg = gel(velg, i), z;
2800 21 : ulong el = elg[1], nz;
2801 : pari_timer ti;
2802 21 : if (DEBUGLEVEL>2) timer_start(&ti);
2803 21 : z = h < el? D_xi_el_ZX_mul(K, elg, ellg, vG_K, d, j0)
2804 21 : : D_xi_el_sl(K, elg, ellg, d, j0);
2805 21 : if (DEBUGLEVEL>2) timer_printf(&ti, "subcyclopclgp:[D_xi_el]");
2806 21 : if (DEBUGLEVEL>3) err_printf("z=%Ps\n", z);
2807 21 : nz = lg(z)-1;
2808 84 : for (k = 1; k <= nz; k++)
2809 63 : y[(j-1)*row+(i-1)*nz+k-1] = get_y(gel(z, k), ellg, d);
2810 21 : set_avma(av2);
2811 : }
2812 : }
2813 7 : }
2814 :
2815 : static long
2816 21 : use_basis(long d_K, long f) { return (d_K<=10 || (d_K<=30 && f<=5000)); }
2817 :
2818 : static long
2819 7 : use_factor(ulong f)
2820 7 : { GEN fa = factoru(f), P = gel(fa, 1); return (P[lg(P)-1]<500); }
2821 :
2822 : /* group structure, destroy gr */
2823 : static GEN
2824 63 : get_str(GEN gr)
2825 : {
2826 63 : GEN z = gel(gr,2);
2827 63 : long i, j, l = lg(z);
2828 154 : for (i = j = 1; i < l; i++)
2829 91 : if (lgefint(gel(z, i)) > 2) gel(z,j++) = gel(z,i);
2830 63 : setlg(z, j); return z;
2831 : }
2832 :
2833 : static GEN
2834 21 : cyc_real_MLL(GEN K, ulong p, long d_pow, long j0, long flag)
2835 : {
2836 21 : ulong d_K = K_get_d(K), f = K_get_f(K), d_chi = K_get_dchi(K);
2837 21 : ulong n, n0 = 1, f0, n_el = d_pow, d = upowuu(p, d_pow), rank = n_el*d_chi;
2838 21 : GEN velg = const_vec(n_el, NULL), vellg = NULL;
2839 21 : GEN oldgr = mkvec2(gen_0, NULL), newgr = mkvec2(gen_0, NULL);
2840 21 : long *y0 = (long*)stack_calloc(sizeof(long)*rank*rank);
2841 :
2842 21 : if (DEBUGLEVEL>1)
2843 0 : err_printf("cyc_real_MLL:p=%ld d_pow=%ld deg(K)=%ld cond(K)=%ld g_K=%ld\n",
2844 : p, d_pow, d_K, f, K_get_g(K));
2845 21 : gel(K, 2) = get_chi(gel(K,1));
2846 21 : if (f-1 <= (d_K<<1)) flag |= USE_F;
2847 21 : else if (use_basis(d_K, f)) flag |= USE_BASIS;
2848 7 : else if (use_factor(f)) flag |= USE_FACTOR;
2849 0 : else flag |= USE_GALOIS_POL;
2850 21 : if (flag&USE_BASIS) K = vec_append(K, xi_data_basis(K));
2851 7 : else if (flag&USE_GALOIS_POL) K = vec_append(K, xi_data_galois(K));
2852 21 : f0 = f%p?f:f/p;
2853 21 : gel(velg, 1) = next_el_real(K, p, d_pow, mkvecsmall2(1, 1), j0, flag);
2854 21 : if (flag&USE_FULL_EL)
2855 : {
2856 0 : for (n=2; n<=n_el; n++)
2857 0 : gel(velg, n) = next_el_real(K, p, d_pow, gel(velg, n+1), j0, flag);
2858 0 : n0 = n_el;
2859 : }
2860 :
2861 21 : for (n=n0; n<=n_el; n++) /* loop while structure is unknown */
2862 : {
2863 21 : pari_sp av2 = avma;
2864 : long n_ell, m, M;
2865 : GEN y;
2866 : pari_timer ti;
2867 21 : if (DEBUGLEVEL>2) timer_start(&ti);
2868 21 : vellg = set_ell_real(K, velg, n, d_chi, d*d, f0, j0);
2869 21 : n_ell = lg(vellg) -1; /* equal to n*d_chi */
2870 21 : if (DEBUGLEVEL>2) timer_printf(&ti, "set_ell_real");
2871 21 : if (DEBUGLEVEL>3) err_printf("vel=%Ps\nvell=%Ps\n", velg, vellg);
2872 21 : if (n_ell==1)
2873 14 : real_MLL1(y0, K, p, d_pow, velg, vellg, j0);
2874 : else
2875 : {
2876 : GEN vG_K;
2877 7 : if (DEBUGLEVEL>2) timer_start(&ti);
2878 7 : vG_K = make_G_K(K, vellg);
2879 7 : if (DEBUGLEVEL>2) timer_printf(&ti, "make_G_K");
2880 7 : if (lgefint(gmael(vellg, n_ell, 1))<=3 || (flag&SAVE_MEMORY))
2881 7 : real_MLL(y0, K, p, d_pow, n, velg, vellg, vG_K, j0);
2882 : else
2883 0 : real_MLLn(y0, K, p, d_pow, n, velg, vellg, vG_K, j0);
2884 : }
2885 21 : set_avma(av2);
2886 21 : y = ary2mat(y0, n_ell);
2887 21 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
2888 21 : y = ZM_snf(y);
2889 21 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
2890 21 : y = make_p_part(y, p, d_pow);
2891 21 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
2892 21 : newgr = structure_MLL(y, d_pow);
2893 21 : if (DEBUGLEVEL>3)
2894 0 : err_printf("d_pow=%ld d_chi=%ld old=%Ps new=%Ps\n",d_pow,d_chi,oldgr,newgr);
2895 21 : if (equalsi(d_pow*d_chi, gel(newgr, 1))) break;
2896 0 : if ((m = find_del_el(&oldgr, newgr, n, n_el, d_chi)))
2897 0 : { M = m = delete_el(velg, m); n--; }
2898 : else
2899 0 : { M = n+1; m = n; }
2900 0 : gel(velg, M) = next_el_real(K, p, d_pow, gel(velg, m), j0, flag);
2901 : }
2902 21 : return get_str(newgr);
2903 : }
2904 :
2905 : static GEN
2906 0 : cyc_buch(long dK, GEN p, long d_pow)
2907 : {
2908 0 : GEN z = Buchquad(stoi(dK), 0.0, 0.0, 0), cyc = gel(z,2);
2909 0 : long i, l = lg(cyc);
2910 0 : if (Z_pval(gel(z,1), p) != d_pow) pari_err_BUG("subcyclopclgp [Buchquad]");
2911 0 : for (i = 1; i < l; i++)
2912 : {
2913 0 : long x = Z_pval(gel(cyc, i), p); if (!x) break;
2914 0 : gel(cyc, i) = utoipos(x);
2915 : }
2916 0 : setlg(cyc, i); return cyc;
2917 : }
2918 :
2919 : static void
2920 0 : verbose_output(GEN K, GEN p, long pow, long j)
2921 : {
2922 0 : long d = K_get_d(K), f = K_get_f(K), s = K_get_s(K), d_chi = K_get_dchi(K);
2923 0 : err_printf("|A_K_psi|=%Ps^%ld, psi=chi^%ld, d_psi=%ld, %s,\n\
2924 : [K:Q]=%ld, [f,H]=[%ld, %Ps]\n",
2925 0 : p,pow*d_chi,j,d_chi,s?"real":"imaginary",d,f,zv_to_ZV(gmael3(K,1,1,1)));
2926 0 : }
2927 :
2928 : static int
2929 35091 : cyc_real_pre(GEN K, GEN xi, ulong p, ulong j, long el)
2930 : {
2931 35091 : pari_sp av = avma;
2932 35091 : ulong i, d_K, x = 1;
2933 35091 : GEN e_chi = get_e_chi(K, j, p, &d_K);
2934 :
2935 35091 : xi++;
2936 1254827 : for (i = 0; i < d_K; i++) x = Fl_mul(x, Fl_powu(xi[i], e_chi[i], el), el);
2937 35091 : return gc_ulong(av, Fl_powu(x, (el-1)/p, el));
2938 : }
2939 :
2940 : /* return vec[-1,[],0], vec[0,[],0], vec[1,[1],0], vec[2,[1,1],0] etc */
2941 : static GEN
2942 26579 : cyc_real_ss(GEN K, GEN xi, ulong p, long j, long pow, long el, ulong pn, long flag)
2943 : {
2944 26579 : ulong d_chi = K_get_dchi(K);
2945 26579 : if (cyc_real_pre(K, xi, pn, j, el) == 1) return NULL; /* not determined */
2946 21273 : if (--pow==0) return mkvec3(gen_0, nullvec(), gen_0); /* trivial */
2947 105 : if (DEBUGLEVEL) verbose_output(K, utoi(p), pow, j);
2948 105 : if (flag&USE_MLL)
2949 : {
2950 21 : pari_sp av = avma;
2951 21 : GEN gr = (K_get_d(K) == 2)? cyc_buch(K_get_f(K), utoi(p), pow)
2952 21 : : cyc_real_MLL(K, p, pow, j, flag);
2953 21 : return gerepilecopy(av, mkvec3(utoipos(d_chi * pow), gr, gen_0));
2954 : }
2955 84 : if (pow==1) return mkvec3(utoi(d_chi), onevec(d_chi), gen_0);
2956 21 : return mkvec3(utoi(pow*d_chi), nullvec(), gen_0);
2957 : }
2958 :
2959 : static GEN
2960 5145 : cyc_real_ll(GEN K, GEN xi, GEN p, long j, long pow, GEN el, GEN pn, long flag)
2961 : {
2962 5145 : pari_sp av = avma;
2963 5145 : ulong i, d_K = K_get_d(K), d_chi = K_get_dchi(K);
2964 5145 : GEN e_chi = get_e_chi_ll(K, j, pn), x = gen_1;
2965 :
2966 5145 : xi++;
2967 246785 : for (i = 0; i < d_K; i++)
2968 241640 : x = Fp_mul(x, Fp_pow(gel(xi, i), gel(e_chi, i), el), el);
2969 5145 : x = Fp_pow(x, diviiexact(subiu(el, 1), pn), el); /* x = x^(el-1)/pn mod el */
2970 5145 : set_avma(av); if (equali1(x)) return NULL; /* not determined */
2971 5145 : if (--pow==0) return mkvec3(gen_0, nullvec(), gen_0); /* trivial */
2972 0 : if (DEBUGLEVEL) verbose_output(K, p, pow, j);
2973 0 : if (flag&USE_MLL)
2974 0 : pari_err_IMPL(stack_sprintf("flag=%ld for large prime", USE_MLL));
2975 0 : if (pow==1) return mkvec3(utoi(d_chi), onevec(d_chi), gen_0);
2976 0 : return mkvec3(utoi(pow*d_chi), nullvec(), gen_0);
2977 : }
2978 :
2979 : /* xi[1+i] = xi^(g^i), 0 <= i <= d-1 */
2980 : static GEN
2981 13797 : xi_conj_s(GEN K, ulong el)
2982 : {
2983 13797 : pari_sp av = avma;
2984 13797 : GEN H = K_get_H(K);
2985 13797 : long d = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g = K_get_g(K);
2986 13797 : long i, gi = 1, z = Fl_powu(pgener_Fl(el), (el-1)/f, el);
2987 13797 : GEN vz = Fl_powers(z, f, el)+1, xi = cgetg(d+1, t_VECSMALL);
2988 :
2989 372953 : for (i=1; i<=d; i++)
2990 : {
2991 359156 : long j, x = 1;
2992 170777852 : for (j=1; j<=h; j++)
2993 170418696 : x = Fl_mul(x, vz[Fl_mul(H[j], gi, f)]-1, el);
2994 359156 : xi[i] = x;
2995 359156 : gi = Fl_mul(gi, g, f);
2996 : }
2997 13797 : return gerepilecopy(av, xi);
2998 : }
2999 :
3000 : static GEN
3001 1673 : xi_conj_l(GEN K, GEN el)
3002 : {
3003 1673 : pari_sp av = avma;
3004 1673 : GEN H = K_get_H(K);
3005 1673 : long d = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g = K_get_g(K);
3006 1673 : long i, gi = 1;
3007 1673 : GEN z = Fp_pow(pgener_Fp(el), diviuexact(subiu(el, 1), f), el);
3008 1673 : GEN vz = Fp_powers(z, f, el)+1, xi = cgetg(d+1, t_VEC);
3009 :
3010 70462 : for (i=1; i<=d; i++)
3011 : {
3012 : long j;
3013 68789 : GEN x = gen_1;
3014 6756043 : for (j=1; j<=h; j++)
3015 6687254 : x = Fp_mul(x, subiu(gel(vz, Fl_mul(H[j], gi, f)), 1), el);
3016 68789 : gel(xi, i) = x;
3017 68789 : gi = Fl_mul(gi, g, f);
3018 : }
3019 1673 : return gerepilecopy(av, xi);
3020 : }
3021 :
3022 : static GEN
3023 10164 : pclgp_cyc_real(GEN K, GEN p, long max_pow, long flag)
3024 : {
3025 10164 : const long NUM_EL = 20;
3026 10164 : GEN C = gel(K, 5);
3027 10164 : long f_K = K_get_f(K), n_conj = K_get_nconj(K);
3028 10164 : long i, pow, n_el, n_done = 0;
3029 10164 : GEN gr = nullvec(), Done = const_vecsmall(n_conj, 0), xi;
3030 10164 : long first = 1;
3031 :
3032 10290 : for (pow=1; pow<=max_pow; pow++)
3033 : {
3034 10290 : GEN pn = powiu(p, pow), fpn = muliu(pn, f_K), el = addiu(fpn, 1);
3035 109753 : for (n_el = 0; n_el < NUM_EL; el = addii(el, fpn))
3036 : {
3037 : ulong uel;
3038 109627 : if (!BPSW_psp(el)) continue;
3039 15470 : n_el++; uel = itou_or_0(el);
3040 15470 : if (uel)
3041 : {
3042 13797 : xi = xi_conj_s(K, uel);
3043 13797 : if (first && n_conj > 10) /* mark trivial chi-part */
3044 : {
3045 8715 : for (i = 1; i <= n_conj; i++)
3046 : {
3047 8512 : if (cyc_real_pre(K, xi, p[2], C[i], uel) == 1) continue;
3048 8260 : Done[i] = 1;
3049 8260 : if (++n_done == n_conj) return gr;
3050 : }
3051 203 : first = 0; continue;
3052 : }
3053 : }
3054 : else
3055 1673 : xi = xi_conj_l(K, el);
3056 55132 : for (i = 1; i <= n_conj; i++)
3057 : {
3058 : GEN z;
3059 50029 : if (Done[i]) continue;
3060 31724 : if (uel)
3061 26579 : z = cyc_real_ss(K, xi, p[2], C[i], pow, uel, itou(pn), flag);
3062 : else
3063 5145 : z = cyc_real_ll(K, xi, p, C[i], pow, el, pn, flag);
3064 31724 : if (!z) continue;
3065 26418 : Done[i] = 1;
3066 26418 : if (!isintzero(gel(z, 1))) gr = vec_append(gr, z);
3067 26418 : if (++n_done == n_conj) return gr;
3068 : }
3069 : }
3070 : }
3071 0 : pari_err_BUG("pclgp_cyc_real: max_pow is not enough");
3072 : return NULL; /*LCOV_EXCL_LINE*/
3073 : }
3074 :
3075 : /* return (el, g_el) */
3076 : static GEN
3077 56 : next_el_imag(GEN elg, long f)
3078 : {
3079 56 : long el = elg[1];
3080 56 : if (odd(f)) f<<=1;
3081 140 : while (!uisprime(el+=f));
3082 56 : return mkvecsmall2(el, pgener_Fl(el));
3083 : }
3084 :
3085 : /* return (ell, g_ell) */
3086 : static GEN
3087 70 : next_ell_imag(GEN ellg, GEN df0l)
3088 : {
3089 70 : GEN ell = gel(ellg, 1);
3090 770 : while (!BPSW_psp(ell = addii(ell, df0l)));
3091 70 : return mkvec2(ell, pgener_Fp(ell));
3092 : }
3093 :
3094 : static GEN
3095 56 : set_ell_imag(GEN velg, long n, long d_chi, GEN df0)
3096 : {
3097 56 : long i, n_ell = n*d_chi;
3098 56 : GEN z = cgetg(n_ell + 1, t_VEC);
3099 56 : GEN df0l = shifti(df0, 1), ellg = mkvec2(gen_1, gen_1);
3100 126 : for (i=1; i<=n; i++) df0l = muliu(df0l, gel(velg, i)[1]);
3101 126 : for (i=1; i<=n_ell; i++) ellg = gel(z, i)= next_ell_imag(ellg, df0l);
3102 56 : return z;
3103 : }
3104 :
3105 : /* U(X)=u(x)+u(X)*X^f+...+f(X)*X^((m-1)f) or u(x)-u(X)*X^f+...
3106 : * U(X)V(X)=u(X)V(X)(1+X^f+...+X^((m-1)f))
3107 : * =w_0+w_1*X+...+w_{f+el-3}*X^(f+el-3)
3108 : * w_i (1 <= i <= f+el-2) are needed.
3109 : * w_{f+el-2}=0 if el-1 == f.
3110 : * W_i = w_i + w_{i+el-1} (1 <= i <= f-1). */
3111 : static GEN
3112 85 : gauss_Flx_mul(ulong f, GEN elg, GEN ellg)
3113 : {
3114 85 : pari_sp av = avma;
3115 85 : ulong el = elg[1], g_el= elg[2];
3116 85 : ulong el_1 = el-1, f2 = f<<1, lv = el_1, lu = f, m = el_1/f;
3117 85 : ulong ell = itou(gel(ellg, 1)), g_ell = itou(gel(ellg, 2));
3118 85 : ulong z_2f = Fl_powu(g_ell, (ell - 1) / f2, ell);
3119 85 : ulong z_el = Fl_powu(g_ell, (ell - 1) / el, ell);
3120 : ulong i, i2, gi;
3121 85 : GEN W = cgetg(f+1, t_VECSMALL), vz_2f, vz_el;
3122 85 : GEN u = cgetg(lu+2, t_VECSMALL), v = cgetg(lv+2, t_VECSMALL), w0;
3123 :
3124 85 : u[1] = evalsigne(1);
3125 85 : v[1] = evalsigne(1);
3126 85 : vz_2f = Fl_powers(z_2f, f2-1, ell);
3127 85 : vz_el = Fl_powers(z_el, el_1, ell);
3128 528459 : for (i=i2=0; i<lu; i++)
3129 : {
3130 : long j2; /* i2=(i*i)%f2, gi=g_el^i */
3131 528374 : j2 = i2?f2-i2:i2;
3132 528374 : u[2+i] = vz_2f[1+j2];
3133 528374 : if ((i2+=i+i+1)>=f2) i2-=f2; /* same as i2%=f2 */
3134 : }
3135 1600537 : for (gi=1,i=i2=0; i<lv; i++)
3136 : {
3137 1600452 : v[2+i] = Fl_mul(vz_2f[1+i2], vz_el[1+gi], ell);
3138 1600452 : gi = Fl_mul(gi, g_el, el);
3139 1600452 : if ((i2+=i+i+1)>=f2) i2%=f2; /* i2-=f2 does not work */
3140 : }
3141 85 : w0 = Flx_mul(u, v, ell) + 1;
3142 85 : if (m==1)
3143 : { /* el_1=f */
3144 0 : for (i=1; i<f; i++) W[i] = Fl_add(w0[i], w0[i+lv], ell);
3145 0 : W[f] = w0[f];
3146 : }
3147 : else
3148 : {
3149 85 : ulong start = 1+f, end = f+el-1;
3150 85 : GEN w = cgetg(end+1, t_VECSMALL);
3151 2128826 : for (i=1; i<end; i++) w[i] = w0[i];
3152 85 : w[end] = 0;
3153 360 : for (i=1; i<m; i++, start+=f)
3154 550 : w = both_odd(f,i)? Flv_shift_sub(w, w0, ell, start, end)
3155 275 : : Flv_shift_add(w, w0, ell, start, end);
3156 528459 : for (i=0; i<f; i++) W[1+i] = Fl_add(w[1+i], w[1+i+lv], ell);
3157 : }
3158 528374 : for (i=i2=1; i<f; i++)
3159 : {
3160 528289 : W[i]=Fl_mul(W[1+i], vz_2f[1+i2], ell);
3161 528289 : if ((i2+=i+i+1)>=f2) i2%=f2;
3162 : }
3163 : /* W[r]=tau_{LL}^{sigma_r}, 1<= r <= f-1 */
3164 85 : return gerepilecopy(av, Flv_to_ZV(W));
3165 : }
3166 :
3167 : static GEN
3168 90 : gauss_ZX_mul(ulong f, GEN elg, GEN ellg)
3169 : {
3170 90 : pari_sp av = avma, av2;
3171 : ulong el, g_el, el_1, f2, lv, lu, m, i, i2, gi;
3172 90 : GEN ell = gel(ellg, 1), g_ell, ell_1, z_2f, z_el, W, vz_2f, vz_el, u, v, w0;
3173 :
3174 90 : if (lgefint(ell) == 3) return gauss_Flx_mul(f, elg, ellg);
3175 5 : g_ell = gel(ellg, 2); ell_1 = subiu(ell, 1);
3176 5 : el = elg[1]; g_el = elg[2]; el_1 = el-1;
3177 5 : f2 = f<<1; lv=el_1; lu=f; m=el_1/f;
3178 5 : z_2f = Fp_pow(g_ell, diviuexact(ell_1, f2), ell);
3179 5 : vz_2f = Fp_powers(z_2f, f2-1, ell);
3180 5 : W = cgetg(f+1, t_VEC);
3181 5 : av2 = avma;
3182 5 : z_el = Fp_pow(g_ell, diviuexact(ell_1, el), ell);
3183 5 : vz_el = Fp_powers(z_el, el_1, ell);
3184 5 : u = cgetg(lu+2, t_POL); u[1] = evalsigne(1) | evalvarn(0);
3185 5 : v = cgetg(lv+2, t_POL); v[1] = evalsigne(1) | evalvarn(0);
3186 35264 : for (gi=1,i=i2=0; i<lu; i++)
3187 : {
3188 : long j2; /* i2=(i*i)%f2, gi=g_el^i */
3189 35259 : j2 = i2?f2-i2:i2;
3190 35259 : gel(u, 2+i) = gel(vz_2f, 1+j2);
3191 35259 : if ((i2+=i+i+1)>=f2) i2-=f2;
3192 : }
3193 82787 : for (gi=1,i=i2=0; i<lv; i++)
3194 : {
3195 82782 : gel(v, 2+i) = Fp_mul(gel(vz_2f, 1+i2), gel(vz_el, 1+gi), ell);
3196 82782 : gi = Fl_mul(gi, g_el, el);
3197 82782 : if ((i2+=i+i+1)>=f2) i2%=f2;
3198 : }
3199 5 : w0 = gerepileupto(av2, FpX_mul(u, v, ell)) + 1; av2 = avma;
3200 5 : if (m==1)
3201 : {
3202 0 : for (i=1; i < f; i++) gel(W,i) = Fp_add(gel(w0, i), gel(w0, i+lv), ell);
3203 0 : gel(W, f) = gel(w0, f);
3204 : }
3205 : else
3206 : {
3207 5 : ulong start = 1+f, end = f+el-1;
3208 5 : GEN w = cgetg(end+1, t_VEC);
3209 118041 : for (i=1; i<end; i++) gel(w, i) = gel(w0, i);
3210 5 : gel(w, end) = gen_0;
3211 15 : for (i=1; i<m; i++, start+=f)
3212 : {
3213 13 : w = both_odd(f,i)? FpV_shift_sub(w, w0, ell, start, end)
3214 10 : : FpV_shift_add(w, w0, ell, start, end);
3215 10 : if ((i & 7) == 0) w = gerepilecopy(av2, w);
3216 : }
3217 35264 : for (i = 1; i <= f; i++) gel(W, i) = addii(gel(w, i), gel(w, i+lv));
3218 : }
3219 35259 : for (i = i2 = 1; i < f; i++)
3220 : {
3221 35254 : gel(W, i) = Fp_mul(gel(W, 1+i), gel(vz_2f, 1+i2), ell);
3222 35254 : if ((i2+=i+i+1) >= f2) i2 %= f2;
3223 : }
3224 5 : return gerepilecopy(av, W); /* W[r]=tau_{LL}^{sigma_r}, 1<= r <= f-1 */
3225 : }
3226 :
3227 : /* fast but consumes memory */
3228 : static GEN
3229 4 : gauss_el_vell(ulong f, GEN elg, GEN vellg, GEN vz_2f)
3230 : {
3231 4 : pari_sp av = avma, av2;
3232 4 : ulong el = elg[1], g_el = elg[2], el_1 = el-1;
3233 4 : ulong lv=el_1, f2=f<<1, lu=f, m=el_1/f;
3234 4 : GEN W = cgetg(f+1, t_VEC), vz_el, u, v, w0, M;
3235 : ulong i, i2, gi;
3236 :
3237 4 : av2 = avma;
3238 4 : vz_el = vz_vell(el, vellg, &M);
3239 4 : u = cgetg(lu+2, t_POL); u[1] = evalsigne(1) | evalvarn(0);
3240 4 : v = cgetg(lv+2, t_POL); v[1] = evalsigne(1) | evalvarn(0);
3241 25554 : for (i=i2=0; i<lu; i++)
3242 : {
3243 : long j2; /* i2=(i*i)%f2, gi=g_el^i */
3244 25550 : j2 = i2?f2-i2:i2;
3245 25550 : gel(u, 2+i) = gel(vz_2f, 1+j2);
3246 25550 : if ((i2+=i+i+1)>=f2) i2%=f2;
3247 : }
3248 86874 : for (gi=1,i=i2=0; i<lv; i++)
3249 : {
3250 86870 : gel(v, 2+i) = Fp_mul(gel(vz_2f, 1+i2), gel(vz_el, 1+gi), M);
3251 86870 : gi = Fl_mul(gi, g_el, el);
3252 86870 : if ((i2+=i+i+1)>=f2) i2%=f2;
3253 : }
3254 4 : M = gclone(M);
3255 4 : w0 = gerepileupto(av2, FpX_mul(u, v, M)) + 1;
3256 4 : u = M; M = icopy(M); gunclone(u);
3257 4 : av2 = avma;
3258 4 : if (m==1)
3259 : { /* el_1=f */
3260 0 : for (i=1; i < f; i++) gel(W,i) = Fp_add(gel(w0, i), gel(w0, i+lv), M);
3261 0 : gel(W, f) = gel(w0, f);
3262 : }
3263 : else
3264 : {
3265 4 : ulong start = 1+f, end = f+el-1;
3266 4 : GEN w = cgetg(end+1, t_VEC);
3267 112420 : for (i=1; i<end; i++) gel(w, i) = gel(w0, i);
3268 4 : gel(w, end) = gen_0;
3269 19 : for (i=1; i<m; i++, start+=f)
3270 : {
3271 22 : w = both_odd(f,i)? FpV_shift_sub(w, w0, M, start, end)
3272 15 : : FpV_shift_add(w, w0, M, start, end);
3273 15 : if ((i & 7) == 0) w = gerepilecopy(av2, w);
3274 : }
3275 25554 : for (i = 1; i <= f; i++) gel(W, i) = Fp_add(gel(w, i), gel(w, i+lv), M);
3276 : }
3277 25550 : for (i = i2 = 1; i < f; i++)
3278 : {
3279 25546 : gel(W, i) = Fp_mul(gel(W, 1+i), gel(vz_2f, 1+i2), M);
3280 25546 : if ((i2+=i+i+1) >= f2) i2 %= f2;
3281 : }
3282 4 : return gerepilecopy(av, W); /* W[r]=tau_{LL}^{sigma_r}, 1<= r <= f-1 */
3283 : }
3284 :
3285 : static GEN
3286 94 : norm_chi(GEN K, GEN TAU, ulong p, long d_pow, GEN ell, long j0)
3287 : {
3288 94 : pari_sp av = avma;
3289 94 : GEN H = K_get_H(K);
3290 94 : ulong d_K, f_K = K_get_f(K), h = K_get_h(K), g_K = K_get_g(K);
3291 94 : ulong i, j, gi, pd = upowuu(p, d_pow), d_chi = K_get_dchi(K);
3292 94 : GEN z = const_vec(d_chi, gen_1);
3293 94 : GEN e_chi = get_e_chi(K, j0, pd, &d_K);
3294 :
3295 1420 : for (gi=1, i=0; i<d_K; i++)
3296 : {
3297 1326 : GEN y = gen_1;
3298 230862 : for (j=1; j<=h; j++)
3299 229536 : y = Fp_mul(y, gel(TAU, Fl_mul(gi, H[j], f_K)), ell);
3300 1326 : gi = Fl_mul(gi, g_K, f_K);
3301 2652 : for (j=1; j<=d_chi; j++)
3302 : {
3303 1326 : GEN y2 = Fp_powu(y, e_chi[(i+j-1)%d_K], ell);
3304 1326 : gel(z, j) = Fp_mul(gel(z, j), y2, ell);
3305 : }
3306 : }
3307 94 : return gerepilecopy(av, z);
3308 : }
3309 :
3310 : static void
3311 2 : imag_MLLn(long *y, GEN K, ulong p, long d_pow, long n,
3312 : GEN velg, GEN vellg, long j0)
3313 : {
3314 2 : long f = K_get_f(K), d = upowuu(p, d_pow), row = lg(vellg)-1, i, j, k, nz;
3315 2 : GEN g, z, M, vz_2f = vz_vell(f << 1, vellg, &M);
3316 6 : for (i=1; i<=n; i++)
3317 : {
3318 4 : pari_sp av = avma;
3319 4 : GEN elg = gel(velg, i);
3320 4 : if (DEBUGLEVEL>1) err_printf("(f,el-1)=(%ld,%ld*%ld)\n", f,(elg[1]-1)/f,f);
3321 4 : g = gauss_el_vell(f, elg, vellg, vz_2f);
3322 4 : z = norm_chi(K, g, p, d_pow, M, j0);
3323 4 : nz = lg(z)-1;
3324 8 : for (k = 1; k <= nz; k++)
3325 12 : for (j = 1; j <= row; j++)
3326 8 : y[(j-1)*row+(i-1)*nz+k-1] = get_y(gel(z, k), gel(vellg, j), d);
3327 4 : set_avma(av);
3328 : }
3329 2 : }
3330 :
3331 : static void
3332 42 : imag_MLL1(long *y, GEN K, ulong p, long d_pow, GEN velg, GEN vellg, long j0)
3333 : {
3334 42 : long f = K_get_f(K), d = upowuu(p, d_pow);
3335 42 : GEN elg = gel(velg, 1), ellg = gel(vellg, 1), ell = gel(ellg, 1), g, z;
3336 :
3337 42 : if (DEBUGLEVEL>1) err_printf("(f,el-1)=(%ld,%ld*%ld)\n", f, (elg[1]-1)/f, f);
3338 42 : g = gauss_ZX_mul(f, elg, ellg);
3339 42 : z = norm_chi(K, g, p, d_pow, ell, j0);
3340 42 : y[0] = get_y(gel(z, 1), ellg, d);
3341 42 : }
3342 :
3343 : static void
3344 12 : imag_MLL(long *y, GEN K, ulong p, long d_pow, long n, GEN velg, GEN vellg,
3345 : long j0)
3346 : {
3347 12 : pari_sp av = avma;
3348 12 : long i, j, f = K_get_f(K), d = upowuu(p, d_pow), row = lg(vellg)-1;
3349 :
3350 36 : for (j=1; j<=row; j++)
3351 : {
3352 24 : GEN ellg = gel(vellg, j), ell = gel(ellg, 1);
3353 72 : for (i=1; i<=n; i++)
3354 : {
3355 48 : GEN elg = gel(velg, i), g, z;
3356 : ulong k, nz;
3357 48 : if (DEBUGLEVEL>1) err_printf("(f,el-1)=(%ld,%ld*%ld)\n",f,(elg[1]-1)/f,f);
3358 48 : g = gauss_ZX_mul(f, elg, ellg);
3359 48 : z = norm_chi(K, g, p, d_pow, ell, j0);
3360 48 : nz = lg(z)-1;
3361 96 : for (k = 1; k <= nz; k++)
3362 48 : y[(j-1)*row+(i-1)*nz+k-1] = get_y(gel(z, k), ellg, d);
3363 48 : set_avma(av);
3364 : }
3365 : }
3366 12 : }
3367 :
3368 : /* return an upper bound >= 0 if one was found, otherwise return -1.
3369 : * set chi-part to be (1) if chi is Teichmuller character.
3370 : * B_{1,omega^(-1)} is not p-adic integer. */
3371 : static GEN
3372 42 : cyc_imag_MLL(GEN K, ulong p, long d_pow, long j, long flag)
3373 : {
3374 42 : long f = K_get_f(K), d_chi = K_get_dchi(K);
3375 42 : long n, n0 = 1, n_el = d_pow, d = upowuu(p, d_pow), rank = n_el*d_chi;
3376 42 : GEN df0, velg = const_vec(n_el, NULL), vellg = NULL;
3377 42 : GEN oldgr = mkvec2(gen_0, NULL), newgr = mkvec2(gen_0, NULL);
3378 42 : long *y0 = (long*)stack_calloc(sizeof(long)*rank*rank);
3379 :
3380 42 : if (DEBUGLEVEL>1)
3381 0 : err_printf("cyc_imag_MLL:p=%ld d_pow=%ld deg(K)=%ld cond(K)=%ld avma=%ld\n",
3382 : p, d_pow, K_get_d(K), f, avma);
3383 42 : df0 = muluu(d, f%p?f:f/p);
3384 42 : gel(velg, 1) = next_el_imag(mkvecsmall2(1, 1), f);
3385 42 : if (flag&USE_FULL_EL)
3386 : {
3387 0 : for (n=2; n<=n_el; n++) gel(velg, n) = next_el_imag(gel(velg, n-1), f);
3388 0 : n0 = n_el;
3389 : }
3390 56 : for (n=n0; n<=n_el; n++) /* loop while structure is unknown */
3391 : {
3392 56 : pari_sp av2 = avma;
3393 : pari_timer ti;
3394 : long n_ell, m, M;
3395 : GEN y;
3396 56 : vellg = set_ell_imag(velg, n, d_chi, df0);
3397 56 : n_ell = lg(vellg)-1; /* equal to n*d_chi */
3398 56 : if (DEBUGLEVEL>2) err_printf("velg=%Ps\nvellg=%Ps\n", velg, vellg);
3399 56 : if (DEBUGLEVEL>2) timer_start(&ti);
3400 56 : if (n_ell==1)
3401 42 : imag_MLL1(y0, K, p, d_pow, velg, vellg, j);
3402 14 : else if (lgefint(gmael(vellg, n, 1))<=3 || (flag&SAVE_MEMORY))
3403 12 : imag_MLL(y0, K, p, d_pow, n, velg, vellg, j);
3404 : else
3405 2 : imag_MLLn(y0, K, p, d_pow, n, velg, vellg, j);
3406 56 : set_avma(av2);
3407 56 : if (DEBUGLEVEL>2) timer_printf(&ti, "gauss sum");
3408 56 : y = ary2mat(y0, n_ell);
3409 56 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
3410 56 : y = ZM_snf(y);
3411 56 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
3412 56 : y = make_p_part(y, p, d_pow);
3413 56 : if (DEBUGLEVEL>3) err_printf("y=%Ps\n", y);
3414 56 : newgr = structure_MLL(y, d_pow);
3415 56 : if (DEBUGLEVEL>3)
3416 0 : err_printf("d_pow=%ld d_chi=%ld old=%Ps new=%Ps\n",d_pow,d_chi,oldgr,newgr);
3417 56 : if (equalsi(d_pow*d_chi, gel(newgr, 1))) break;
3418 14 : if ((m = find_del_el(&oldgr, newgr, n, n_el, d_chi)))
3419 0 : { M = m = delete_el(velg, m); n--; }
3420 : else
3421 14 : { M = n+1; m = n; }
3422 14 : gel(velg, M) = next_el_imag(gel(velg, m), f);
3423 : }
3424 42 : return get_str(newgr);
3425 : }
3426 :
3427 : /* When |A_psi|=p^e, A_psi=(p^e1,...,p^er) (psi=chi^j),
3428 : * return vec[e, [e1, ... ,er], 1].
3429 : * If gr str is not determined, return vec[e, [], 1].
3430 : * If |A_chi|=1, return vec[0, [], 1].
3431 : * If |A_chi|=p, return vec[1, [1], 1].
3432 : * If e is not determined, return vec[-1, [], 1].
3433 : * If psi is Teichmuller, return vec[0, [], 1].
3434 : * B_{1,omega^(-1)} is not p-adic integer. */
3435 : static GEN
3436 26334 : cyc_imag(GEN K, GEN B, GEN p, long j, GEN powp, long flag)
3437 : {
3438 26334 : pari_sp av = avma;
3439 26334 : GEN MinPol = gel(K, 3), Chi = gel(K, 2), B1, B2, gr;
3440 26334 : long x, d_K = K_get_d(K), f_K = K_get_f(K), d_chi = K_get_dchi(K);
3441 :
3442 26334 : if (f_K == d_K+1 && equaliu(p, f_K) && j == 1) /* Teichmuller */
3443 77 : return mkvec3(gen_0, nullvec(), gen_1);
3444 26257 : B1 = FpX_rem(ZX_ber_conj(B, j, d_K), MinPol, powp);
3445 26257 : B2 = FpX_rem(ZX_ber_den(Chi, j, d_K), MinPol, powp);
3446 26257 : if (degpol(B1)<0 || degpol(B2)<0)
3447 : {
3448 0 : set_avma(av);
3449 0 : return mkvec3(gen_m1, nullvec(), gen_1); /* B=0(mod p^pow) */
3450 : }
3451 26257 : x = ZX_pval(B1, p) - ZX_pval(B2, p);
3452 26257 : set_avma(av);
3453 26257 : if (x<0) pari_err_BUG("subcyclopclgp [Bernoulli number]");
3454 26257 : if (DEBUGLEVEL && x) verbose_output(K, p, x, j);
3455 26257 : if (x==0) return mkvec3(gen_0, nullvec(), gen_1); /* trivial */
3456 588 : if (x==1) return mkvec3(utoi(d_chi), onevec(d_chi), gen_1);
3457 140 : if ((flag&USE_MLL)==0) return mkvec3(utoi(x*d_chi), nullvec(), gen_1);
3458 42 : gr = d_K == 2? cyc_buch(-f_K, p, x): cyc_imag_MLL(K, itou(p), x, j, flag);
3459 42 : return gerepilecopy(av, mkvec3(utoipos(d_chi * x), gr, gen_1));
3460 : }
3461 :
3462 : /* handle representatives of all injective characters, d_chi=[Q_p(zeta_d):Q_p],
3463 : * d=d_K */
3464 : static GEN
3465 10080 : pclgp_cyc_imag(GEN K, GEN p, long start_pow, long max_pow, long flag)
3466 : {
3467 10080 : GEN C = gel(K, 5), Chi = gel(K, 2);
3468 10080 : long n_conj = K_get_nconj(K), d_K = K_get_d(K), f_K = K_get_f(K);
3469 10080 : long i, pow, n_done = 0;
3470 10080 : GEN gr = nullvec(), Done = const_vecsmall(n_conj, 0);
3471 10080 : GEN B = zx_ber_num(Chi, f_K, d_K), B_num;
3472 :
3473 10080 : if (lgefint(p)==3 && n_conj>10) /* mark trivial chi-part by pre-calculation */
3474 : {
3475 595 : ulong up = itou(p);
3476 595 : GEN minpol = ZX_to_Flx(gel(K, 3), up);
3477 7350 : for (i=1; i<=n_conj; i++)
3478 : {
3479 7168 : pari_sp av = avma;
3480 : long degB;
3481 7168 : B_num = Flx_rem(Flx_ber_conj(B, C[i], d_K, up), minpol, up);
3482 7168 : degB = degpol(B_num);
3483 7168 : set_avma(av);
3484 7168 : if (degB<0) continue;
3485 6937 : Done[i] = 1;
3486 6937 : if (++n_done == n_conj) return gr;
3487 : }
3488 : }
3489 9667 : for (pow = start_pow; pow<=max_pow; pow++)
3490 : {
3491 9667 : GEN powp = powiu(p, pow);
3492 27503 : for (i = 1; i <= n_conj; i++)
3493 : {
3494 : GEN z;
3495 27503 : if (Done[i]) continue;
3496 26334 : z = cyc_imag(K, B, p, C[i], powp, flag);
3497 26334 : if (equalim1(gel(z, 1))) continue;
3498 26334 : Done[i] = 1;
3499 26334 : if (!isintzero(gel(z, 1))) gr = vec_append(gr, z);
3500 26334 : if (++n_done == n_conj) return gr;
3501 : }
3502 : }
3503 0 : pari_err_BUG("pclgp_cyc_imag: max_pow is not enough");
3504 : return NULL; /*LCOV_EXCL_LINE*/
3505 : }
3506 :
3507 : static GEN
3508 392 : gather_part(GEN g, long sgn)
3509 : {
3510 392 : long i, j, l = lg(g), ord = 0, flag = 1;
3511 392 : GEN z2 = cgetg(l, t_VEC);
3512 1778 : for (i = j = 1; i < l; i++)
3513 : {
3514 1386 : GEN t = gel(g,i);
3515 1386 : if (equaliu(gel(t, 3), sgn))
3516 : {
3517 693 : ord += itou(gel(t, 1));
3518 693 : if (lg(gel(t, 2)) == 1) flag = 0;
3519 693 : gel(z2, j++) = gel(t, 2);
3520 : }
3521 : }
3522 392 : if (flag==0 || ord==0) z2 = nullvec();
3523 : else
3524 : {
3525 126 : setlg(z2, j); z2 = shallowconcat1(z2);
3526 126 : ZV_sort_inplace(z2); vecreverse_inplace(z2);
3527 : }
3528 392 : return mkvec2(utoi(ord), z2);
3529 : }
3530 :
3531 : #ifdef DEBUG
3532 : static void
3533 : handling(GEN K)
3534 : {
3535 : long d_K = K_get_d(K), f_K = K_get_f(K), s_K = K_get_s(K), g_K = K_get_g(K);
3536 : long d_chi = K_get_dchi(K);
3537 : err_printf(" handling %s cyclic subfield K,\
3538 : deg(K)=%ld, cond(K)=%ld g_K=%ld d_chi=%ld H=%Ps\n",
3539 : s_K? "a real": "an imaginary",d_K,f_K,g_K,d_chi,zv_to_ZV(gmael3(K,1,1,1)));
3540 : }
3541 : #endif
3542 :
3543 : /* HH a t_VECSMALL listing group generators
3544 : * Aoki and Fukuda, LNCS vol.4076 (2006), 56-74. */
3545 : static GEN
3546 161 : pclgp(GEN p0, long f, GEN HH, long degF, long flag)
3547 : {
3548 : long start_pow, max_pow, ip, lp, i, n_f;
3549 161 : GEN vH1, z, vData, cycGH, vp = typ(p0) == t_INT? mkvec(p0): p0;
3550 :
3551 161 : vH1 = GHinit(f, HH, &cycGH); n_f = lg(vH1)-1;
3552 : #ifdef DEBUG
3553 : err_printf("F is %s, deg(F)=%ld, ", srh_1(HH)? "real": "imaginary", degF);
3554 : err_printf("cond(F)=%ld, G(F/Q)=%Ps\n",f, cycGH);
3555 : err_printf("F has %ld cyclic subfield%s except for Q.\n", n_f,n_f>1?"s":"");
3556 : #endif
3557 :
3558 161 : lp = lg(vp); z = cgetg(lp, t_MAT);
3559 357 : for (ip = 1; ip < lp; ip++)
3560 : {
3561 196 : pari_sp av = avma;
3562 196 : long n_sub=0, n_chi=0;
3563 196 : GEN gr=nullvec(), p = gel(vp, ip), zi;
3564 : /* find conductor e of cyclic subfield K and set the subgroup HE of (Z/eZ)^*
3565 : * corresponding to K */
3566 196 : set_p_f(p, f, &start_pow, &max_pow);
3567 196 : vData = const_vec(degF, NULL);
3568 :
3569 16982 : for (i=1; i<=n_f; i++) /* prescan. set Teichmuller */
3570 : {
3571 16863 : GEN H1 = gel(vH1, i);
3572 16863 : long d_K = _get_d(H1), f_K = _get_f(H1), g_K = _get_g(H1);
3573 :
3574 16863 : if (f_K == d_K+1 && equaliu(p, f_K)) /* found K=Q(zeta_p) */
3575 : {
3576 : pari_timer ti;
3577 77 : GEN pnmax = powiu(p, max_pow), vNewton, C, MinPol;
3578 77 : long d_chi = 1, n_conj = eulerphiu(d_K);
3579 77 : ulong pmodd = umodiu(p, d_K);
3580 :
3581 77 : C = set_C(pmodd, d_K, d_chi, n_conj);
3582 77 : MinPol = set_minpol_teich(g_K, p, max_pow);
3583 77 : if (DEBUGLEVEL>3) timer_start(&ti);
3584 77 : vNewton = FpX_Newton(MinPol, d_K+1, pnmax);
3585 77 : if (DEBUGLEVEL>3)
3586 0 : timer_printf(&ti, "FpX_Newton: teich: %ld %ld", degpol(MinPol), d_K);
3587 77 : gel(vData, d_K) = mkvec4(MinPol, vNewton, C,
3588 : mkvecsmall2(d_chi, n_conj));
3589 77 : break;
3590 : }
3591 : }
3592 :
3593 20440 : for (i=1; i<=n_f; i++) /* handle all cyclic K */
3594 : {
3595 20244 : GEN H1 = gel(vH1, i), K, z1, Chi;
3596 20244 : long d_K = _get_d(H1), s_K = _get_s(H1);
3597 : pari_sp av2;
3598 :
3599 20244 : if ((flag&SKIP_PROPER) && degF != d_K) continue;
3600 20244 : if (!gel(vData, d_K))
3601 : {
3602 : pari_timer ti;
3603 819 : GEN pnmax = powiu(p, max_pow), vNewton, C, MinPol;
3604 819 : ulong pmodd = umodiu(p, d_K);
3605 819 : long d_chi = order_f_x(d_K, pmodd), n_conj = eulerphiu(d_K)/d_chi;
3606 :
3607 819 : C = set_C(pmodd, d_K, d_chi, n_conj);
3608 819 : MinPol = set_minpol(d_K, p, max_pow, n_conj);
3609 819 : if (DEBUGLEVEL>3) timer_start(&ti);
3610 : /* vNewton[2+i] = vNewton[2+i+d_K]. We need vNewton[2+i] for
3611 : * 0 <= i < d_K. But vNewton[2+d_K-1] may be 0 and will be deleted.
3612 : * So we need vNewton[2+d_K] not to delete vNewton[2+d_K-1]. */
3613 819 : vNewton = FpX_Newton(MinPol, d_K+1, pnmax);
3614 819 : if (DEBUGLEVEL>3)
3615 0 : timer_printf(&ti, "FpX_Newton: %ld %ld", degpol(MinPol), d_K);
3616 819 : gel(vData, d_K) = mkvec4(MinPol, vNewton, C,
3617 : mkvecsmall2(d_chi, n_conj));
3618 : }
3619 20244 : av2 = avma;
3620 20244 : Chi = s_K? NULL: get_chi(H1);
3621 20244 : K = shallowconcat(mkvec2(H1, Chi), gel(vData, d_K));
3622 : #ifdef DEBUG
3623 : handling(K);
3624 : #endif
3625 20244 : if (s_K && !(flag&NO_PLUS_PART))
3626 10164 : z1 = pclgp_cyc_real(K, p, max_pow, flag);
3627 10080 : else if (!s_K && !(flag&NO_MINUS_PART))
3628 10080 : z1 = pclgp_cyc_imag(K, p, start_pow, max_pow, flag);
3629 0 : else { set_avma(av2); continue; }
3630 20244 : n_sub++; n_chi += gmael(vData, d_K, 4)[2]; /* += n_conj */
3631 20244 : if (lg(z1) == 1) set_avma(av2);
3632 658 : else gr = gerepilecopy(av2, shallowconcat(gr, z1));
3633 : }
3634 196 : zi = mkcol(p);
3635 196 : zi = vec_append(zi, (flag&NO_PLUS_PART)?nullvec():gather_part(gr, 0));
3636 196 : zi = vec_append(zi, (flag&NO_MINUS_PART)?nullvec():gather_part(gr, 1));
3637 196 : zi = shallowconcat(zi, mkcol3(cycGH, utoi(n_sub), utoi(n_chi)));
3638 196 : gel(z, ip) = gerepilecopy(av, zi);
3639 : }
3640 161 : return typ(p0) == t_INT? shallowtrans(gel(z,1)): shallowtrans(z);
3641 : }
3642 :
3643 : static GEN
3644 413 : reduce_gcd(GEN x1, GEN x2)
3645 : {
3646 413 : GEN d = gcdii(x1, x2);
3647 413 : x1 = diviiexact(x1, d);
3648 413 : x2 = diviiexact(x2, d);
3649 413 : return mkvec2(x1, x2);
3650 : }
3651 :
3652 : /* norm of x0 (= pol of zeta_d with deg <= d-1) by g of order n
3653 : * x0^{1+g+g^2+...+g^(n-1)} */
3654 : static GEN
3655 49 : ber_norm_cyc(GEN x0, long g, long n, long d)
3656 : {
3657 49 : pari_sp av = avma;
3658 49 : long i, ei, di, fi = 0, l = ulogint(n, 2);
3659 49 : GEN xi = x0;
3660 49 : ei = 1L << l; di = n / ei;
3661 203 : for (i = 1; i <= l; i++)
3662 : {
3663 154 : if (odd(di)) fi += ei;
3664 154 : ei = 1L << (l-i); di = n / ei;
3665 154 : xi = ZX_mod_Xnm1(ZX_mul(xi, ber_conj(xi, Fl_powu(g, ei, d), d)), d);
3666 154 : if (odd(di))
3667 42 : xi = ZX_mod_Xnm1(ZX_mul(xi, ber_conj(x0, Fl_powu(g, fi, d), d)), d);
3668 : }
3669 49 : return gerepilecopy(av, xi);
3670 : }
3671 :
3672 : /* x0 a ZX of deg < d */
3673 : static GEN
3674 21 : ber_norm_by_cyc(GEN x0, long d, GEN MinPol)
3675 : {
3676 21 : pari_sp av=avma;
3677 21 : GEN x = x0, z = znstar(utoi(d)), cyc = gel(z, 2), gen = gel(z, 3);
3678 21 : long i, l = lg(cyc);
3679 : pari_timer ti;
3680 :
3681 21 : if (DEBUGLEVEL>1) timer_start(&ti);
3682 70 : for (i = 1; i < l; i++)
3683 49 : x = ber_norm_cyc(x, itou(gmael(gen, i, 2)), itou(gel(cyc, i)), d);
3684 21 : if (DEBUGLEVEL>1) timer_printf(&ti, "ber_norm_by_cyc [ber_norm_cyc]");
3685 21 : x = ZX_rem(x, MinPol); /* slow */
3686 21 : if (DEBUGLEVEL>1) timer_printf(&ti, "ber_norm_by_cyc [ZX_rem]");
3687 21 : if (lg(x) != 3) pari_err_BUG("subcyclohminus [norm of Bernoulli number]");
3688 21 : return gerepilecopy(av, gel(x, 2));
3689 : }
3690 :
3691 : /* MinPol = polcyclo(d_K, 0).
3692 : * MinPol = fac*cofac (mod p).
3693 : * B is zv.
3694 : * K : H1, MinPol, [fac, cofac], C, [d_chi, n_conj] */
3695 : static long
3696 98 : ber_norm_by_val(GEN K, GEN B, GEN p)
3697 : {
3698 98 : pari_sp av = avma;
3699 98 : GEN MinPol = gel(K, 2), C = gel(K, 4);
3700 98 : GEN vfac = gel(K, 3), fac = gel(vfac, 1), cofac = gel(vfac, 2);
3701 98 : long d_chi = K_get_dchi(K), n_conj = K_get_nconj(K), d_K = K_get_d(K);
3702 98 : long i, r, n_done = 0, x = 0, dcofac = degpol(cofac);
3703 : GEN pr, Done;
3704 :
3705 98 : Done = const_vecsmall(n_conj, 0);
3706 98 : if (lgefint(p)==3)
3707 : { /* mark trivial chi-part by pre-calculation */
3708 98 : ulong up = itou(p);
3709 98 : GEN facs = ZX_to_Flx(fac, up);
3710 196 : for (i = 1; i <= n_conj; i++)
3711 : {
3712 98 : pari_sp av2 = avma;
3713 98 : GEN B_conj = Flx_rem(Flx_ber_conj(B, C[i], d_K, up), facs, up);
3714 98 : long degB = degpol(B_conj);
3715 98 : set_avma(av2); if (degB < 0) continue;
3716 0 : Done[i] = 1; if (++n_done == n_conj) return gc_long(av, x);
3717 : }
3718 : }
3719 : else
3720 : {
3721 0 : for (i = 1; i <= n_conj; i++)
3722 : {
3723 0 : pari_sp av2 = avma;
3724 0 : GEN B_conj = FpX_rem(FpX_ber_conj(B, C[i], d_K, p), fac, p);
3725 0 : long degB = degpol(B_conj);
3726 0 : set_avma(av2); if (degB < 0) continue;
3727 0 : Done[i] = 1; if (++n_done == n_conj) return gc_long(av, x);
3728 : }
3729 : }
3730 252 : for (pr = p, r = 2; r; r <<= 1)
3731 : {
3732 : GEN polr;
3733 252 : pr = sqri(pr); /* p^r */
3734 252 : polr = (dcofac==0)? FpX_red(MinPol, pr)
3735 252 : : gel(ZpX_liftfact(MinPol, vfac, pr, p, r), 1);
3736 406 : for (i = 1; i <= n_conj; i++)
3737 : {
3738 252 : pari_sp av2 = avma;
3739 : GEN B_conj;
3740 : long degB;
3741 252 : if (Done[i]) continue;
3742 252 : B_conj = FpX_rem(FpX_ber_conj(B, C[i], d_K, pr), polr, pr);
3743 252 : degB = degpol(B_conj);
3744 252 : set_avma(av2); if (degB < 0) continue;
3745 98 : x += d_chi * ZX_pval(B_conj, p);
3746 98 : Done[i] = 1; if (++n_done == n_conj) return gc_long(av, x);
3747 : }
3748 : }
3749 : pari_err_BUG("ber_norm_by_val"); return 0;/*LCOV_EXCL_LINE*/
3750 : }
3751 :
3752 : /* n > 2, p = odd prime not dividing n, e > 0, pe = p^e; d = n*p^e
3753 : * return generators of the subgroup H of (Z/dZ)^* corresponding to
3754 : * Q(zeta_{p^e}): H = {1<=a<=d | gcd(a,n)=1, a=1(mod p^e)} */
3755 : static GEN
3756 0 : znstar_subgr(ulong n, ulong pe, ulong d)
3757 : {
3758 0 : GEN z = znstar(utoi(n)), g = gel(z, 3), G;
3759 0 : long i, l = lg(g);
3760 0 : G = cgetg(l, t_VECSMALL);
3761 0 : for (i=1; i<l; i++) G[i] = u_chinese_coprime(itou(gmael(g,i,2)), 1, n, pe, d);
3762 0 : return mkvec2(gel(z,2), G);
3763 : }
3764 :
3765 : /* K is a cyclic extension of degree n*p^e (n>=4 is even).
3766 : * x a ZX of deg < n*p^e. */
3767 : static long
3768 0 : ber_norm_with_val(GEN x, long n, ulong p, ulong e)
3769 : {
3770 0 : pari_sp av = avma;
3771 0 : long i, j, r, degx, pe = upowuu(p, e), d = n*pe;
3772 0 : GEN z, gr, gen, y = cgetg(pe+2, t_POL), MinPol = polcyclo(n, 0);
3773 0 : y[1] = evalsigne(1) | evalvarn(0);
3774 0 : z = znstar_subgr(n, pe, d);
3775 0 : gr = gel(z, 1); gen = gel(z, 2); r = lg(gr)-1;
3776 0 : for (i=1; i<=r; i++)
3777 0 : x = ber_norm_cyc(x, itou(gel(gen, i)), itou(gel(gr, i)), d);
3778 0 : degx = degpol(x);
3779 0 : for (j=0; j<pe; j++)
3780 : {
3781 0 : GEN t = pol_zero(n), z;
3782 0 : long a = j; /* a=i*pe+j */
3783 0 : for (i=0; i<n; i++)
3784 : {
3785 0 : if (a>degx) break;
3786 0 : gel(t, 2+a%n) = gel(x, 2+a);
3787 0 : a += pe;
3788 : }
3789 0 : z = ZX_rem(ZX_renormalize(t, 2+n), MinPol);
3790 0 : if (degpol(z)<0) gel(y, 2+j) = gen_0;
3791 0 : else if (degpol(z)==0) gel(y, 2+j) = gel(z, 2);
3792 0 : else pari_err_BUG("ber_norm_subgr");
3793 : }
3794 0 : y = ZX_renormalize(y, pe+2);
3795 0 : if (e>1) y = ZX_rem(y, polcyclo(pe, 0));
3796 0 : return gc_long(av, ZX_p_val(y, p, e));
3797 : }
3798 :
3799 : /* K is a cyclic extension of degree 2*p^e. x a ZX of deg < 2*p^e. In most
3800 : * cases, deg(x)=2*p^e-1. But deg(x) can be any value in [0, 2*p^e-1]. */
3801 : static long
3802 301 : ber_norm_with_val2(GEN x, ulong p, ulong e)
3803 : {
3804 301 : pari_sp av = avma;
3805 301 : long i, d = degpol(x), pe = upowuu(p, e);
3806 301 : GEN y = pol_zero(pe);
3807 301 : if (d == 2*pe-1)
3808 : {
3809 38416 : for (i = 0; i < pe; i++)
3810 76230 : gel(y, 2+i) = odd(i)? subii(gel(x, 2+i+pe), gel(x, 2+i))
3811 38115 : : subii(gel(x, 2+i), gel(x, 2+i+pe));
3812 : }
3813 : else
3814 : {
3815 0 : for (i = 0; i < pe && i <= d; i++)
3816 0 : gel(y, 2+i) = odd(i)? negi(gel(x, 2+i)): gel(x, 2+i);
3817 0 : for (i = pe; i <= d; i++)
3818 0 : gel(y, 2+i-pe) = odd(i)? subii(gel(y, 2+i-pe), gel(x, 2+i))
3819 0 : : addii(gel(y, 2+i-pe), gel(x, 2+i));
3820 : }
3821 301 : y = ZX_renormalize(y, 2+pe);
3822 301 : if (e > 1) y = ZX_rem(y, polcyclo(pe, 0));
3823 301 : return gc_long(av, ZX_p_val(y, p, e));
3824 : }
3825 :
3826 : /* K : H1, MinPol, [fac, cofac], C, [d_chi, n_conj] */
3827 : static GEN
3828 812 : ber_cyc5(GEN K, GEN p)
3829 : {
3830 812 : pari_sp av = avma;
3831 812 : GEN MinPol = gel(K, 2), H = K_get_H(K);
3832 812 : long d = K_get_d(K), f = K_get_f(K), h = K_get_h(K), g = K_get_g(K);
3833 812 : GEN x, x1, x2, y, B = const_vecsmall(d+1, 0);
3834 812 : long i, j, gi, e, f2 = f>>1, dMinPol = degpol(MinPol), chi2 = -1, *B2 = B+2;
3835 :
3836 : /* get_chi inlined here to save memory */
3837 18111989 : for (j=1; j<=h; j++) /* i = 0 */
3838 : {
3839 18111177 : if (H[j] == 2) chi2 = 0;
3840 18111177 : if (H[j] <= f2) B2[0]++; /* Chi[H[j]] = 0 */
3841 : }
3842 97314 : for (i = 1, gi = g; i < d; i++)
3843 : {
3844 93017085 : for (j=1; j<=h; j++)
3845 : {
3846 92920583 : long t = Fl_mul(gi, H[j], f); /* Chi[t] = i */
3847 92920583 : if (t == 2) chi2 = i;
3848 92920583 : if (t <= f2) B2[i]++;
3849 : }
3850 96502 : gi = Fl_mul(gi, g, f);
3851 : }
3852 812 : y = zx_to_ZX(zx_renormalize(B, d+2));
3853 :
3854 812 : if (p)
3855 : {
3856 : ulong n;
3857 399 : e = u_pvalrem(d, p, &n);
3858 399 : if (e == 0)
3859 98 : x1 = utoi(ber_norm_by_val(K, B, p));
3860 301 : else if (n > 2)
3861 0 : x1 = utoi(ber_norm_with_val(y, n, itou(p), e));
3862 : else
3863 301 : x1 = utoi(ber_norm_with_val2(y, itou(p), e));
3864 : }
3865 : else
3866 : {
3867 413 : if (dMinPol > 100)
3868 21 : x1 = ber_norm_by_cyc(y, d, MinPol);
3869 : else
3870 392 : x1 = ZX_resultant(MinPol, ZX_rem(y, MinPol));
3871 : }
3872 :
3873 812 : if (chi2 < 0) /* chi2 = Chi[2] */
3874 0 : x2 = shifti(gen_1, 2*dMinPol);
3875 812 : else if (chi2 == 0)
3876 21 : x2 = shifti(gen_1, dMinPol);
3877 : else
3878 : {
3879 791 : long e = d/ugcd(chi2, d);
3880 791 : x2 = powiu(polcyclo_eval(e, gen_2), eulerphiu(d)/eulerphiu(e));
3881 791 : x2 = shifti(x2, dMinPol);
3882 : }
3883 812 : if (p) x = stoi(itou(x1)-Z_pval(x2, p)); else x = reduce_gcd(x1, x2);
3884 812 : return gerepilecopy(av, x);
3885 : }
3886 :
3887 : /* Hirabayashi-Yoshino, Manuscripta Math. vol.60, 423-436 (1988), Theorem 1
3888 : *
3889 : * F is a subfield of Q(zeta_f)
3890 : * f=p^a => Q=1
3891 : * If F=Q(zeta_f), Q=1 <=> f=p^a
3892 : * If f=4*p^a, p^a*q^b (p,q are odd primes), Q=2 <=> [Q(zeta_f):F] is odd */
3893 : static long
3894 21 : unit_index(ulong d, ulong f)
3895 : {
3896 : ulong r, d_f;
3897 21 : GEN fa = factoru(f), P = gel(fa, 1), E = gel(fa, 2); r = lg(P)-1;
3898 21 : if (r==1) return 1; /* f=P^a */
3899 7 : d_f = eulerphiu_fact(fa);
3900 7 : if (d==d_f) return 2; /* F=Q(zeta_f) */
3901 0 : if (r==2 && ((P[1]==2 && E[1]==2) || P[1]>2)) return odd(d_f/d)?2:1;
3902 0 : return 0;
3903 : }
3904 :
3905 : /* Compute relative class number h of the subfield K of Q(zeta_f)
3906 : * corresponding to the subgroup HH of (Z/fZ)^*.
3907 : * If p!=NULL, then return valuation(h,p). */
3908 : static GEN
3909 119 : rel_class_num(long f, GEN HH, long degF, GEN p)
3910 : {
3911 : long i, n_f, W, Q;
3912 119 : GEN vH1, vData, x, z = gen_1, z1 = gen_0, z2 = mkvec2(gen_1, gen_1);
3913 :
3914 119 : vH1 = GHinit(f, HH, NULL); n_f = lg(vH1)-1;
3915 119 : vData = const_vec(degF, NULL);
3916 1652 : for (i=1; i<=n_f; i++)
3917 : {
3918 1533 : GEN H1 = gel(vH1, i), K;
3919 1533 : long d_K = _get_d(H1), s = _get_s(H1);
3920 :
3921 1533 : if (s) continue; /* F is real */
3922 : #ifdef DEBUG
3923 : err_printf(" handling %s cyclic subfield K, deg(K)=%ld, cond(K)=%ld\n",
3924 : s? "a real": "an imaginary", d_K, _get_f(H1));
3925 : #endif
3926 812 : if (!gel(vData, d_K))
3927 : {
3928 : GEN C, MinPol, fac, cofac;
3929 : ulong d_chi, n_conj;
3930 497 : MinPol = polcyclo(d_K,0);
3931 497 : if (p && umodui(d_K, p))
3932 98 : {
3933 98 : ulong pmodd = umodiu(p, d_K);
3934 98 : GEN MinPol_p = FpX_red(MinPol, p);
3935 98 : d_chi = order_f_x(d_K, pmodd);
3936 98 : n_conj = eulerphiu(d_K)/d_chi;
3937 98 : if (n_conj==1) fac = MinPol_p; /* polcyclo(d_K) is irred mod p */
3938 0 : else fac = FpX_one_cyclo(d_K, p);
3939 98 : cofac = FpX_div(MinPol_p, fac, p);
3940 98 : C = set_C(pmodd, d_K, d_chi, n_conj);
3941 : }
3942 : else
3943 : {
3944 399 : fac = cofac = C = NULL;
3945 399 : d_chi = n_conj = 0;
3946 : }
3947 497 : gel(vData, d_K) = mkvec5(MinPol, mkvec2(fac, cofac), C,
3948 : NULL, mkvecsmall2(d_chi, n_conj));
3949 : }
3950 812 : K = vec_prepend(gel(vData, d_K), H1);
3951 812 : z = ber_cyc5(K, p);
3952 812 : if (p) z1 = addii(z1, z);
3953 : else
3954 : {
3955 413 : gel(z2, 1) = mulii(gel(z2, 1), gel(z, 1));
3956 413 : gel(z2, 2) = mulii(gel(z2, 2), gel(z, 2));
3957 : }
3958 : }
3959 119 : W = root_of_1(f, HH);
3960 119 : if (p) return addiu(z1, z_pval(W, p));
3961 21 : Q = unit_index(degF, f);
3962 21 : x = dvmdii(muliu(gel(z2,1), 2 * W), gel(z2,2), &z1);
3963 21 : if (signe(z1)) pari_err_BUG("subcyclohminus [norm of Bernoulli number]");
3964 21 : if (!Q && mpodd(x)) Q = 2; /* FIXME: can this happen ? */
3965 21 : if (Q == 1) x = shifti(x, -1);
3966 21 : return mkvec2(x, utoi(Q));
3967 : }
3968 :
3969 : static void
3970 336 : checkp(const char *fun, long degF, GEN p)
3971 : {
3972 336 : if (!BPSW_psp(p)) pari_err_PRIME(fun, p);
3973 329 : if (equaliu(p, 2)) pari_err_DOMAIN(fun,"p","=", gen_2, p);
3974 315 : if (degF && dvdsi(degF, p)) errpdiv(fun, p, degF);
3975 301 : }
3976 :
3977 : /* if flag is set, handle quadratic fields specially (don't set H) */
3978 : static long
3979 427 : subcyclo_init(const char *fun, GEN FH, long *pdegF, GEN *pH, long flag)
3980 : {
3981 427 : long f = 0, degF = 2;
3982 427 : GEN F = NULL, H = NULL;
3983 427 : if (typ(FH) == t_POL)
3984 : {
3985 70 : degF = degpol(FH);
3986 70 : if (degF < 1 || !RgX_is_ZX(FH)) pari_err_TYPE(fun, FH);
3987 70 : if (flag && degF == 2)
3988 : {
3989 49 : F = coredisc(ZX_disc(FH));
3990 49 : if (is_bigint(F))
3991 0 : pari_err_IMPL(stack_sprintf("conductor f > %lu in %s", ULONG_MAX, fun));
3992 49 : f = itos(F); if (f == 1) degF = 1;
3993 : }
3994 : else
3995 : {
3996 21 : GEN z, bnf = Buchall(pol_x(fetch_var()), 0, DEFAULTPREC);
3997 21 : z = rnfconductor(bnf, FH); H = gel(z,3);
3998 21 : f = subcyclo_nH(fun, gel(z,2), &H);
3999 21 : delete_var();
4000 21 : H = znstar_generate(f, H); /* group elements */
4001 : }
4002 : }
4003 : else
4004 : {
4005 357 : long l = lg(FH), fH;
4006 357 : if (typ(FH) == t_INT) F = FH;
4007 273 : else if (typ(FH) == t_VEC && (l == 2 || l == 3))
4008 : {
4009 273 : F = gel(FH, 1);
4010 273 : if (l == 3) H = gel(FH, 2);
4011 : }
4012 0 : else pari_err_TYPE(fun, FH);
4013 357 : f = subcyclo_nH(fun, F, &H);
4014 350 : H = znstar_generate(f, H); /* group elements */
4015 350 : fH = znstar_conductor(H);
4016 350 : if (fH == 1) degF = 1;
4017 : else
4018 : {
4019 350 : if (fH != f) { H = znstar_reduce_modulus(H, fH); f = fH; }
4020 350 : degF = eulerphiu(f) / zv_prod(gel(H, 2));
4021 : }
4022 : }
4023 420 : *pH = H; *pdegF = degF; return f;
4024 : }
4025 :
4026 : GEN
4027 210 : subcyclopclgp(GEN FH, GEN p, long flag)
4028 : {
4029 210 : pari_sp av = avma;
4030 : GEN H;
4031 210 : long degF, f = subcyclo_init("subcyclopclgp", FH, °F, &H, 0);
4032 203 : if (typ(p) == t_VEC)
4033 : {
4034 28 : long i, l = lg(p);
4035 77 : for (i = 1; i < l; i++) checkp("subcyclopclgp", degF, gel(p, i));
4036 14 : if (f == 1) { set_avma(av); return const_vec(l-1, nullvec()); }
4037 : }
4038 : else
4039 : {
4040 175 : checkp("subcyclopclgp", degF, p);
4041 154 : if (f == 1) { set_avma(av); return nullvec(); }
4042 : }
4043 168 : if (flag >= USE_BASIS) pari_err_FLAG("subcyclopclgp");
4044 161 : return gerepilecopy(av, pclgp(p, f, H, degF, flag));
4045 : }
4046 :
4047 : static GEN
4048 98 : subcycloiwasawa_i(GEN FH, GEN P, long n)
4049 : {
4050 : long B, p, f, degF;
4051 : GEN H;
4052 98 : const char *fun = "subcycloiwasawa";
4053 :
4054 98 : if (typ(P) != t_INT) pari_err_TYPE(fun, P);
4055 98 : if (n < 0) pari_err_DOMAIN(fun, "n", "<", gen_0, stoi(n));
4056 98 : B = 1L << (BITS_IN_LONG/4);
4057 98 : if (is_bigint(P) || cmpiu(P, B) > 0)
4058 0 : pari_err_IMPL(stack_sprintf("prime p > %ld in %s", B, fun));
4059 98 : p = itos(P);
4060 98 : if (p <= 1 || !uisprime(p)) pari_err_PRIME(fun, P);
4061 98 : f = subcyclo_init(fun, FH, °F, &H, 1);
4062 98 : if (degF == 1) return NULL;
4063 98 : if (degF == 2)
4064 : {
4065 49 : long m = ((f & 3) == 0)? f / 4: f;
4066 49 : if (H && !srh_1(H)) m = -m;
4067 49 : if (!n) return quadlambda(p, m);
4068 28 : return m < 0? imagquadstkpol(p, m, n): realquadstkpol(p, m, n);
4069 : }
4070 49 : if (p == 2) pari_err_DOMAIN(fun, "p", "=", gen_2, gen_2);
4071 49 : if (srh_1(H)) return NULL;
4072 49 : if (degF % p == 0) errpdiv("abeliwasawa", P, degF);
4073 49 : return abeliwasawa(p, f, H, degF, n);
4074 : }
4075 : GEN
4076 98 : subcycloiwasawa(GEN FH, GEN P, long n)
4077 : {
4078 98 : pari_sp av = avma;
4079 98 : GEN z = subcycloiwasawa_i(FH, P, n);
4080 98 : if (!z) { set_avma(av); return n? nullvec(): mkvec(gen_0); }
4081 98 : return gerepilecopy(av, z);
4082 : }
4083 :
4084 : GEN
4085 119 : subcyclohminus(GEN FH, GEN P)
4086 : {
4087 119 : const char *fun = "subcyclohminus";
4088 119 : pari_sp av = avma;
4089 : GEN H;
4090 119 : long degF, f = subcyclo_init(fun, FH, °F, &H, 0);
4091 119 : if (P)
4092 : {
4093 98 : if (typ(P) != t_INT) pari_err_TYPE(fun, P);
4094 98 : if (isintzero(P)) P = NULL; else checkp(fun, 0, P);
4095 : }
4096 119 : if (degF == 1 || srh_1(H) == 1) return gen_1;
4097 119 : return gerepilecopy(av, rel_class_num(f, H, degF, P));
4098 : }
|
