Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 : #include "pari.h"
15 : #include "paripriv.h"
16 :
17 : #define DEBUGLEVEL DEBUGLEVEL_alg
18 :
19 : #define dbg_printf(lvl) if (DEBUGLEVEL >= (lvl) + 3) err_printf
20 :
21 : /********************************************************************/
22 : /** **/
23 : /** ASSOCIATIVE ALGEBRAS, CENTRAL SIMPLE ALGEBRAS **/
24 : /** contributed by Aurel Page (2014) **/
25 : /** **/
26 : /********************************************************************/
27 : static GEN alg_subalg(GEN al, GEN basis);
28 : static GEN alg_maximal_primes(GEN al, GEN P);
29 : static GEN algnatmultable(GEN al, long D);
30 : static GEN _tablemul_ej(GEN mt, GEN x, long j);
31 : static GEN _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p);
32 : static GEN _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p);
33 : static ulong algtracei(GEN mt, ulong p, ulong expo, ulong modu);
34 : static GEN alg_pmaximal(GEN al, GEN p);
35 : static GEN alg_maximal(GEN al);
36 : static GEN algtracematrix(GEN al);
37 : static GEN algtableinit_i(GEN mt0, GEN p);
38 : static GEN algbasisrightmultable(GEN al, GEN x);
39 : static GEN algabstrace(GEN al, GEN x);
40 : static GEN algbasismul(GEN al, GEN x, GEN y);
41 : static GEN algbasismultable(GEN al, GEN x);
42 : static GEN algbasismultable_Flm(GEN mt, GEN x, ulong m);
43 :
44 : static void H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d);
45 : static GEN H_add(GEN x, GEN y);
46 : static GEN H_charpoly(GEN x, long v, long abs);
47 : static GEN H_divl_i(GEN x, GEN y);
48 : static GEN H_inv(GEN x);
49 : static GEN H_mul(GEN x, GEN y);
50 : static GEN H_neg(GEN x);
51 : static GEN H_norm(GEN x, long abs);
52 : static GEN H_random(GEN b);
53 : static GEN H_sqr(GEN x);
54 : static GEN H_tomatrix(GEN x, long abs);
55 : static GEN H_trace(GEN x, long abs);
56 : static GEN mk_C();
57 :
58 : static int
59 911713 : checkalg_i(GEN al)
60 : {
61 : GEN mt, rnf;
62 : long t;
63 911713 : if (typ(al) != t_VEC || lg(al) != 12) return 0;
64 911496 : mt = alg_get_multable(al);
65 911496 : if (typ(mt) != t_VEC || lg(mt) == 1 || typ(gel(mt,1)) != t_MAT) return 0;
66 911475 : rnf = alg_get_splittingfield(al);
67 911475 : if (isintzero(rnf) || !gequal0(alg_get_char(al)))
68 446387 : return 1;
69 465088 : if (typ(gel(al,2)) != t_VEC || lg(gel(al,2)) == 1) return 0;
70 : /* not checkrnf_i: beware placeholder from alg_csa_table */
71 465081 : t = typ(rnf);
72 465081 : return t==t_COMPLEX || t==t_REAL || (t==t_VEC && lg(rnf)==13);
73 : }
74 : void
75 1041066 : checkalg(GEN al)
76 : {
77 1041066 : if (al && !checkalg_i(al))
78 112 : pari_err_TYPE("checkalg [please apply alginit()]",al);
79 1040954 : }
80 :
81 : static int
82 180992 : checklat_i(GEN al, GEN lat)
83 : {
84 : long N,i,j;
85 : GEN m,t,c;
86 180992 : if (typ(lat)!=t_VEC || lg(lat) != 3) return 0;
87 180992 : t = gel(lat,2);
88 180992 : if (typ(t) != t_INT && typ(t) != t_FRAC) return 0;
89 180992 : if (gsigne(t)<=0) return 0;
90 180992 : m = gel(lat,1);
91 180992 : if (typ(m) != t_MAT) return 0;
92 180992 : N = alg_get_absdim(al);
93 180992 : if (lg(m)-1 != N || lg(gel(m,1))-1 != N) return 0;
94 1628886 : for (i=1; i<=N; i++)
95 13031067 : for (j=1; j<=N; j++) {
96 11583173 : c = gcoeff(m,i,j);
97 11583173 : if (typ(c) != t_INT) return 0;
98 11583173 : if (j<i && signe(gcoeff(m,i,j))) return 0;
99 11583173 : if (i==j && !signe(gcoeff(m,i,j))) return 0;
100 : }
101 180985 : return 1;
102 : }
103 180992 : void checklat(GEN al, GEN lat)
104 180992 : { if (!checklat_i(al,lat)) pari_err_TYPE("checklat [please apply alglathnf()]", lat); }
105 :
106 : /** ACCESSORS **/
107 : long
108 5948557 : alg_type(GEN al)
109 : {
110 : long t;
111 5948557 : if (!al) return al_REAL;
112 5818406 : t = typ(alg_get_splittingfield(al));
113 5818406 : if (t==t_REAL || t==t_COMPLEX) return al_REAL;
114 5814990 : if (isintzero(alg_get_splittingfield(al)) || !gequal0(alg_get_char(al))) return al_TABLE;
115 3952388 : switch(typ(gmael(al,2,1))) {
116 933436 : case t_MAT: return al_CSA;
117 3018917 : case t_INT:
118 : case t_FRAC:
119 : case t_POL:
120 3018917 : case t_POLMOD: return al_CYCLIC;
121 35 : default: return al_NULL;
122 : }
123 : return -1; /*LCOV_EXCL_LINE*/
124 : }
125 : long
126 224 : algtype(GEN al)
127 224 : { return checkalg_i(al)? alg_type(al): al_NULL; }
128 :
129 : /* absdim == dim for al_TABLE. */
130 : static long
131 238 : algreal_dim(GEN al)
132 : {
133 238 : switch(lg(alg_get_multable(al))) {
134 154 : case 2: case 3: return 1;
135 77 : case 5: return 4;
136 7 : default: pari_err_TYPE("algreal_dim", al);
137 : }
138 : return -1; /*LCOV_EXCL_LINE*/
139 : }
140 : long
141 226119 : alg_get_dim(GEN al)
142 : {
143 : long d;
144 226119 : if (!al) return 4;
145 226119 : switch(alg_type(al)) {
146 11660 : case al_TABLE: return lg(alg_get_multable(al))-1;
147 214347 : case al_CSA: return lg(alg_get_relmultable(al))-1;
148 77 : case al_CYCLIC: d = alg_get_degree(al); return d*d;
149 28 : case al_REAL: return algreal_dim(al);
150 7 : default: pari_err_TYPE("alg_get_dim", al);
151 : }
152 : return -1; /*LCOV_EXCL_LINE*/
153 : }
154 :
155 : long
156 1703827 : alg_get_absdim(GEN al)
157 : {
158 1703827 : if (!al) return 4;
159 1657109 : switch(alg_type(al)) {
160 776031 : case al_TABLE: case al_REAL: return lg(alg_get_multable(al))-1;
161 113351 : case al_CSA: return alg_get_dim(al)*nf_get_degree(alg_get_center(al));
162 767720 : case al_CYCLIC:
163 767720 : return rnf_get_absdegree(alg_get_splittingfield(al))*alg_get_degree(al);
164 7 : default: pari_err_TYPE("alg_get_absdim", al);
165 : }
166 : return -1;/*LCOV_EXCL_LINE*/
167 : }
168 :
169 : long
170 2450 : algdim(GEN al, long abs)
171 : {
172 2450 : checkalg(al);
173 2429 : if (abs) return alg_get_absdim(al);
174 2198 : return alg_get_dim(al);
175 : }
176 :
177 : /* only cyclic */
178 : GEN
179 14308 : alg_get_auts(GEN al)
180 : {
181 14308 : long ta = alg_type(al);
182 14308 : if (ta != al_CYCLIC && ta != al_REAL)
183 0 : pari_err_TYPE("alg_get_auts [noncyclic algebra]", al);
184 14308 : return gel(al,2);
185 : }
186 : GEN
187 112 : alg_get_aut(GEN al)
188 : {
189 112 : long ta = alg_type(al);
190 112 : if (ta != al_CYCLIC && ta != al_REAL)
191 7 : pari_err_TYPE("alg_get_aut [noncyclic algebra]", al);
192 105 : return gel(alg_get_auts(al),1);
193 : }
194 : GEN
195 42 : algaut(GEN al) { checkalg(al); return alg_get_aut(al); }
196 : GEN
197 14329 : alg_get_b(GEN al)
198 : {
199 14329 : long ta = alg_type(al);
200 14329 : if (ta != al_CYCLIC && ta != al_REAL)
201 7 : pari_err_TYPE("alg_get_b [noncyclic algebra]", al);
202 14322 : return gel(al,3);
203 : }
204 : GEN
205 56 : algb(GEN al) { checkalg(al); return alg_get_b(al); }
206 :
207 : /* only CSA */
208 : GEN
209 216475 : alg_get_relmultable(GEN al)
210 : {
211 216475 : if (alg_type(al) != al_CSA)
212 14 : pari_err_TYPE("alg_get_relmultable [algebra not given via mult. table]", al);
213 216461 : return gel(al,2);
214 : }
215 : GEN
216 49 : algrelmultable(GEN al) { checkalg(al); return alg_get_relmultable(al); }
217 : GEN
218 56 : alg_get_splittingdata(GEN al)
219 : {
220 56 : if (alg_type(al) != al_CSA)
221 14 : pari_err_TYPE("alg_get_splittingdata [algebra not given via mult. table]",al);
222 42 : return gel(al,3);
223 : }
224 : GEN
225 56 : algsplittingdata(GEN al) { checkalg(al); return alg_get_splittingdata(al); }
226 : GEN
227 4102 : alg_get_splittingbasis(GEN al)
228 : {
229 4102 : if (alg_type(al) != al_CSA)
230 0 : pari_err_TYPE("alg_get_splittingbasis [algebra not given via mult. table]",al);
231 4102 : return gmael(al,3,2);
232 : }
233 : GEN
234 4102 : alg_get_splittingbasisinv(GEN al)
235 : {
236 4102 : if (alg_type(al) != al_CSA)
237 0 : pari_err_TYPE("alg_get_splittingbasisinv [algebra not given via mult. table]",al);
238 4102 : return gmael(al,3,3);
239 : }
240 :
241 : /* only cyclic and CSA */
242 : GEN
243 14996078 : alg_get_splittingfield(GEN al) { return gel(al,1); }
244 : GEN
245 119 : algsplittingfield(GEN al)
246 : {
247 : long ta;
248 119 : checkalg(al);
249 119 : ta = alg_type(al);
250 119 : if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
251 7 : pari_err_TYPE("alg_get_splittingfield [use alginit]",al);
252 112 : return alg_get_splittingfield(al);
253 : }
254 : long
255 1217074 : alg_get_degree(GEN al)
256 : {
257 : long ta;
258 1217074 : ta = alg_type(al);
259 1217074 : if (ta == al_REAL) return algreal_dim(al)==1? 1 : 2;
260 1216990 : if (ta != al_CYCLIC && ta != al_CSA)
261 21 : pari_err_TYPE("alg_get_degree [use alginit]",al);
262 1216969 : return rnf_get_degree(alg_get_splittingfield(al));
263 : }
264 : long
265 322 : algdegree(GEN al)
266 : {
267 322 : checkalg(al);
268 315 : return alg_get_degree(al);
269 : }
270 :
271 : GEN
272 302961 : alg_get_center(GEN al)
273 : {
274 : long ta;
275 302961 : ta = alg_type(al);
276 302961 : if (ta == al_REAL)
277 : {
278 21 : if (algreal_dim(al) != 4) return alg_get_splittingfield(al);
279 7 : return stor(1,3);
280 : }
281 302940 : if (ta != al_CSA && ta != al_CYCLIC)
282 7 : pari_err_TYPE("alg_get_center [use alginit]",al);
283 302933 : return rnf_get_nf(alg_get_splittingfield(al));
284 : }
285 : GEN
286 70 : alg_get_splitpol(GEN al)
287 : {
288 70 : long ta = alg_type(al);
289 70 : if (ta != al_CYCLIC && ta != al_CSA)
290 0 : pari_err_TYPE("alg_get_splitpol [use alginit]",al);
291 70 : return rnf_get_pol(alg_get_splittingfield(al));
292 : }
293 : GEN
294 72038 : alg_get_abssplitting(GEN al)
295 : {
296 72038 : long ta = alg_type(al), prec;
297 72038 : if (ta != al_CYCLIC && ta != al_CSA)
298 0 : pari_err_TYPE("alg_get_abssplitting [use alginit]",al);
299 72038 : prec = nf_get_prec(alg_get_center(al));
300 72038 : return rnf_build_nfabs(alg_get_splittingfield(al), prec);
301 : }
302 : GEN
303 1204 : alg_get_hasse_i(GEN al)
304 : {
305 1204 : long ta = alg_type(al);
306 1204 : if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
307 7 : pari_err_TYPE("alg_get_hasse_i [use alginit]",al);
308 1197 : if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
309 1190 : return gel(al,4);
310 : }
311 : GEN
312 231 : alghassei(GEN al) { checkalg(al); return alg_get_hasse_i(al); }
313 : GEN
314 1988 : alg_get_hasse_f(GEN al)
315 : {
316 1988 : long ta = alg_type(al);
317 : GEN hf;
318 1988 : if (ta != al_CYCLIC && ta != al_CSA)
319 7 : pari_err_TYPE("alg_get_hasse_f [use alginit]",al);
320 1981 : if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
321 1974 : hf = gel(al,5);
322 1974 : if (typ(hf) == t_INT) /* could be computed on the fly */
323 28 : pari_err(e_MISC, "Hasse invariants were not computed for this algebra");
324 1946 : return hf;
325 : }
326 : GEN
327 336 : alghassef(GEN al) { checkalg(al); return alg_get_hasse_f(al); }
328 :
329 : /* all types */
330 : GEN
331 2744 : alg_get_basis(GEN al) { return gel(al,7); }
332 : GEN
333 91 : algbasis(GEN al) { checkalg(al); return alg_get_basis(al); }
334 : GEN
335 61881 : alg_get_invbasis(GEN al) { return gel(al,8); }
336 : GEN
337 63 : alginvbasis(GEN al) { checkalg(al); return alg_get_invbasis(al); }
338 : GEN
339 2536343 : alg_get_multable(GEN al) { return gel(al,9); }
340 : GEN
341 245 : algmultable(GEN al) { checkalg(al); return alg_get_multable(al); }
342 : GEN
343 6139647 : alg_get_char(GEN al) { if (!al) return gen_0; return gel(al,10); }
344 : GEN
345 112 : algchar(GEN al) { checkalg(al); return alg_get_char(al); }
346 : GEN
347 248527 : alg_get_tracebasis(GEN al) { return gel(al,11); }
348 :
349 : /* lattices */
350 : GEN
351 244314 : alglat_get_primbasis(GEN lat) { return gel(lat,1); }
352 : GEN
353 289905 : alglat_get_scalar(GEN lat) { return gel(lat,2); }
354 :
355 : /** ADDITIONAL **/
356 :
357 : /* is N=smooth*prime? */
358 14682 : static int Z_easyfactor(GEN N, ulong lim)
359 : {
360 : GEN fa;
361 14682 : if (lgefint(N) <= 3) return 1;
362 13629 : fa = absZ_factor_limit(N, lim);
363 13629 : return BPSW_psp(veclast(gel(fa,1)));
364 : }
365 :
366 : /* no garbage collection */
367 : static GEN
368 1106 : backtrackfacto(GEN y0, long n, GEN red, GEN pl, GEN nf, GEN data, int (*test)(GEN,GEN), GEN* fa, GEN N, GEN I)
369 : {
370 : long b, i;
371 1106 : ulong lim = 1UL << 17;
372 1106 : long *v = new_chunk(n+1);
373 1106 : pari_sp av = avma;
374 1106 : for (b = 0;; b += (2*b)/(3*n) + 1)
375 316 : {
376 : GEN ny, y1, y2;
377 1422 : set_avma(av);
378 4242 : for (i = 1; i <= n; i++) v[i] = -b;
379 1422 : v[n]--;
380 : for(;;)
381 : {
382 15035 : i = n;
383 15629 : while (i > 0)
384 15313 : { if (v[i] == b) v[i--] = -b; else { v[i]++; break; } }
385 15035 : if (i==0) break;
386 :
387 14719 : y1 = y0;
388 31835 : for (i = 1; i <= n; i++) y1 = nfadd(nf, y1, ZC_z_mul(gel(red,i), v[i]));
389 14719 : if (!nfchecksigns(nf, y1, pl)) continue;
390 :
391 14682 : ny = absi_shallow(nfnorm(nf, y1));
392 14682 : if (!signe(ny)) continue;
393 14682 : ny = diviiexact(ny, gcdii(ny, N));
394 14682 : if (!Z_easyfactor(ny, lim)) continue;
395 :
396 1661 : y2 = idealdivexact(nf, y1, idealadd(nf,y1,I));
397 1661 : *fa = idealfactor(nf, y2);
398 1661 : if (!data || test(data,*fa)) return y1;
399 : }
400 : }
401 : }
402 :
403 : /* if data == NULL, the test is skipped */
404 : /* in the test, the factorization does not contain the known factors */
405 : static GEN
406 1106 : factoredextchinesetest(GEN nf, GEN x, GEN y, GEN pl, GEN* fa, GEN data, int (*test)(GEN,GEN))
407 : {
408 1106 : pari_sp av = avma;
409 : long n,i;
410 1106 : GEN x1, y0, y1, red, N, I, P = gel(x,1), E = gel(x,2);
411 1106 : n = nf_get_degree(nf);
412 1106 : x = idealchineseinit(nf, mkvec2(x,pl));
413 1106 : x1 = gel(x,1);
414 1106 : red = lg(x1) == 1? matid(n): gmael(x1,1,1);
415 1106 : y0 = idealchinese(nf, x, y);
416 :
417 1106 : E = shallowcopy(E);
418 1106 : if (!gequal0(y0))
419 5839 : for (i=1; i<lg(E); i++)
420 : {
421 4733 : long v = nfval(nf,y0,gel(P,i));
422 4733 : if (cmpsi(v, gel(E,i)) < 0) gel(E,i) = stoi(v);
423 : }
424 : /* N and I : known factors */
425 1106 : I = factorbackprime(nf, P, E);
426 1106 : N = idealnorm(nf,I);
427 :
428 1106 : y1 = backtrackfacto(y0, n, red, pl, nf, data, test, fa, N, I);
429 :
430 : /* restore known factors */
431 5839 : for (i=1; i<lg(E); i++) gel(E,i) = stoi(nfval(nf,y1,gel(P,i)));
432 1106 : *fa = famat_reduce(famat_mul_shallow(*fa, mkmat2(P, E)));
433 1106 : return gc_all(av, 2, &y1, fa);
434 : }
435 :
436 : static GEN
437 833 : factoredextchinese(GEN nf, GEN x, GEN y, GEN pl, GEN* fa)
438 833 : { return factoredextchinesetest(nf,x,y,pl,fa,NULL,NULL); }
439 :
440 : /** OPERATIONS ON ASSOCIATIVE ALGEBRAS algebras.c **/
441 :
442 : /*
443 : Convention:
444 : (K/F,sigma,b) = sum_{i=0..n-1} u^i*K
445 : t*u = u*sigma(t)
446 :
447 : Natural basis:
448 : 1<=i<=d*n^2
449 : b_i = u^((i-1)/(dn))*ZKabs.((i-1)%(dn)+1)
450 :
451 : Integral basis:
452 : Basis of some order.
453 :
454 : al:
455 : 1- rnf of the cyclic splitting field of degree n over the center nf of degree d
456 : 2- VEC of aut^i 1<=i<=n if n>1, or i=0 if n=1
457 : 3- b in nf
458 : 4- infinite hasse invariants (mod n) : VECSMALL of size r1, values only 0 or n/2 (if integral)
459 : 5- finite hasse invariants (mod n) : VEC[VEC of primes, VECSMALL of hasse inv mod n]
460 : 6- nf of the splitting field (absolute)
461 : 7* dn^2*dn^2 matrix expressing the integral basis in terms of the natural basis
462 : 8* dn^2*dn^2 matrix expressing the natural basis in terms of the integral basis
463 : 9* VEC of dn^2 matrices giving the dn^2*dn^2 left multiplication tables of the integral basis
464 : 10* characteristic of the base field (used only for algebras given by a multiplication table)
465 : 11* trace of basis elements
466 :
467 : If al is given by a multiplication table (al_TABLE), only the * fields are present.
468 : */
469 :
470 : /* assumes same center and same variable */
471 : /* currently only works for coprime degrees */
472 : GEN
473 84 : algtensor(GEN al1, GEN al2, long flag) {
474 84 : pari_sp av = avma;
475 : long v, k, d1, d2;
476 : GEN nf, P1, P2, aut1, aut2, b1, b2, C, rnf, aut, b, x1, x2, al, rnfpol;
477 :
478 84 : checkalg(al1);
479 70 : checkalg(al2);
480 63 : if (alg_type(al1) != al_CYCLIC || alg_type(al2) != al_CYCLIC)
481 21 : pari_err_IMPL("tensor of noncyclic algebras"); /* TODO: do it. */
482 :
483 42 : nf = alg_get_center(al1);
484 42 : if (!gequal(alg_get_center(al2),nf))
485 7 : pari_err_OP("tensor product [not the same center]", al1, al2);
486 :
487 35 : P1=alg_get_splitpol(al1); aut1=alg_get_aut(al1); b1=alg_get_b(al1);
488 35 : P2=alg_get_splitpol(al2); aut2=alg_get_aut(al2); b2=alg_get_b(al2);
489 35 : v=varn(P1);
490 :
491 35 : d1=alg_get_degree(al1);
492 35 : d2=alg_get_degree(al2);
493 35 : if (ugcd(d1,d2) != 1)
494 7 : pari_err_IMPL("tensor of cyclic algebras of noncoprime degrees"); /* TODO */
495 :
496 28 : if (d1==1) return gcopy(al2);
497 21 : if (d2==1) return gcopy(al1);
498 :
499 14 : C = nfcompositum(nf, P1, P2, 3);
500 14 : rnfpol = gel(C,1);
501 14 : if (!(flag & al_FACTOR)) rnfpol = mkvec2(rnfpol, stoi(1<<20));
502 14 : rnf = rnfinit(nf, rnfpol);
503 : /* TODO use integral basis of P1 and P2 to get that of C */
504 14 : x1 = gel(C,2);
505 14 : x2 = gel(C,3);
506 14 : k = itos(gel(C,4));
507 14 : aut = gadd(gsubst(aut2,v,x2),gmulsg(k,gsubst(aut1,v,x1)));
508 14 : b = nfmul(nf,nfpow_u(nf,b1,d2),nfpow_u(nf,b2,d1));
509 14 : al = alg_cyclic(rnf, aut, b, flag);
510 14 : return gerepilecopy(av,al);
511 : }
512 :
513 : /* M an n x d Flm of rank d, n >= d. Initialize Mx = y solver */
514 : static GEN
515 4454 : Flm_invimage_init(GEN M, ulong p)
516 : {
517 4454 : GEN v = Flm_indexrank(M, p), perm = gel(v,1);
518 4454 : GEN MM = rowpermute(M, perm); /* square invertible */
519 4454 : return mkvec2(Flm_inv(MM,p), perm);
520 : }
521 : /* assume Mx = y has a solution, v = Flm_invimage_init(M,p); return x */
522 : static GEN
523 245283 : Flm_invimage_pre(GEN v, GEN y, ulong p)
524 : {
525 245283 : GEN inv = gel(v,1), perm = gel(v,2);
526 245283 : return Flm_Flc_mul(inv, vecsmallpermute(y, perm), p);
527 : }
528 :
529 : GEN
530 6349 : algradical(GEN al)
531 : {
532 6349 : pari_sp av = avma;
533 : GEN I, x, traces, K, MT, P, mt;
534 : long l,i,ni, n;
535 : ulong modu, expo, p;
536 6349 : checkalg(al);
537 6349 : if (alg_type(al) != al_TABLE) return gen_0;
538 6258 : P = alg_get_char(al);
539 6258 : mt = alg_get_multable(al);
540 6258 : n = alg_get_absdim(al);
541 6258 : dbg_printf(1)("algradical: char=%Ps, dim=%d\n", P, n);
542 6258 : traces = algtracematrix(al);
543 6258 : if (!signe(P))
544 : {
545 525 : dbg_printf(2)(" char 0, computing kernel...\n");
546 525 : K = ker(traces);
547 525 : dbg_printf(2)(" ...done.\n");
548 525 : ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
549 70 : return gerepileupto(av, K);
550 : }
551 5733 : dbg_printf(2)(" char>0, computing kernel...\n");
552 5733 : K = FpM_ker(traces, P);
553 5733 : dbg_printf(2)(" ...done.\n");
554 5733 : ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
555 3768 : if (abscmpiu(P,n)>0) return gerepileupto(av, K);
556 :
557 : /* tough case, p <= n. Ronyai's algorithm */
558 2418 : p = P[2]; l = 1;
559 2418 : expo = p; modu = p*p;
560 2418 : dbg_printf(2)(" char>0, hard case.\n");
561 4895 : while (modu<=(ulong)n) { l++; modu *= p; }
562 2418 : MT = ZMV_to_FlmV(mt, modu);
563 2418 : I = ZM_to_Flm(K,p); /* I_0 */
564 6536 : for (i=1; i<=l; i++) {/*compute I_i, expo = p^i, modu = p^(l+1) > n*/
565 : long j, lig,col;
566 4454 : GEN v = cgetg(ni+1, t_VECSMALL);
567 4454 : GEN invI = Flm_invimage_init(I, p);
568 4454 : dbg_printf(2)(" computing I_%d:\n", i);
569 4454 : traces = cgetg(ni+1,t_MAT);
570 29356 : for (j = 1; j <= ni; j++)
571 : {
572 24902 : GEN M = algbasismultable_Flm(MT, gel(I,j), modu);
573 24902 : uel(v,j) = algtracei(M, p,expo,modu);
574 : }
575 29356 : for (col=1; col<=ni; col++)
576 : {
577 24902 : GEN t = cgetg(n+1,t_VECSMALL); gel(traces,col) = t;
578 24902 : x = gel(I, col); /*col-th basis vector of I_{i-1}*/
579 270185 : for (lig=1; lig<=n; lig++)
580 : {
581 245283 : GEN y = _tablemul_ej_Fl(MT,x,lig,p);
582 245283 : GEN z = Flm_invimage_pre(invI, y, p);
583 245283 : uel(t,lig) = Flv_dotproduct(v, z, p);
584 : }
585 : }
586 4454 : dbg_printf(2)(" computing kernel...\n");
587 4454 : K = Flm_ker(traces, p);
588 4454 : dbg_printf(2)(" ...done.\n");
589 4454 : ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
590 4118 : I = Flm_mul(I,K,p);
591 4118 : expo *= p;
592 : }
593 2082 : return Flm_to_ZM(I);
594 : }
595 :
596 : /* compute the multiplication table of the element x, where mt is a
597 : * multiplication table in an arbitrary ring */
598 : static GEN
599 476 : Rgmultable(GEN mt, GEN x)
600 : {
601 476 : long i, l = lg(x);
602 476 : GEN z = NULL;
603 6188 : for (i = 1; i < l; i++)
604 : {
605 5712 : GEN c = gel(x,i);
606 5712 : if (!gequal0(c))
607 : {
608 714 : GEN M = RgM_Rg_mul(gel(mt,i),c);
609 714 : z = z? RgM_add(z, M): M;
610 : }
611 : }
612 476 : return z;
613 : }
614 :
615 : static GEN
616 56 : change_Rgmultable(GEN mt, GEN P, GEN Pi)
617 : {
618 : GEN mt2;
619 56 : long lmt = lg(mt), i;
620 56 : mt2 = cgetg(lmt,t_VEC);
621 532 : for (i=1;i<lmt;i++) {
622 476 : GEN mti = Rgmultable(mt,gel(P,i));
623 476 : gel(mt2,i) = RgM_mul(Pi, RgM_mul(mti,P));
624 : }
625 56 : return mt2;
626 : }
627 :
628 : /* S: lift (basis of quotient) ; Si: proj */
629 : static GEN
630 21673 : alg_quotient0(GEN al, GEN S, GEN Si, long nq, GEN p, long maps)
631 : {
632 21673 : GEN mt = cgetg(nq+1,t_VEC), P, Pi, d;
633 : long i;
634 21673 : dbg_printf(3)(" alg_quotient0: char=%Ps, dim=%d, dim I=%d\n", p, alg_get_absdim(al), lg(S)-1);
635 85799 : for (i=1; i<=nq; i++) {
636 64126 : GEN mti = algbasismultable(al,gel(S,i));
637 64126 : if (signe(p)) gel(mt,i) = FpM_mul(Si, FpM_mul(mti,S,p), p);
638 6076 : else gel(mt,i) = RgM_mul(Si, RgM_mul(mti,S));
639 : }
640 21673 : if (!signe(p) && !isint1(Q_denom(mt))) {
641 42 : dbg_printf(3)(" bad case: denominator=%Ps\n", Q_denom(mt));
642 42 : P = Q_remove_denom(Si,&d);
643 42 : P = ZM_hnf(P);
644 42 : P = RgM_Rg_div(P,d); /* P: new basis (Z-basis of image of order in al) */
645 42 : Pi = RgM_inv(P);
646 42 : mt = change_Rgmultable(mt,P,Pi);
647 42 : Si = RgM_mul(Pi,Si);
648 42 : S = RgM_mul(S,P);
649 : }
650 21673 : al = algtableinit_i(mt,p);
651 21673 : if (maps) al = mkvec3(al,Si,S); /* algebra, proj, lift */
652 21673 : return al;
653 : }
654 :
655 : /* quotient of an algebra by a nontrivial two-sided ideal */
656 : GEN
657 3523 : alg_quotient(GEN al, GEN I, long maps)
658 : {
659 3523 : pari_sp av = avma;
660 : GEN p, IS, ISi, S, Si;
661 : long n, ni;
662 :
663 3523 : checkalg(al);
664 3523 : if (alg_type(al) != al_TABLE) pari_err_TYPE("alg_quotient [not a table algebra]", al);
665 3516 : p = alg_get_char(al);
666 3516 : n = alg_get_absdim(al);
667 3516 : ni = lg(I)-1;
668 :
669 : /* force first vector of complement to be the identity */
670 3516 : IS = shallowconcat(I, gcoeff(alg_get_multable(al),1,1));
671 3516 : if (signe(p)) {
672 3488 : IS = FpM_suppl(IS,p);
673 3488 : ISi = FpM_inv(IS,p);
674 : }
675 : else {
676 28 : IS = suppl(IS);
677 28 : ISi = RgM_inv(IS);
678 : }
679 3516 : S = vecslice(IS, ni+1, n);
680 3516 : Si = rowslice(ISi, ni+1, n);
681 3516 : return gerepilecopy(av, alg_quotient0(al, S, Si, n-ni, p, maps));
682 : }
683 :
684 : static GEN
685 28397 : image_keep_first(GEN m, GEN p) /* assume first column is nonzero or m==0, no GC */
686 : {
687 : GEN ir, icol, irow, M, c, x;
688 : long i;
689 28397 : if (gequal0(gel(m,1))) return zeromat(nbrows(m),0);
690 :
691 28383 : if (signe(p)) ir = FpM_indexrank(m,p);
692 1708 : else ir = indexrank(m);
693 :
694 28383 : icol = gel(ir,2);
695 28383 : if (icol[1]==1) return extract0(m,icol,NULL);
696 :
697 7 : irow = gel(ir,1);
698 7 : M = extract0(m, irow, icol);
699 7 : c = extract0(gel(m,1), irow, NULL);
700 7 : if (signe(p)) x = FpM_FpC_invimage(M,c,p);
701 0 : else x = inverseimage(M,c); /* TODO modulo a small prime */
702 :
703 7 : for (i=1; i<lg(x); i++)
704 : {
705 7 : if (!gequal0(gel(x,i)))
706 : {
707 7 : icol[i] = 1;
708 7 : vecsmall_sort(icol);
709 7 : return extract0(m,icol,NULL);
710 : }
711 : }
712 :
713 : return NULL; /* LCOV_EXCL_LINE */
714 : }
715 :
716 : /* z[1],...z[nz] central elements such that z[1]A + z[2]A + ... + z[nz]A = A
717 : * is a direct sum. idempotents ==> first basis element is identity */
718 : GEN
719 8738 : alg_centralproj(GEN al, GEN z, long maps)
720 : {
721 8738 : pari_sp av = avma;
722 : GEN S, U, Ui, alq, p;
723 8738 : long i, iu, lz = lg(z), ta;
724 :
725 8738 : checkalg(al);
726 8738 : ta = alg_type(al);
727 8738 : if (ta != al_TABLE) pari_err_TYPE("algcentralproj [not a table algebra]", al);
728 8731 : if (typ(z) != t_VEC) pari_err_TYPE("alcentralproj",z);
729 8724 : p = alg_get_char(al);
730 8724 : dbg_printf(3)(" alg_centralproj: char=%Ps, dim=%d, #z=%d\n", p, alg_get_absdim(al), lz-1);
731 8724 : S = cgetg(lz,t_VEC); /* S[i] = Im(z_i) */
732 26895 : for (i=1; i<lz; i++)
733 : {
734 18171 : GEN mti = algbasismultable(al, gel(z,i));
735 18171 : gel(S,i) = image_keep_first(mti,p);
736 : }
737 8724 : U = shallowconcat1(S); /* U = [Im(z_1)|Im(z_2)|...|Im(z_nz)], n x n */
738 8724 : if (lg(U)-1 < alg_get_absdim(al)) pari_err_TYPE("alcentralproj [z[i]'s not surjective]",z);
739 8717 : if (signe(p)) Ui = FpM_inv(U,p);
740 854 : else Ui = RgM_inv(U);
741 : if (!Ui) pari_err_BUG("alcentralproj"); /*LCOV_EXCL_LINE*/
742 :
743 8717 : alq = cgetg(lz,t_VEC);
744 26874 : for (iu=0,i=1; i<lz; i++)
745 : {
746 18157 : long nq = lg(gel(S,i))-1, ju = iu + nq;
747 18157 : GEN Si = rowslice(Ui, iu+1, ju);
748 18157 : gel(alq, i) = alg_quotient0(al,gel(S,i),Si,nq,p,maps);
749 18157 : iu = ju;
750 : }
751 8717 : return gerepilecopy(av, alq);
752 : }
753 :
754 : /* al is an al_TABLE */
755 : static GEN
756 19777 : algtablecenter(GEN al)
757 : {
758 19777 : pari_sp av = avma;
759 : long n, i, j, k, ic;
760 : GEN C, cij, mt, p;
761 :
762 19777 : n = alg_get_absdim(al);
763 19777 : mt = alg_get_multable(al);
764 19777 : p = alg_get_char(al);
765 19777 : C = cgetg(n+1,t_MAT);
766 94606 : for (j=1; j<=n; j++)
767 : {
768 74829 : gel(C,j) = cgetg(n*n-n+1,t_COL);
769 74829 : ic = 1;
770 604363 : for (i=2; i<=n; i++) {
771 529534 : if (signe(p)) cij = FpC_sub(gmael(mt,i,j),gmael(mt,j,i),p);
772 56294 : else cij = RgC_sub(gmael(mt,i,j),gmael(mt,j,i));
773 7448598 : for (k=1; k<=n; k++, ic++) gcoeff(C,ic,j) = gel(cij, k);
774 : }
775 : }
776 19777 : if (signe(p)) return gerepileupto(av, FpM_ker(C,p));
777 1764 : else return gerepileupto(av, ker(C));
778 : }
779 :
780 : GEN
781 4886 : algcenter(GEN al)
782 : {
783 4886 : checkalg(al);
784 4886 : if (alg_type(al)==al_TABLE) return algtablecenter(al);
785 49 : return alg_get_center(al);
786 : }
787 :
788 : /* Only in positive characteristic. Assumes that al is semisimple. */
789 : GEN
790 4995 : algprimesubalg(GEN al)
791 : {
792 4995 : pari_sp av = avma;
793 : GEN p, Z, F, K;
794 : long nz, i;
795 4995 : checkalg(al);
796 4995 : p = alg_get_char(al);
797 4995 : if (!signe(p)) pari_err_DOMAIN("algprimesubalg","characteristic","=",gen_0,p);
798 :
799 4981 : Z = algtablecenter(al);
800 4981 : nz = lg(Z)-1;
801 4981 : if (nz==1) return Z;
802 :
803 3602 : F = cgetg(nz+1, t_MAT);
804 17105 : for (i=1; i<=nz; i++) {
805 13503 : GEN zi = gel(Z,i);
806 13503 : gel(F,i) = FpC_sub(algpow(al,zi,p),zi,p);
807 : }
808 3602 : K = FpM_ker(F,p);
809 3602 : return gerepileupto(av, FpM_mul(Z,K,p));
810 : }
811 :
812 : static GEN
813 15083 : out_decompose(GEN t, GEN Z, GEN P, GEN p)
814 : {
815 15083 : GEN ali = gel(t,1), projm = gel(t,2), liftm = gel(t,3), pZ;
816 15083 : if (signe(p)) pZ = FpM_image(FpM_mul(projm,Z,p),p);
817 1617 : else pZ = image(RgM_mul(projm,Z));
818 15083 : return mkvec5(ali, projm, liftm, pZ, P);
819 : }
820 : /* fa factorization of charpol(x) */
821 : static GEN
822 7580 : alg_decompose_from_facto(GEN al, GEN x, GEN fa, GEN Z, long mini)
823 : {
824 7580 : long k = lgcols(fa)-1, k2 = mini? 1: k/2;
825 7580 : GEN v1 = rowslice(fa,1,k2);
826 7580 : GEN v2 = rowslice(fa,k2+1,k);
827 7580 : GEN alq, P, Q, p = alg_get_char(al);
828 7580 : dbg_printf(3)(" alg_decompose_from_facto\n");
829 7580 : if (signe(p)) {
830 6754 : P = FpXV_factorback(gel(v1,1), gel(v1,2), p, 0);
831 6754 : Q = FpXV_factorback(gel(v2,1), gel(v2,2), p, 0);
832 6754 : P = FpX_mul(P, FpXQ_inv(P,Q,p), p);
833 : }
834 : else {
835 826 : P = factorback(v1);
836 826 : Q = factorback(v2);
837 826 : P = RgX_mul(P, RgXQ_inv(P,Q));
838 : }
839 7580 : P = algpoleval(al, P, x);
840 7580 : if (signe(p)) Q = FpC_sub(col_ei(lg(P)-1,1), P, p);
841 826 : else Q = gsub(gen_1, P);
842 7580 : if (gequal0(P) || gequal0(Q)) return NULL;
843 7580 : alq = alg_centralproj(al, mkvec2(P,Q), 1);
844 :
845 7580 : P = out_decompose(gel(alq,1), Z, P, p); if (mini) return P;
846 7503 : Q = out_decompose(gel(alq,2), Z, Q, p);
847 7503 : return mkvec2(P,Q);
848 : }
849 :
850 : static GEN
851 12095 : random_pm1(long n)
852 : {
853 12095 : GEN z = cgetg(n+1,t_VECSMALL);
854 : long i;
855 53245 : for (i = 1; i <= n; i++) z[i] = random_bits(5)%3 - 1;
856 12095 : return z;
857 : }
858 :
859 : static GEN alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt);
860 : /* Try to split al using x's charpoly. Return gen_0 if simple, NULL if failure.
861 : * And a splitting otherwise
862 : * If pt_primelt!=NULL, compute a primitive element of the center when simple */
863 : static GEN
864 14168 : try_fact(GEN al, GEN x, GEN zx, GEN Z, GEN Zal, long mini, GEN* pt_primelt)
865 : {
866 14168 : GEN z, dec0, dec1, cp = algcharpoly(Zal,zx,0,1), fa, p = alg_get_char(al);
867 : long nfa, e;
868 14168 : dbg_printf(3)(" try_fact: zx=%Ps\n", zx);
869 14168 : if (signe(p)) fa = FpX_factor(cp,p);
870 1491 : else fa = factor(cp);
871 14168 : dbg_printf(3)(" charpoly=%Ps\n", fa);
872 14168 : nfa = nbrows(fa);
873 14168 : if (nfa == 1) {
874 6588 : if (signe(p)) e = gel(fa,2)[1];
875 665 : else e = itos(gcoeff(fa,1,2));
876 6588 : if (e == 1) {
877 3745 : if (pt_primelt != NULL) *pt_primelt = mkvec2(x, cp);
878 3745 : return gen_0;
879 : }
880 2843 : else return NULL;
881 : }
882 7580 : dec0 = alg_decompose_from_facto(al, x, fa, Z, mini);
883 7580 : if (!dec0) return NULL;
884 7580 : if (!mini) return dec0;
885 77 : dec1 = alg_decompose(gel(dec0,1), gel(dec0,4), 1, pt_primelt);
886 77 : z = gel(dec0,5);
887 77 : if (!isintzero(dec1)) {
888 7 : if (signe(p)) z = FpM_FpC_mul(gel(dec0,3),dec1,p);
889 7 : else z = RgM_RgC_mul(gel(dec0,3),dec1);
890 : }
891 77 : return z;
892 : }
893 : static GEN
894 7 : randcol(long n, GEN b)
895 : {
896 7 : GEN N = addiu(shifti(b,1), 1);
897 : long i;
898 7 : GEN res = cgetg(n+1,t_COL);
899 63 : for (i=1; i<=n; i++)
900 : {
901 56 : pari_sp av = avma;
902 56 : gel(res,i) = gerepileuptoint(av, subii(randomi(N),b));
903 : }
904 7 : return res;
905 : }
906 : /* Return gen_0 if already simple. mini: only returns a central idempotent
907 : * corresponding to one simple factor
908 : * if pt_primelt!=NULL, sets it to a primitive element of the center when simple */
909 : static GEN
910 20717 : alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt)
911 : {
912 : pari_sp av;
913 : GEN Zal, x, zx, rand, dec0, B, p;
914 20717 : long i, nz = lg(Z)-1;
915 :
916 20717 : if (nz == 1) {
917 9392 : if (pt_primelt != 0) *pt_primelt = mkvec2(zerocol(alg_get_dim(al)), pol_x(0));
918 9392 : return gen_0;
919 : }
920 11325 : p = alg_get_char(al);
921 11325 : dbg_printf(2)(" alg_decompose: char=%Ps, dim=%d, dim Z=%d\n", p, alg_get_absdim(al), nz);
922 11325 : Zal = alg_subalg(al,Z);
923 11325 : Z = gel(Zal,2);
924 11325 : Zal = gel(Zal,1);
925 11325 : av = avma;
926 :
927 11325 : rand = random_pm1(nz);
928 11325 : zx = zc_to_ZC(rand);
929 11325 : if (signe(p)) {
930 10191 : zx = FpC_red(zx,p);
931 10191 : x = ZM_zc_mul(Z,rand);
932 10191 : x = FpC_red(x,p);
933 : }
934 1134 : else x = RgM_zc_mul(Z,rand);
935 11325 : dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
936 11325 : if (dec0) return dec0;
937 2773 : set_avma(av);
938 :
939 2843 : for (i=2; i<=nz; i++)
940 : {
941 2836 : dec0 = try_fact(al,gel(Z,i),col_ei(nz,i),Z,Zal,mini,pt_primelt);
942 2836 : if (dec0) return dec0;
943 70 : set_avma(av);
944 : }
945 7 : B = int2n(10);
946 : for (;;)
947 0 : {
948 7 : GEN x = randcol(nz,B), zx = ZM_ZC_mul(Z,x);
949 7 : dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
950 7 : if (dec0) return dec0;
951 0 : set_avma(av);
952 : }
953 : }
954 :
955 : static GEN
956 17119 : alg_decompose_total(GEN al, GEN Z, long maps)
957 : {
958 : GEN dec, sc, p;
959 : long i;
960 :
961 17119 : dec = alg_decompose(al, Z, 0, NULL);
962 17119 : if (isintzero(dec))
963 : {
964 9616 : if (maps) {
965 7208 : long n = alg_get_absdim(al);
966 7208 : al = mkvec3(al, matid(n), matid(n));
967 : }
968 9616 : return mkvec(al);
969 : }
970 7503 : p = alg_get_char(al); if (!signe(p)) p = NULL;
971 7503 : sc = cgetg(lg(dec), t_VEC);
972 22509 : for (i=1; i<lg(sc); i++) {
973 15006 : GEN D = gel(dec,i), a = gel(D,1), Za = gel(D,4);
974 15006 : GEN S = alg_decompose_total(a, Za, maps);
975 15006 : gel(sc,i) = S;
976 15006 : if (maps)
977 : {
978 11170 : GEN projm = gel(D,2), liftm = gel(D,3);
979 11170 : long j, lS = lg(S);
980 30564 : for (j=1; j<lS; j++)
981 : {
982 19394 : GEN Sj = gel(S,j), p2 = gel(Sj,2), l2 = gel(Sj,3);
983 19394 : if (p) p2 = FpM_mul(p2, projm, p);
984 1449 : else p2 = RgM_mul(p2, projm);
985 19394 : if (p) l2 = FpM_mul(liftm, l2, p);
986 1449 : else l2 = RgM_mul(liftm, l2);
987 19394 : gel(Sj,2) = p2;
988 19394 : gel(Sj,3) = l2;
989 : }
990 : }
991 : }
992 7503 : return shallowconcat1(sc);
993 : }
994 :
995 : static GEN
996 11381 : alg_subalg(GEN al, GEN basis)
997 : {
998 11381 : GEN invbasis, mt, p = alg_get_char(al);
999 11381 : long i, j, n = lg(basis)-1;
1000 :
1001 11381 : if (!signe(p)) p = NULL;
1002 11381 : basis = shallowmatconcat(mkvec2(col_ei(n,1), basis));
1003 11381 : if (p)
1004 : {
1005 10226 : basis = image_keep_first(basis,p);
1006 10226 : invbasis = FpM_inv(basis,p);
1007 : }
1008 : else
1009 : { /* FIXME use an integral variant of image_keep_first */
1010 1155 : basis = QM_ImQ_hnf(basis);
1011 1155 : invbasis = RgM_inv(basis);
1012 : }
1013 11381 : mt = cgetg(n+1,t_VEC);
1014 11381 : gel(mt,1) = matid(n);
1015 38469 : for (i = 2; i <= n; i++)
1016 : {
1017 27088 : GEN mtx = cgetg(n+1,t_MAT), x = gel(basis,i);
1018 27088 : gel(mtx,1) = col_ei(n,i);
1019 174930 : for (j = 2; j <= n; j++)
1020 : {
1021 147842 : GEN xy = algmul(al, x, gel(basis,j));
1022 147842 : if (p) gel(mtx,j) = FpM_FpC_mul(invbasis, xy, p);
1023 36218 : else gel(mtx,j) = RgM_RgC_mul(invbasis, xy);
1024 : }
1025 27088 : gel(mt,i) = mtx;
1026 : }
1027 11381 : return mkvec2(algtableinit_i(mt,p), basis);
1028 : }
1029 :
1030 : GEN
1031 70 : algsubalg(GEN al, GEN basis)
1032 : {
1033 70 : pari_sp av = avma;
1034 : GEN p;
1035 70 : checkalg(al);
1036 70 : if (alg_type(al) == al_REAL) pari_err_TYPE("algsubalg [real algebra]", al);
1037 63 : if (typ(basis) != t_MAT) pari_err_TYPE("algsubalg",basis);
1038 56 : p = alg_get_char(al);
1039 56 : if (signe(p)) basis = RgM_to_FpM(basis,p);
1040 56 : return gerepilecopy(av, alg_subalg(al,basis));
1041 : }
1042 :
1043 : static int
1044 12200 : cmp_algebra(GEN x, GEN y)
1045 : {
1046 : long d;
1047 12200 : d = gel(x,1)[1] - gel(y,1)[1]; if (d) return d < 0? -1: 1;
1048 10933 : d = gel(x,1)[2] - gel(y,1)[2]; if (d) return d < 0? -1: 1;
1049 10933 : return cmp_universal(gel(x,2), gel(y,2));
1050 : }
1051 :
1052 : GEN
1053 5100 : algsimpledec_ss(GEN al, long maps)
1054 : {
1055 5100 : pari_sp av = avma;
1056 : GEN Z, p, r, res, perm;
1057 : long i, l, n;
1058 5100 : checkalg(al);
1059 5100 : p = alg_get_char(al);
1060 5100 : dbg_printf(1)("algsimpledec_ss: char=%Ps, dim=%d\n", p, alg_get_absdim(al));
1061 5100 : if (signe(p)) Z = algprimesubalg(al);
1062 273 : else if (alg_type(al)!=al_TABLE) Z = gen_0;
1063 252 : else Z = algtablecenter(al);
1064 :
1065 5100 : if (lg(Z) == 2) {/* dim Z = 1 */
1066 2987 : n = alg_get_absdim(al);
1067 2987 : set_avma(av);
1068 2987 : if (!maps) return mkveccopy(al);
1069 2840 : retmkvec(mkvec3(gcopy(al), matid(n), matid(n)));
1070 : }
1071 2113 : res = alg_decompose_total(al, Z, maps);
1072 2113 : l = lg(res); r = cgetg(l, t_VEC);
1073 11729 : for (i = 1; i < l; i++)
1074 : {
1075 9616 : GEN A = maps? gmael(res,i,1): gel(res,i);
1076 9616 : gel(r,i) = mkvec2(mkvecsmall2(alg_get_dim(A), lg(algtablecenter(A))),
1077 : alg_get_multable(A));
1078 : }
1079 2113 : perm = gen_indexsort(r, (void*)cmp_algebra, &cmp_nodata);
1080 2113 : return gerepilecopy(av, vecpermute(res, perm));
1081 : }
1082 :
1083 : GEN
1084 784 : algsimpledec(GEN al, long maps)
1085 : {
1086 784 : pari_sp av = avma;
1087 : int ss;
1088 784 : GEN rad, dec, res, proj=NULL, lift=NULL;
1089 784 : rad = algradical(al);
1090 784 : ss = gequal0(rad);
1091 784 : if (!ss)
1092 : {
1093 42 : al = alg_quotient(al, rad, maps);
1094 42 : if (maps) {
1095 14 : proj = gel(al,2);
1096 14 : lift = gel(al,3);
1097 14 : al = gel(al,1);
1098 : }
1099 : }
1100 784 : dec = algsimpledec_ss(al, maps);
1101 784 : if (!ss && maps) /* update maps */
1102 : {
1103 14 : GEN p = alg_get_char(al);
1104 : long i;
1105 42 : for (i=1; i<lg(dec); i++)
1106 : {
1107 28 : if (signe(p))
1108 : {
1109 14 : gmael(dec,i,2) = FpM_mul(gmael(dec,i,2), proj, p);
1110 14 : gmael(dec,i,3) = FpM_mul(lift, gmael(dec,i,3), p);
1111 : }
1112 : else
1113 : {
1114 14 : gmael(dec,i,2) = RgM_mul(gmael(dec,i,2), proj);
1115 14 : gmael(dec,i,3) = RgM_mul(lift, gmael(dec,i,3));
1116 : }
1117 : }
1118 : }
1119 784 : res = mkvec2(rad, dec);
1120 784 : return gerepilecopy(av,res);
1121 : }
1122 :
1123 : static GEN alg_idempotent(GEN al, long n, long d);
1124 : static GEN
1125 6482 : try_split(GEN al, GEN x, long n, long d)
1126 : {
1127 6482 : GEN cp, p = alg_get_char(al), fa, e, pol, exp, P, Q, U, u, mx, mte, ire;
1128 6482 : long nfa, i, smalldim = alg_get_absdim(al)+1, dim, smalli = 0;
1129 6482 : cp = algcharpoly(al,x,0,1);
1130 6482 : fa = FpX_factor(cp,p);
1131 6482 : nfa = nbrows(fa);
1132 6482 : if (nfa == 1) return NULL;
1133 3052 : pol = gel(fa,1);
1134 3052 : exp = gel(fa,2);
1135 :
1136 : /* charpoly is always a d-th power */
1137 9254 : for (i=1; i<lg(exp); i++) {
1138 6209 : if (exp[i]%d) pari_err(e_MISC, "the algebra must be simple (try_split 1)");
1139 6202 : exp[i] /= d;
1140 : }
1141 3045 : cp = FpXV_factorback(gel(fa,1), gel(fa,2), p, 0);
1142 :
1143 : /* find smallest Fp-dimension of a characteristic space */
1144 9247 : for (i=1; i<lg(pol); i++) {
1145 6202 : dim = degree(gel(pol,i))*exp[i];
1146 6202 : if (dim < smalldim) {
1147 3115 : smalldim = dim;
1148 3115 : smalli = i;
1149 : }
1150 : }
1151 3045 : i = smalli;
1152 3045 : if (smalldim != n) return NULL;
1153 : /* We could also compute e*al*e and try again with this smaller algebra */
1154 : /* Fq-rank 1 = Fp-rank n idempotent: success */
1155 :
1156 : /* construct idempotent */
1157 3031 : mx = algbasismultable(al,x);
1158 3031 : P = gel(pol,i);
1159 3031 : P = FpX_powu(P, exp[i], p);
1160 3031 : Q = FpX_div(cp, P, p);
1161 3031 : e = algpoleval(al, Q, mkvec2(x,mx));
1162 3031 : U = FpXQ_inv(Q, P, p);
1163 3031 : u = algpoleval(al, U, mkvec2(x,mx));
1164 3031 : e = algbasismul(al, e, u);
1165 3031 : mte = algbasisrightmultable(al,e);
1166 3031 : ire = FpM_indexrank(mte,p);
1167 3031 : if (lg(gel(ire,1))-1 != smalldim*d) pari_err(e_MISC, "the algebra must be simple (try_split 2)");
1168 :
1169 3024 : return mkvec3(e,mte,ire);
1170 : }
1171 :
1172 : /*
1173 : * Given a simple algebra al of dimension d^2 over its center of degree n,
1174 : * find an idempotent e in al with rank n (which is minimal).
1175 : */
1176 : static GEN
1177 3038 : alg_idempotent(GEN al, long n, long d)
1178 : {
1179 3038 : pari_sp av = avma;
1180 3038 : long i, N = alg_get_absdim(al);
1181 3038 : GEN e, p = alg_get_char(al), x;
1182 6377 : for(i=2; i<=N; i++) {
1183 6321 : x = col_ei(N,i);
1184 6321 : e = try_split(al, x, n, d);
1185 6307 : if (e) return e;
1186 3339 : set_avma(av);
1187 : }
1188 : for(;;) {
1189 161 : x = random_FpC(N,p);
1190 161 : e = try_split(al, x, n, d);
1191 161 : if (e) return e;
1192 105 : set_avma(av);
1193 : }
1194 : }
1195 :
1196 : static GEN
1197 3857 : try_descend(GEN M, GEN B, GEN p, long m, long n, long d)
1198 : {
1199 3857 : GEN B2 = cgetg(m+1,t_MAT), b;
1200 3857 : long i, j, k=0;
1201 11011 : for (i=1; i<=d; i++)
1202 : {
1203 7154 : k++;
1204 7154 : b = gel(B,i);
1205 7154 : gel(B2,k) = b;
1206 17248 : for (j=1; j<n; j++)
1207 : {
1208 10094 : k++;
1209 10094 : b = FpM_FpC_mul(M,b,p);
1210 10094 : gel(B2,k) = b;
1211 : }
1212 : }
1213 3857 : if (!signe(FpM_det(B2,p))) return NULL;
1214 3437 : return FpM_inv(B2,p);
1215 : }
1216 :
1217 : /* Given an m*m matrix M with irreducible charpoly over F of degree n,
1218 : * let K = F(M), which is a field, and write m=d*n.
1219 : * Compute the d-dimensional K-vector space structure on V=F^m induced by M.
1220 : * Return [B,C] where:
1221 : * - B is m*d matrix over F giving a K-basis b_1,...,b_d of V
1222 : * - C is d*m matrix over F[x] expressing the canonical F-basis of V on the b_i
1223 : * Currently F = Fp TODO extend this. */
1224 : static GEN
1225 3437 : descend_i(GEN M, long n, GEN p)
1226 : {
1227 : GEN B, C;
1228 : long m,d,i;
1229 : pari_sp av;
1230 3437 : m = lg(M)-1;
1231 3437 : d = m/n;
1232 3437 : B = cgetg(d+1,t_MAT);
1233 3437 : av = avma;
1234 :
1235 : /* try a subset of the canonical basis */
1236 9751 : for (i=1; i<=d; i++)
1237 6314 : gel(B,i) = col_ei(m,n*(i-1)+1);
1238 3437 : C = try_descend(M,B,p,m,n,d);
1239 3437 : if (C) return mkvec2(B,C);
1240 385 : set_avma(av);
1241 :
1242 : /* try smallish elements */
1243 1155 : for (i=1; i<=d; i++)
1244 770 : gel(B,i) = FpC_red(zc_to_ZC(random_pm1(m)),p);
1245 385 : C = try_descend(M,B,p,m,n,d);
1246 385 : if (C) return mkvec2(B,C);
1247 35 : set_avma(av);
1248 :
1249 : /* try random elements */
1250 : for (;;)
1251 : {
1252 105 : for (i=1; i<=d; i++)
1253 70 : gel(B,i) = random_FpC(m,p);
1254 35 : C = try_descend(M,B,p,m,n,d);
1255 35 : if (C) return mkvec2(B,C);
1256 0 : set_avma(av);
1257 : }
1258 : }
1259 : static GEN
1260 15568 : RgC_contract(GEN C, long n, long v) /* n>1 */
1261 : {
1262 : GEN C2, P;
1263 : long m, d, i, j;
1264 15568 : m = lg(C)-1;
1265 15568 : d = m/n;
1266 15568 : C2 = cgetg(d+1,t_COL);
1267 43344 : for (i=1; i<=d; i++)
1268 : {
1269 27776 : P = pol_xn(n-1,v);
1270 105728 : for (j=1; j<=n; j++)
1271 77952 : gel(P,j+1) = gel(C,n*(i-1)+j);
1272 27776 : P = normalizepol(P);
1273 27776 : gel(C2,i) = P;
1274 : }
1275 15568 : return C2;
1276 : }
1277 : static GEN
1278 3437 : RgM_contract(GEN A, long n, long v) /* n>1 */
1279 : {
1280 3437 : GEN A2 = cgetg(lg(A),t_MAT);
1281 : long i;
1282 19005 : for (i=1; i<lg(A2); i++)
1283 15568 : gel(A2,i) = RgC_contract(gel(A,i),n,v);
1284 3437 : return A2;
1285 : }
1286 : static GEN
1287 3437 : descend(GEN M, long n, GEN p, long v)
1288 : {
1289 3437 : GEN res = descend_i(M,n,p);
1290 3437 : gel(res,2) = RgM_contract(gel(res,2),n,v);
1291 3437 : return res;
1292 : }
1293 :
1294 : /* isomorphism of Fp-vector spaces M_d(F_p^n) -> (F_p)^(d^2*n) */
1295 : static GEN
1296 29939 : Fq_mat2col(GEN M, long d, long n)
1297 : {
1298 29939 : long N = d*d*n, i, j, k;
1299 29939 : GEN C = cgetg(N+1, t_COL);
1300 90160 : for (i=1; i<=d; i++)
1301 191632 : for (j=1; j<=d; j++)
1302 400526 : for (k=0; k<n; k++)
1303 269115 : gel(C,n*(d*(i-1)+j-1)+k+1) = polcoef_i(gcoeff(M,i,j),k,-1);
1304 29939 : return C;
1305 : }
1306 :
1307 : static GEN
1308 3752 : alg_finite_csa_split(GEN al, long v)
1309 : {
1310 : GEN Z, e, mte, ire, primelt, b, T, M, proje, lifte, extre, p, B, C, mt, mx, map, mapi, T2, ro;
1311 3752 : long n, d, N = alg_get_absdim(al), i;
1312 3752 : p = alg_get_char(al);
1313 : /* compute the center */
1314 3752 : Z = algcenter(al);
1315 : /* TODO option to give the center as input instead of computing it */
1316 3752 : n = lg(Z)-1;
1317 :
1318 : /* compute a minimal rank idempotent e */
1319 3752 : if (n==N) {
1320 707 : d = 1;
1321 707 : e = col_ei(N,1);
1322 707 : mte = matid(N);
1323 707 : ire = mkvec2(identity_perm(n),identity_perm(n));
1324 : }
1325 : else {
1326 3045 : d = usqrt(N/n);
1327 3045 : if (d*d*n != N) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 1)");
1328 3038 : e = alg_idempotent(al,n,d);
1329 3024 : mte = gel(e,2);
1330 3024 : ire = gel(e,3);
1331 3024 : e = gel(e,1);
1332 : }
1333 :
1334 : /* identify the center */
1335 3731 : if (n==1)
1336 : {
1337 287 : T = pol_x(v);
1338 287 : primelt = gen_0;
1339 : }
1340 : else
1341 : {
1342 3444 : b = alg_decompose(al, Z, 1, &primelt);
1343 3444 : if (!gequal0(b)) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 2)");
1344 3437 : T = gel(primelt,2);
1345 3437 : primelt = gel(primelt,1);
1346 3437 : setvarn(T,v);
1347 : }
1348 :
1349 : /* use the ffinit polynomial */
1350 3724 : if (n>1)
1351 : {
1352 3437 : T2 = init_Fq(p,n,v);
1353 3437 : setvarn(T,fetch_var_higher());
1354 3437 : ro = FpXQX_roots(T2,T,p);
1355 3437 : ro = gel(ro,1);
1356 3437 : primelt = algpoleval(al,ro,primelt);
1357 3437 : T = T2;
1358 : }
1359 :
1360 : /* descend al*e to a vector space over the center */
1361 : /* lifte: al*e -> al ; proje: al*e -> al */
1362 3724 : lifte = shallowextract(mte,gel(ire,2));
1363 3724 : extre = shallowmatextract(mte,gel(ire,1),gel(ire,2));
1364 3724 : extre = FpM_inv(extre,p);
1365 3724 : proje = rowpermute(mte,gel(ire,1));
1366 3724 : proje = FpM_mul(extre,proje,p);
1367 3724 : if (n==1)
1368 : {
1369 287 : B = lifte;
1370 287 : C = proje;
1371 : }
1372 : else
1373 : {
1374 3437 : M = algbasismultable(al,primelt);
1375 3437 : M = FpM_mul(M,lifte,p);
1376 3437 : M = FpM_mul(proje,M,p);
1377 3437 : B = descend(M,n,p,v);
1378 3437 : C = gel(B,2);
1379 3437 : B = gel(B,1);
1380 3437 : B = FpM_mul(lifte,B,p);
1381 3437 : C = FqM_mul(C,proje,T,p);
1382 : }
1383 :
1384 : /* compute the isomorphism */
1385 3724 : mt = alg_get_multable(al);
1386 3724 : map = cgetg(N+1,t_VEC);
1387 3724 : M = cgetg(N+1,t_MAT);
1388 33663 : for (i=1; i<=N; i++)
1389 : {
1390 29939 : mx = gel(mt,i);
1391 29939 : mx = FpM_mul(mx,B,p);
1392 29939 : mx = FqM_mul(C,mx,T,p);
1393 29939 : gel(map,i) = mx;
1394 29939 : gel(M,i) = Fq_mat2col(mx,d,n);
1395 : }
1396 3724 : mapi = FpM_inv(M,p);
1397 3724 : if (!mapi) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 3)");
1398 3717 : return mkvec3(T,map,mapi);
1399 : }
1400 :
1401 : GEN
1402 3766 : algsplit(GEN al, long v)
1403 : {
1404 3766 : pari_sp av = avma;
1405 : GEN res, T, map, mapi, ff, p;
1406 : long i,j,k,li,lj;
1407 3766 : checkalg(al);
1408 3759 : p = alg_get_char(al);
1409 3759 : if (gequal0(p))
1410 7 : pari_err_IMPL("splitting a characteristic 0 algebra over its center");
1411 3752 : res = alg_finite_csa_split(al, v);
1412 3717 : T = gel(res,1);
1413 3717 : map = gel(res,2);
1414 3717 : mapi = gel(res,3);
1415 3717 : ff = Tp_to_FF(T,p);
1416 33593 : for (i=1; i<lg(map); i++)
1417 : {
1418 29876 : li = lg(gel(map,i));
1419 89908 : for (j=1; j<li; j++)
1420 : {
1421 60032 : lj = lg(gmael(map,i,j));
1422 190876 : for (k=1; k<lj; k++)
1423 130844 : gmael3(map,i,j,k) = Fq_to_FF(gmael3(map,i,j,k),ff);
1424 : }
1425 : }
1426 :
1427 3717 : return gerepilecopy(av, mkvec2(map,mapi));
1428 : }
1429 :
1430 : /* multiplication table sanity checks */
1431 : static GEN
1432 38920 : check_mt_noid(GEN mt, GEN p)
1433 : {
1434 : long i, l;
1435 38920 : GEN MT = cgetg_copy(mt, &l);
1436 38920 : if (typ(MT) != t_VEC || l == 1) return NULL;
1437 187372 : for (i = 1; i < l; i++)
1438 : {
1439 148501 : GEN M = gel(mt,i);
1440 148501 : if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
1441 148473 : if (p) M = RgM_to_FpM(M,p);
1442 148473 : gel(MT,i) = M;
1443 : }
1444 38871 : return MT;
1445 : }
1446 : static GEN
1447 38416 : check_mt(GEN mt, GEN p)
1448 : {
1449 : long i;
1450 : GEN MT;
1451 38416 : MT = check_mt_noid(mt, p);
1452 38416 : if (!MT || !ZM_isidentity(gel(MT,1))) return NULL;
1453 145330 : for (i=2; i<lg(MT); i++)
1454 106942 : if (ZC_is_ei(gmael(MT,i,1)) != i) return NULL;
1455 38388 : return MT;
1456 : }
1457 :
1458 : static GEN
1459 175 : check_relmt(GEN nf, GEN mt)
1460 : {
1461 175 : long i, l = lg(mt), j, k;
1462 175 : GEN MT = gcopy(mt), a, b, d;
1463 175 : if (typ(MT) != t_VEC || l == 1) return NULL;
1464 693 : for (i = 1; i < l; i++)
1465 : {
1466 539 : GEN M = gel(MT,i);
1467 539 : if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
1468 2758 : for (k = 1; k < l; k++)
1469 13643 : for (j = 1; j < l; j++)
1470 : {
1471 11424 : a = gcoeff(M,j,k);
1472 11424 : if (typ(a)==t_INT) continue;
1473 1771 : b = algtobasis(nf,a);
1474 1771 : d = Q_denom(b);
1475 1771 : if (!isint1(d))
1476 14 : pari_err_DOMAIN("alg_csa_table", "denominator(mt)", "!=", gen_1, mt);
1477 1757 : gcoeff(M,j,k) = lift(basistoalg(nf,b));
1478 : }
1479 525 : if (i > 1 && RgC_is_ei(gel(M,1)) != i) return NULL; /* i = 1 checked at end */
1480 518 : gel(MT,i) = M;
1481 : }
1482 154 : if (!RgM_isidentity(gel(MT,1))) return NULL;
1483 154 : return MT;
1484 : }
1485 :
1486 : int
1487 511 : algisassociative(GEN mt0, GEN p)
1488 : {
1489 511 : pari_sp av = avma;
1490 : long i, j, k, n;
1491 : GEN M, mt;
1492 :
1493 511 : if (checkalg_i(mt0)) { p = alg_get_char(mt0); mt0 = alg_get_multable(mt0); }
1494 511 : if (typ(p) != t_INT) pari_err_TYPE("algisassociative",p);
1495 504 : mt = check_mt_noid(mt0, isintzero(p)? NULL: p);
1496 504 : if (!mt) pari_err_TYPE("algisassociative (mult. table)", mt0);
1497 469 : if (!ZM_isidentity(gel(mt,1))) return gc_bool(av,0);
1498 455 : n = lg(mt)-1;
1499 455 : M = cgetg(n+1,t_MAT);
1500 3542 : for (j=1; j<=n; j++) gel(M,j) = cgetg(n+1,t_COL);
1501 3542 : for (i=1; i<=n; i++)
1502 : {
1503 3087 : GEN mi = gel(mt,i);
1504 35182 : for (j=1; j<=n; j++) gcoeff(M,i,j) = gel(mi,j); /* ei.ej */
1505 : }
1506 3073 : for (i=2; i<=n; i++) {
1507 2625 : GEN mi = gel(mt,i);
1508 28973 : for (j=2; j<=n; j++) {
1509 368291 : for (k=2; k<=n; k++) {
1510 : GEN x, y;
1511 341943 : if (signe(p)) {
1512 242039 : x = _tablemul_ej_Fp(mt,gcoeff(M,i,j),k,p);
1513 242039 : y = FpM_FpC_mul(mi,gcoeff(M,j,k),p);
1514 : }
1515 : else {
1516 99904 : x = _tablemul_ej(mt,gcoeff(M,i,j),k);
1517 99904 : y = RgM_RgC_mul(mi,gcoeff(M,j,k));
1518 : }
1519 : /* not cmp_universal: must not fail on 0 == Mod(0,2) for instance */
1520 341943 : if (!gequal(x,y)) return gc_bool(av,0);
1521 : }
1522 : }
1523 : }
1524 448 : return gc_bool(av,1);
1525 : }
1526 :
1527 : int
1528 371 : algiscommutative(GEN al) /* assumes e_1 = 1 */
1529 : {
1530 : long i,j,k,N,sp;
1531 : GEN mt,a,b,p;
1532 371 : checkalg(al);
1533 371 : if (alg_type(al) != al_TABLE) return alg_get_degree(al)==1;
1534 308 : N = alg_get_absdim(al);
1535 308 : mt = alg_get_multable(al);
1536 308 : p = alg_get_char(al);
1537 308 : sp = signe(p);
1538 1449 : for (i=2; i<=N; i++)
1539 9464 : for (j=2; j<=N; j++)
1540 85820 : for (k=1; k<=N; k++) {
1541 77553 : a = gcoeff(gel(mt,i),k,j);
1542 77553 : b = gcoeff(gel(mt,j),k,i);
1543 77553 : if (sp) {
1544 73423 : if (cmpii(Fp_red(a,p), Fp_red(b,p))) return 0;
1545 : }
1546 4130 : else if (gcmp(a,b)) return 0;
1547 : }
1548 252 : return 1;
1549 : }
1550 :
1551 : int
1552 371 : algissemisimple(GEN al)
1553 : {
1554 371 : pari_sp av = avma;
1555 : GEN rad;
1556 371 : checkalg(al);
1557 371 : if (alg_type(al) != al_TABLE) return 1;
1558 308 : rad = algradical(al);
1559 308 : set_avma(av);
1560 308 : return gequal0(rad);
1561 : }
1562 :
1563 : /* ss : known to be semisimple */
1564 : int
1565 280 : algissimple(GEN al, long ss)
1566 : {
1567 280 : pari_sp av = avma;
1568 : GEN Z, dec, p;
1569 280 : checkalg(al);
1570 280 : if (alg_type(al) != al_TABLE) return 1;
1571 224 : if (!ss && !algissemisimple(al)) return 0;
1572 :
1573 182 : p = alg_get_char(al);
1574 182 : if (signe(p)) Z = algprimesubalg(al);
1575 91 : else Z = algtablecenter(al);
1576 :
1577 182 : if (lg(Z) == 2) {/* dim Z = 1 */
1578 105 : set_avma(av);
1579 105 : return 1;
1580 : }
1581 77 : dec = alg_decompose(al, Z, 1, NULL);
1582 77 : set_avma(av);
1583 77 : return gequal0(dec);
1584 : }
1585 :
1586 : static long
1587 329 : is_place_emb(GEN nf, GEN pl)
1588 : {
1589 : long r, r1, r2;
1590 329 : if (typ(pl) != t_INT) pari_err_TYPE("is_place_emb", pl);
1591 315 : if (signe(pl)<=0) pari_err_DOMAIN("is_place_emb", "pl", "<=", gen_0, pl);
1592 308 : nf_get_sign(nf,&r1,&r2); r = r1+r2;
1593 308 : if (cmpiu(pl,r)>0) pari_err_DOMAIN("is_place_emb", "pl", ">", utoi(r), pl);
1594 294 : return itou(pl);
1595 : }
1596 :
1597 : static long
1598 294 : alghasse_emb(GEN al, long emb)
1599 : {
1600 294 : GEN nf = alg_get_center(al);
1601 294 : long r1 = nf_get_r1(nf);
1602 294 : return (emb <= r1)? alg_get_hasse_i(al)[emb]: 0;
1603 : }
1604 :
1605 : static long
1606 413 : alghasse_pr(GEN al, GEN pr)
1607 : {
1608 413 : GEN hf = alg_get_hasse_f(al);
1609 406 : long i = tablesearch(gel(hf,1), pr, &cmp_prime_ideal);
1610 406 : return i? gel(hf,2)[i]: 0;
1611 : }
1612 :
1613 : static long
1614 777 : alghasse_0(GEN al, GEN pl)
1615 : {
1616 : long ta;
1617 : GEN pr, nf;
1618 777 : ta = alg_type(al);
1619 777 : if (ta == al_REAL) return algreal_dim(al)!=1;
1620 756 : if (!pl)
1621 7 : pari_err(e_MISC, "must provide a place pl");
1622 749 : if (ta == al_CSA)
1623 7 : pari_err_IMPL("computation of Hasse invariants over table CSA");
1624 742 : if ((pr = get_prid(pl))) return alghasse_pr(al, pr);
1625 329 : nf = alg_get_center(al);
1626 329 : return alghasse_emb(al, is_place_emb(nf, pl));
1627 : }
1628 : GEN
1629 252 : alghasse(GEN al, GEN pl)
1630 : {
1631 : long h;
1632 252 : checkalg(al);
1633 252 : if (alg_type(al) == al_TABLE) pari_err_TYPE("alghasse [use alginit]",al);
1634 245 : h = alghasse_0(al,pl);
1635 189 : return sstoQ(h, alg_get_degree(al));
1636 : }
1637 :
1638 : /* h >= 0, d >= 0 */
1639 : static long
1640 819 : indexfromhasse(long h, long d) { return d/ugcd(h,d); }
1641 :
1642 : long
1643 819 : algindex(GEN al, GEN pl)
1644 : {
1645 : long d, res, i, l, ta;
1646 : GEN hi, hf;
1647 :
1648 819 : checkalg(al);
1649 812 : ta = alg_type(al);
1650 812 : if (ta == al_TABLE) pari_err_TYPE("algindex [use alginit]",al);
1651 805 : if (ta == al_REAL) return algreal_dim(al)==1 ? 1 : 2;
1652 721 : d = alg_get_degree(al);
1653 721 : if (pl) return indexfromhasse(alghasse_0(al,pl), d);
1654 :
1655 : /* else : global index */
1656 189 : res = 1;
1657 189 : hi = alg_get_hasse_i(al); l = lg(hi);
1658 322 : for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hi[i],d));
1659 189 : hf = gel(alg_get_hasse_f(al), 2); l = lg(hf);
1660 336 : for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hf[i],d));
1661 182 : return res;
1662 : }
1663 :
1664 : int
1665 224 : algisdivision(GEN al, GEN pl)
1666 : {
1667 224 : checkalg(al);
1668 224 : if (alg_type(al) == al_TABLE) {
1669 21 : if (!algissimple(al,0)) return 0;
1670 14 : if (algiscommutative(al)) return 1;
1671 7 : pari_err_IMPL("algisdivision for table algebras");
1672 : }
1673 203 : return algindex(al,pl) == alg_get_degree(al);
1674 : }
1675 :
1676 : int
1677 406 : algissplit(GEN al, GEN pl)
1678 : {
1679 406 : checkalg(al);
1680 406 : if (alg_type(al) == al_TABLE) pari_err_TYPE("algissplit [use alginit]", al);
1681 392 : return algindex(al,pl) == 1;
1682 : }
1683 :
1684 : int
1685 203 : algisramified(GEN al, GEN pl) { return !algissplit(al,pl); }
1686 :
1687 : GEN
1688 105 : algramifiedplaces(GEN al)
1689 : {
1690 105 : pari_sp av = avma;
1691 : GEN ram, hf, hi, Lpr;
1692 : long r1, count, i, ta;
1693 105 : checkalg(al);
1694 105 : ta = alg_type(al);
1695 105 : if (ta != al_CSA && ta != al_CYCLIC)
1696 14 : pari_err_TYPE("algramifiedplaces [not a central simple algebra"
1697 : " over a number field]", al);
1698 91 : r1 = nf_get_r1(alg_get_center(al));
1699 91 : hi = alg_get_hasse_i(al);
1700 91 : hf = alg_get_hasse_f(al);
1701 84 : Lpr = gel(hf,1);
1702 84 : hf = gel(hf,2);
1703 84 : ram = cgetg(r1+lg(Lpr), t_VEC);
1704 84 : count = 0;
1705 280 : for (i=1; i<=r1; i++)
1706 196 : if (hi[i]) {
1707 91 : count++;
1708 91 : gel(ram,count) = stoi(i);
1709 : }
1710 272 : for (i=1; i<lg(Lpr); i++)
1711 188 : if (hf[i]) {
1712 77 : count++;
1713 77 : gel(ram,count) = gel(Lpr,i);
1714 : }
1715 84 : setlg(ram, count+1);
1716 84 : return gerepilecopy(av, ram);
1717 : }
1718 :
1719 : /** OPERATIONS ON ELEMENTS operations.c **/
1720 :
1721 : static long
1722 1142032 : alg_model0(GEN al, GEN x)
1723 : {
1724 1142032 : long t, N = alg_get_absdim(al), lx = lg(x), d, n, D, i;
1725 1142032 : if (typ(x) == t_MAT) return al_MATRIX;
1726 1095909 : if (typ(x) != t_COL) return al_INVALID;
1727 1095846 : if (N == 1) {
1728 3010 : if (lx != 2) return al_INVALID;
1729 2989 : switch(typ(gel(x,1)))
1730 : {
1731 1967 : case t_INT: case t_FRAC: return al_TRIVIAL; /* cannot distinguish basis and alg from size */
1732 1015 : case t_POL: case t_POLMOD: return al_ALGEBRAIC;
1733 7 : default: return al_INVALID;
1734 : }
1735 : }
1736 :
1737 1092836 : switch(alg_type(al)) {
1738 660011 : case al_TABLE:
1739 660011 : if (lx != N+1) return al_INVALID;
1740 659990 : return al_BASIS;
1741 346746 : case al_CYCLIC:
1742 346746 : d = alg_get_degree(al);
1743 346746 : if (lx == N+1) return al_BASIS;
1744 94872 : if (lx == d+1) return al_ALGEBRAIC;
1745 14 : return al_INVALID;
1746 86079 : case al_CSA:
1747 86079 : D = alg_get_dim(al);
1748 86079 : n = nf_get_degree(alg_get_center(al));
1749 86079 : if (n == 1) {
1750 1323 : if (lx != D+1) return al_INVALID;
1751 4144 : for (i=1; i<=D; i++) {
1752 3437 : t = typ(gel(x,i));
1753 3437 : if (t == t_POL || t == t_POLMOD) return al_ALGEBRAIC;
1754 : /* TODO t_COL for coefficients in basis form ? */
1755 : }
1756 707 : return al_BASIS;
1757 : }
1758 : else {
1759 84756 : if (lx == N+1) return al_BASIS;
1760 22582 : if (lx == D+1) return al_ALGEBRAIC;
1761 7 : return al_INVALID;
1762 : }
1763 : }
1764 : return al_INVALID; /* LCOV_EXCL_LINE */
1765 : }
1766 :
1767 : static void
1768 1141892 : checkalgx(GEN x, long model)
1769 : {
1770 : long t, i;
1771 1141892 : switch(model) {
1772 974745 : case al_BASIS:
1773 9886971 : for (i=1; i<lg(x); i++) {
1774 8912233 : t = typ(gel(x,i));
1775 8912233 : if (t != t_INT && t != t_FRAC)
1776 7 : pari_err_TYPE("checkalgx", gel(x,i));
1777 : }
1778 974738 : return;
1779 121024 : case al_TRIVIAL:
1780 : case al_ALGEBRAIC:
1781 409566 : for (i=1; i<lg(x); i++) {
1782 288549 : t = typ(gel(x,i));
1783 288549 : if (t != t_INT && t != t_FRAC && t != t_POL && t != t_POLMOD)
1784 : /* TODO t_COL ? */
1785 7 : pari_err_TYPE("checkalgx", gel(x,i));
1786 : }
1787 121017 : return;
1788 : }
1789 : }
1790 :
1791 : long
1792 1142032 : alg_model(GEN al, GEN x)
1793 : {
1794 1142032 : long res = alg_model0(al, x);
1795 1142032 : if (res == al_INVALID) pari_err_TYPE("alg_model", x);
1796 1141892 : checkalgx(x, res); return res;
1797 : }
1798 :
1799 : static long
1800 462630 : H_model0(GEN x)
1801 : {
1802 : long i;
1803 462630 : switch(typ(x))
1804 : {
1805 15218 : case t_INT:
1806 : case t_FRAC:
1807 : case t_REAL:
1808 : case t_COMPLEX:
1809 15218 : return H_SCALAR;
1810 10157 : case t_MAT:
1811 10157 : return H_MATRIX;
1812 437143 : case t_COL:
1813 437143 : if (lg(x)!=5) return H_INVALID;
1814 2185603 : for (i=1; i<=4; i++) if (!is_real_t(typ(gel(x,i)))) return H_INVALID;
1815 437115 : return H_QUATERNION;
1816 112 : default:
1817 112 : return al_INVALID;
1818 : }
1819 : }
1820 :
1821 : static long
1822 462630 : H_model(GEN x)
1823 : {
1824 462630 : long res = H_model0(x);
1825 462630 : if (res == H_INVALID) pari_err_TYPE("H_model", x);
1826 462490 : return res;
1827 : }
1828 :
1829 : static GEN
1830 756 : alC_add_i(GEN al, GEN x, GEN y, long lx)
1831 : {
1832 756 : GEN A = cgetg(lx, t_COL);
1833 : long i;
1834 2296 : for (i=1; i<lx; i++) gel(A,i) = algadd(al, gel(x,i), gel(y,i));
1835 749 : return A;
1836 : }
1837 : static GEN
1838 406 : alM_add(GEN al, GEN x, GEN y)
1839 : {
1840 406 : long lx = lg(x), l, j;
1841 : GEN z;
1842 406 : if (lg(y) != lx) pari_err_DIM("alM_add (rows)");
1843 392 : if (lx == 1) return cgetg(1, t_MAT);
1844 385 : z = cgetg(lx, t_MAT); l = lgcols(x);
1845 385 : if (lgcols(y) != l) pari_err_DIM("alM_add (columns)");
1846 1127 : for (j = 1; j < lx; j++) gel(z,j) = alC_add_i(al, gel(x,j), gel(y,j), l);
1847 371 : return z;
1848 : }
1849 : static GEN
1850 17745 : H_add(GEN x, GEN y)
1851 : {
1852 17745 : long tx = H_model(x), ty = H_model(y);
1853 17724 : if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_add", x, y);
1854 17710 : if (tx>ty) { swap(x,y); lswap(tx,ty); }
1855 17710 : switch (tx)
1856 : {
1857 105 : case H_MATRIX: /* both H_MATRIX */ return alM_add(NULL, x, y);
1858 16681 : case H_QUATERNION: /* both H_QUATERNION */ return gadd(x,y);
1859 924 : case H_SCALAR:
1860 924 : if (ty == H_SCALAR) return gadd(x,y);
1861 : else /* ty == H_QUATERNION */
1862 : {
1863 217 : pari_sp av = avma;
1864 217 : GEN res = gcopy(y), im;
1865 217 : gel(res,1) = gadd(gel(res,1), real_i(x));
1866 217 : im = imag_i(x);
1867 217 : if (im != gen_0) gel(res,2) = gadd(gel(res,2), im);
1868 217 : return gerepileupto(av, res);
1869 : }
1870 : }
1871 : return NULL; /*LCOV_EXCL_LINE*/
1872 : }
1873 : GEN
1874 54845 : algadd(GEN al, GEN x, GEN y)
1875 : {
1876 54845 : pari_sp av = avma;
1877 : long tx, ty;
1878 : GEN p;
1879 54845 : checkalg(al);
1880 54845 : if (alg_type(al)==al_REAL) return H_add(x,y);
1881 37100 : tx = alg_model(al,x);
1882 37093 : ty = alg_model(al,y);
1883 37093 : p = alg_get_char(al);
1884 37093 : if (signe(p)) return FpC_add(x,y,p);
1885 36960 : if (tx==ty) {
1886 36078 : if (tx!=al_MATRIX) return gadd(x,y);
1887 301 : return gerepilecopy(av, alM_add(al,x,y));
1888 : }
1889 882 : if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
1890 882 : if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
1891 882 : return gerepileupto(av, gadd(x,y));
1892 : }
1893 :
1894 : static GEN
1895 98 : H_neg(GEN x)
1896 : {
1897 98 : (void)H_model(x);
1898 70 : return gneg(x);
1899 : }
1900 :
1901 : GEN
1902 245 : algneg(GEN al, GEN x)
1903 : {
1904 245 : checkalg(al);
1905 245 : if (alg_type(al)==al_REAL) return H_neg(x);
1906 147 : (void)alg_model(al,x);
1907 140 : return gneg(x);
1908 : }
1909 :
1910 : static GEN
1911 210 : alC_sub_i(GEN al, GEN x, GEN y, long lx)
1912 : {
1913 : long i;
1914 210 : GEN A = cgetg(lx, t_COL);
1915 630 : for (i=1; i<lx; i++) gel(A,i) = algsub(al, gel(x,i), gel(y,i));
1916 210 : return A;
1917 : }
1918 : static GEN
1919 126 : alM_sub(GEN al, GEN x, GEN y)
1920 : {
1921 126 : long lx = lg(x), l, j;
1922 : GEN z;
1923 126 : if (lg(y) != lx) pari_err_DIM("alM_sub (rows)");
1924 119 : if (lx == 1) return cgetg(1, t_MAT);
1925 112 : z = cgetg(lx, t_MAT); l = lgcols(x);
1926 112 : if (lgcols(y) != l) pari_err_DIM("alM_sub (columns)");
1927 315 : for (j = 1; j < lx; j++) gel(z,j) = alC_sub_i(al, gel(x,j), gel(y,j), l);
1928 105 : return z;
1929 : }
1930 : GEN
1931 1120 : algsub(GEN al, GEN x, GEN y)
1932 : {
1933 : long tx, ty;
1934 1120 : pari_sp av = avma;
1935 : GEN p;
1936 1120 : checkalg(al);
1937 1120 : if (alg_type(al)==al_REAL) return gerepileupto(av, algadd(NULL,x,gneg(y)));
1938 966 : tx = alg_model(al,x);
1939 959 : ty = alg_model(al,y);
1940 959 : p = alg_get_char(al);
1941 959 : if (signe(p)) return FpC_sub(x,y,p);
1942 868 : if (tx==ty) {
1943 546 : if (tx != al_MATRIX) return gsub(x,y);
1944 126 : return gerepilecopy(av, alM_sub(al,x,y));
1945 : }
1946 322 : if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
1947 322 : if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
1948 322 : return gerepileupto(av, gsub(x,y));
1949 : }
1950 :
1951 : static GEN
1952 1659 : algalgmul_cyc(GEN al, GEN x, GEN y)
1953 : {
1954 1659 : pari_sp av = avma;
1955 1659 : long n = alg_get_degree(al), i, k;
1956 : GEN xalg, yalg, res, rnf, auts, sum, b, prod, autx;
1957 1659 : rnf = alg_get_splittingfield(al);
1958 1659 : auts = alg_get_auts(al);
1959 1659 : b = alg_get_b(al);
1960 :
1961 1659 : xalg = cgetg(n+1, t_COL);
1962 4935 : for (i=0; i<n; i++)
1963 3276 : gel(xalg,i+1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
1964 :
1965 1659 : yalg = cgetg(n+1, t_COL);
1966 4935 : for (i=0; i<n; i++) gel(yalg,i+1) = rnfbasistoalg(rnf,gel(y,i+1));
1967 :
1968 1659 : res = cgetg(n+1,t_COL);
1969 4935 : for (k=0; k<n; k++) {
1970 3276 : gel(res,k+1) = gmul(gel(xalg,k+1),gel(yalg,1));
1971 5166 : for (i=1; i<=k; i++) {
1972 1890 : autx = poleval(gel(xalg,k-i+1),gel(auts,i));
1973 1890 : prod = gmul(autx,gel(yalg,i+1));
1974 1890 : gel(res,k+1) = gadd(gel(res,k+1), prod);
1975 : }
1976 :
1977 3276 : sum = gen_0;
1978 5166 : for (; i<n; i++) {
1979 1890 : autx = poleval(gel(xalg,k+n-i+1),gel(auts,i));
1980 1890 : prod = gmul(autx,gel(yalg,i+1));
1981 1890 : sum = gadd(sum,prod);
1982 : }
1983 3276 : sum = gmul(b,sum);
1984 :
1985 3276 : gel(res,k+1) = gadd(gel(res,k+1),sum);
1986 : }
1987 :
1988 1659 : return gerepilecopy(av, res);
1989 : }
1990 :
1991 : static GEN
1992 212751 : _tablemul(GEN mt, GEN x, GEN y)
1993 : {
1994 212751 : pari_sp av = avma;
1995 212751 : long D = lg(mt)-1, i;
1996 212751 : GEN res = NULL;
1997 2069921 : for (i=1; i<=D; i++) {
1998 1857170 : GEN c = gel(x,i);
1999 1857170 : if (!gequal0(c)) {
2000 1012816 : GEN My = RgM_RgC_mul(gel(mt,i),y);
2001 1012816 : GEN t = RgC_Rg_mul(My,c);
2002 1012816 : res = res? RgC_add(res,t): t;
2003 : }
2004 : }
2005 212751 : if (!res) { set_avma(av); return zerocol(D); }
2006 211848 : return gerepileupto(av, res);
2007 : }
2008 :
2009 : static GEN
2010 263319 : _tablemul_Fp(GEN mt, GEN x, GEN y, GEN p)
2011 : {
2012 263319 : pari_sp av = avma;
2013 263319 : long D = lg(mt)-1, i;
2014 263319 : GEN res = NULL;
2015 2654754 : for (i=1; i<=D; i++) {
2016 2391435 : GEN c = gel(x,i);
2017 2391435 : if (signe(c)) {
2018 456216 : GEN My = FpM_FpC_mul(gel(mt,i),y,p);
2019 456216 : GEN t = FpC_Fp_mul(My,c,p);
2020 456216 : res = res? FpC_add(res,t,p): t;
2021 : }
2022 : }
2023 263319 : if (!res) { set_avma(av); return zerocol(D); }
2024 262780 : return gerepileupto(av, res);
2025 : }
2026 :
2027 : /* x*ej */
2028 : static GEN
2029 99904 : _tablemul_ej(GEN mt, GEN x, long j)
2030 : {
2031 99904 : pari_sp av = avma;
2032 99904 : long D = lg(mt)-1, i;
2033 99904 : GEN res = NULL;
2034 1563793 : for (i=1; i<=D; i++) {
2035 1463889 : GEN c = gel(x,i);
2036 1463889 : if (!gequal0(c)) {
2037 116718 : GEN My = gel(gel(mt,i),j);
2038 116718 : GEN t = RgC_Rg_mul(My,c);
2039 116718 : res = res? RgC_add(res,t): t;
2040 : }
2041 : }
2042 99904 : if (!res) { set_avma(av); return zerocol(D); }
2043 99764 : return gerepileupto(av, res);
2044 : }
2045 : static GEN
2046 242039 : _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p)
2047 : {
2048 242039 : pari_sp av = avma;
2049 242039 : long D = lg(mt)-1, i;
2050 242039 : GEN res = NULL;
2051 4364787 : for (i=1; i<=D; i++) {
2052 4122748 : GEN c = gel(x,i);
2053 4122748 : if (!gequal0(c)) {
2054 289954 : GEN My = gel(gel(mt,i),j);
2055 289954 : GEN t = FpC_Fp_mul(My,c,p);
2056 289954 : res = res? FpC_add(res,t,p): t;
2057 : }
2058 : }
2059 242039 : if (!res) { set_avma(av); return zerocol(D); }
2060 241927 : return gerepileupto(av, res);
2061 : }
2062 :
2063 : static GEN
2064 245283 : _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p)
2065 : {
2066 245283 : pari_sp av = avma;
2067 245283 : long D = lg(mt)-1, i;
2068 245283 : GEN res = NULL;
2069 3945192 : for (i=1; i<=D; i++) {
2070 3699909 : ulong c = x[i];
2071 3699909 : if (c) {
2072 394117 : GEN My = gel(gel(mt,i),j);
2073 394117 : GEN t = Flv_Fl_mul(My,c, p);
2074 394117 : res = res? Flv_add(res,t, p): t;
2075 : }
2076 : }
2077 245283 : if (!res) { set_avma(av); return zero_Flv(D); }
2078 245283 : return gerepileupto(av, res);
2079 : }
2080 :
2081 : static GEN
2082 686 : algalgmul_csa(GEN al, GEN x, GEN y)
2083 : {
2084 686 : GEN z, nf = alg_get_center(al);
2085 : long i;
2086 686 : z = _tablemul(alg_get_relmultable(al), x, y);
2087 2485 : for (i=1; i<lg(z); i++)
2088 1799 : gel(z,i) = basistoalg(nf,gel(z,i));
2089 686 : return z;
2090 : }
2091 :
2092 : /* assumes x and y in algebraic form */
2093 : static GEN
2094 2345 : algalgmul(GEN al, GEN x, GEN y)
2095 : {
2096 2345 : switch(alg_type(al))
2097 : {
2098 1659 : case al_CYCLIC: return algalgmul_cyc(al, x, y);
2099 686 : case al_CSA: return algalgmul_csa(al, x, y);
2100 : }
2101 : return NULL; /*LCOV_EXCL_LINE*/
2102 : }
2103 :
2104 : static GEN
2105 475384 : algbasismul(GEN al, GEN x, GEN y)
2106 : {
2107 475384 : GEN mt = alg_get_multable(al), p = alg_get_char(al);
2108 475384 : if (signe(p)) return _tablemul_Fp(mt, x, y, p);
2109 212065 : return _tablemul(mt, x, y);
2110 : }
2111 :
2112 : /* x[i,]*y. Assume lg(x) > 1 and 0 < i < lgcols(x) */
2113 : static GEN
2114 119651 : alMrow_alC_mul_i(GEN al, GEN x, GEN y, long i, long lx)
2115 : {
2116 119651 : pari_sp av = avma;
2117 119651 : GEN c = algmul(al,gcoeff(x,i,1),gel(y,1)), ZERO;
2118 : long k;
2119 119651 : ZERO = zerocol(alg_get_absdim(al));
2120 273308 : for (k = 2; k < lx; k++)
2121 : {
2122 153657 : GEN t = algmul(al, gcoeff(x,i,k), gel(y,k));
2123 153657 : if (!gequal(t,ZERO)) c = algadd(al, c, t);
2124 : }
2125 119651 : return gerepilecopy(av, c);
2126 : }
2127 : /* return x * y, 1 < lx = lg(x), l = lgcols(x) */
2128 : static GEN
2129 54502 : alM_alC_mul_i(GEN al, GEN x, GEN y, long lx, long l)
2130 : {
2131 54502 : GEN z = cgetg(l,t_COL);
2132 : long i;
2133 174153 : for (i=1; i<l; i++) gel(z,i) = alMrow_alC_mul_i(al,x,y,i,lx);
2134 54502 : return z;
2135 : }
2136 : static GEN
2137 25627 : alM_mul(GEN al, GEN x, GEN y)
2138 : {
2139 25627 : long j, l, lx=lg(x), ly=lg(y);
2140 : GEN z;
2141 25627 : if (ly==1) return cgetg(1,t_MAT);
2142 25529 : if (lx != lgcols(y)) pari_err_DIM("alM_mul");
2143 25508 : if (lx==1) return zeromat(0, ly-1);
2144 25501 : l = lgcols(x); z = cgetg(ly,t_MAT);
2145 80003 : for (j=1; j<ly; j++) gel(z,j) = alM_alC_mul_i(al,x,gel(y,j),lx,l);
2146 25501 : return z;
2147 : }
2148 :
2149 : static void
2150 205583 : H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d)
2151 : {
2152 205583 : switch(H_model(x))
2153 : {
2154 5173 : case H_SCALAR:
2155 5173 : *a = real_i(x);
2156 5173 : *b = imag_i(x);
2157 5173 : *c = gen_0;
2158 5173 : *d = gen_0;
2159 5173 : return;
2160 200410 : case H_QUATERNION:
2161 200410 : *a = gel(x,1);
2162 200410 : *b = gel(x,2);
2163 200410 : *c = gel(x,3);
2164 200410 : *d = gel(x,4);
2165 200410 : return;
2166 : default: *a = *b = *c = *d = NULL; return; /*LCOV_EXCL_LINE*/
2167 : }
2168 : }
2169 : static GEN
2170 108101 : H_mul(GEN x, GEN y)
2171 : {
2172 108101 : pari_sp av = avma;
2173 : GEN a,b,c,d,u,v,w,z;
2174 108101 : long tx = H_model(x), ty = H_model(y);
2175 108087 : if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_mul", x, y);
2176 108080 : if (tx == H_MATRIX) /* both H_MATRIX */ return alM_mul(NULL, x, y);
2177 103789 : if (tx == H_SCALAR && ty == H_SCALAR) return gmul(x,y);
2178 102592 : H_compo(x,&a,&b,&c,&d);
2179 102592 : H_compo(y,&u,&v,&w,&z);
2180 102592 : return gerepilecopy(av,mkcol4(
2181 : gsub(gmul(a,u), gadd(gadd(gmul(b,v),gmul(c,w)),gmul(d,z))),
2182 : gsub(gadd(gmul(a,v),gadd(gmul(b,u),gmul(c,z))), gmul(d,w)),
2183 : gsub(gadd(gmul(a,w),gadd(gmul(c,u),gmul(d,v))), gmul(b,z)),
2184 : gsub(gadd(gmul(a,z),gadd(gmul(b,w),gmul(d,u))), gmul(c,v))
2185 : ));
2186 : }
2187 :
2188 : GEN
2189 497528 : algmul(GEN al, GEN x, GEN y)
2190 : {
2191 497528 : pari_sp av = avma;
2192 : long tx, ty;
2193 497528 : checkalg(al);
2194 497528 : if (alg_type(al)==al_REAL) return H_mul(x,y);
2195 389679 : tx = alg_model(al,x);
2196 389665 : ty = alg_model(al,y);
2197 389665 : if (tx==al_MATRIX) {
2198 20832 : if (ty==al_MATRIX) return alM_mul(al,x,y);
2199 7 : pari_err_TYPE("algmul", y);
2200 : }
2201 368833 : if (signe(alg_get_char(al))) return algbasismul(al,x,y);
2202 211652 : if (tx==al_TRIVIAL) retmkcol(gmul(gel(x,1),gel(y,1)));
2203 211547 : if (tx==al_ALGEBRAIC && ty==al_ALGEBRAIC) return algalgmul(al,x,y);
2204 210021 : if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
2205 210021 : if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
2206 210021 : return gerepileupto(av,algbasismul(al,x,y));
2207 : }
2208 :
2209 : static GEN
2210 329 : H_sqr(GEN x)
2211 : {
2212 329 : pari_sp av = avma;
2213 329 : long tx = H_model(x);
2214 : GEN a,b,c,d;
2215 308 : if (tx == H_SCALAR) return gsqr(x);
2216 224 : if (tx == H_MATRIX) return H_mul(x,x);
2217 119 : H_compo(x,&a,&b,&c,&d);
2218 119 : return gerepilecopy(av, mkcol4(
2219 : gsub(gsqr(a), gadd(gsqr(b),gadd(gsqr(c),gsqr(d)))),
2220 : gshift(gmul(a,b),1),
2221 : gshift(gmul(a,c),1),
2222 : gshift(gmul(a,d),1)
2223 : ));
2224 : }
2225 :
2226 : GEN
2227 107265 : algsqr(GEN al, GEN x)
2228 : {
2229 107265 : pari_sp av = avma;
2230 : long tx;
2231 107265 : checkalg(al);
2232 107230 : if (alg_type(al)==al_REAL) return H_sqr(x);
2233 106901 : tx = alg_model(al,x);
2234 106831 : if (tx==al_MATRIX) return gerepilecopy(av,alM_mul(al,x,x));
2235 106320 : if (signe(alg_get_char(al))) return algbasismul(al,x,x);
2236 3213 : if (tx==al_TRIVIAL) retmkcol(gsqr(gel(x,1)));
2237 2863 : if (tx==al_ALGEBRAIC) return algalgmul(al,x,x);
2238 2044 : return gerepileupto(av,algbasismul(al,x,x));
2239 : }
2240 :
2241 : static GEN
2242 9380 : algmtK2Z_cyc(GEN al, GEN m)
2243 : {
2244 9380 : pari_sp av = avma;
2245 9380 : GEN nf = alg_get_abssplitting(al), res, mt, rnf = alg_get_splittingfield(al), c, dc;
2246 9380 : long n = alg_get_degree(al), N = nf_get_degree(nf), Nn, i, j, i1, j1;
2247 9380 : Nn = N*n;
2248 9380 : res = zeromatcopy(Nn,Nn);
2249 42364 : for (i=0; i<n; i++)
2250 196168 : for (j=0; j<n; j++) {
2251 163184 : c = gcoeff(m,i+1,j+1);
2252 163184 : if (!gequal0(c)) {
2253 32984 : c = rnfeltreltoabs(rnf,c);
2254 32984 : c = algtobasis(nf,c);
2255 32984 : c = Q_remove_denom(c,&dc);
2256 32984 : mt = zk_multable(nf,c);
2257 32984 : if (dc) mt = ZM_Z_div(mt,dc);
2258 302162 : for (i1=1; i1<=N; i1++)
2259 2945936 : for (j1=1; j1<=N; j1++)
2260 2676758 : gcoeff(res,i*N+i1,j*N+j1) = gcoeff(mt,i1,j1);
2261 : }
2262 : }
2263 9380 : return gerepilecopy(av,res);
2264 : }
2265 :
2266 : static GEN
2267 945 : algmtK2Z_csa(GEN al, GEN m)
2268 : {
2269 945 : pari_sp av = avma;
2270 945 : GEN nf = alg_get_center(al), res, mt, c, dc;
2271 945 : long d2 = alg_get_dim(al), n = nf_get_degree(nf), D, i, j, i1, j1;
2272 945 : D = d2*n;
2273 945 : res = zeromatcopy(D,D);
2274 5502 : for (i=0; i<d2; i++)
2275 31122 : for (j=0; j<d2; j++) {
2276 26565 : c = gcoeff(m,i+1,j+1);
2277 26565 : if (!gequal0(c)) {
2278 3906 : c = algtobasis(nf,c);
2279 3906 : c = Q_remove_denom(c,&dc);
2280 3906 : mt = zk_multable(nf,c);
2281 3906 : if (dc) mt = ZM_Z_div(mt,dc);
2282 13076 : for (i1=1; i1<=n; i1++)
2283 32564 : for (j1=1; j1<=n; j1++)
2284 23394 : gcoeff(res,i*n+i1,j*n+j1) = gcoeff(mt,i1,j1);
2285 : }
2286 : }
2287 945 : return gerepilecopy(av,res);
2288 : }
2289 :
2290 : /* assumes al is a CSA or CYCLIC */
2291 : static GEN
2292 10325 : algmtK2Z(GEN al, GEN m)
2293 : {
2294 10325 : switch(alg_type(al))
2295 : {
2296 9380 : case al_CYCLIC: return algmtK2Z_cyc(al, m);
2297 945 : case al_CSA: return algmtK2Z_csa(al, m);
2298 : }
2299 : return NULL; /*LCOV_EXCL_LINE*/
2300 : }
2301 :
2302 : /* left multiplication table, as a vector space of dimension n over the splitting field (by right multiplication) */
2303 : static GEN
2304 11998 : algalgmultable_cyc(GEN al, GEN x)
2305 : {
2306 11998 : pari_sp av = avma;
2307 11998 : long n = alg_get_degree(al), i, j;
2308 : GEN res, rnf, auts, b, pol;
2309 11998 : rnf = alg_get_splittingfield(al);
2310 11998 : auts = alg_get_auts(al);
2311 11998 : b = alg_get_b(al);
2312 11998 : pol = rnf_get_pol(rnf);
2313 :
2314 11998 : res = zeromatcopy(n,n);
2315 50288 : for (i=0; i<n; i++)
2316 38290 : gcoeff(res,i+1,1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
2317 :
2318 50288 : for (i=0; i<n; i++) {
2319 106386 : for (j=1; j<=i; j++)
2320 68096 : gcoeff(res,i+1,j+1) = gmodulo(poleval(gcoeff(res,i-j+1,1),gel(auts,j)),pol);
2321 106386 : for (; j<n; j++)
2322 68096 : gcoeff(res,i+1,j+1) = gmodulo(gmul(b,poleval(gcoeff(res,n+i-j+1,1),gel(auts,j))), pol);
2323 : }
2324 :
2325 50288 : for (i=0; i<n; i++)
2326 38290 : gcoeff(res,i+1,1) = gmodulo(gcoeff(res,i+1,1),pol);
2327 :
2328 11998 : return gerepilecopy(av, res);
2329 : }
2330 :
2331 : static GEN
2332 1393 : elementmultable(GEN mt, GEN x)
2333 : {
2334 1393 : pari_sp av = avma;
2335 1393 : long D = lg(mt)-1, i;
2336 1393 : GEN z = NULL;
2337 7448 : for (i=1; i<=D; i++)
2338 : {
2339 6055 : GEN c = gel(x,i);
2340 6055 : if (!gequal0(c))
2341 : {
2342 2163 : GEN M = RgM_Rg_mul(gel(mt,i),c);
2343 2163 : z = z? RgM_add(z, M): M;
2344 : }
2345 : }
2346 1393 : if (!z) { set_avma(av); return zeromatcopy(D,D); }
2347 1393 : return gerepileupto(av, z);
2348 : }
2349 : /* mt a t_VEC of Flm modulo m */
2350 : static GEN
2351 24902 : algbasismultable_Flm(GEN mt, GEN x, ulong m)
2352 : {
2353 24902 : pari_sp av = avma;
2354 24902 : long D = lg(gel(mt,1))-1, i;
2355 24902 : GEN z = NULL;
2356 270185 : for (i=1; i<=D; i++)
2357 : {
2358 245283 : ulong c = x[i];
2359 245283 : if (c)
2360 : {
2361 34130 : GEN M = Flm_Fl_mul(gel(mt,i),c, m);
2362 34130 : z = z? Flm_add(z, M, m): M;
2363 : }
2364 : }
2365 24902 : if (!z) { set_avma(av); return zero_Flm(D,D); }
2366 24902 : return gerepileupto(av, z);
2367 : }
2368 : static GEN
2369 227045 : elementabsmultable_Z(GEN mt, GEN x)
2370 : {
2371 227045 : long i, l = lg(x);
2372 227045 : GEN z = NULL;
2373 2337306 : for (i = 1; i < l; i++)
2374 : {
2375 2110261 : GEN c = gel(x,i);
2376 2110261 : if (signe(c))
2377 : {
2378 816294 : GEN M = ZM_Z_mul(gel(mt,i),c);
2379 816294 : z = z? ZM_add(z, M): M;
2380 : }
2381 : }
2382 227045 : return z;
2383 : }
2384 : static GEN
2385 116175 : elementabsmultable(GEN mt, GEN x)
2386 : {
2387 116175 : GEN d, z = elementabsmultable_Z(mt, Q_remove_denom(x,&d));
2388 116175 : return (z && d)? ZM_Z_div(z, d): z;
2389 : }
2390 : static GEN
2391 110870 : elementabsmultable_Fp(GEN mt, GEN x, GEN p)
2392 : {
2393 110870 : GEN z = elementabsmultable_Z(mt, x);
2394 110870 : return z? FpM_red(z, p): z;
2395 : }
2396 : static GEN
2397 227045 : algbasismultable(GEN al, GEN x)
2398 : {
2399 227045 : pari_sp av = avma;
2400 227045 : GEN z, p = alg_get_char(al), mt = alg_get_multable(al);
2401 227045 : z = signe(p)? elementabsmultable_Fp(mt, x, p): elementabsmultable(mt, x);
2402 227045 : if (!z)
2403 : {
2404 713 : long D = lg(mt)-1;
2405 713 : set_avma(av); return zeromat(D,D);
2406 : }
2407 226332 : return gerepileupto(av, z);
2408 : }
2409 :
2410 : static GEN
2411 1393 : algalgmultable_csa(GEN al, GEN x)
2412 : {
2413 1393 : GEN nf = alg_get_center(al), m;
2414 : long i,j;
2415 1393 : m = elementmultable(alg_get_relmultable(al), x);
2416 7448 : for (i=1; i<lg(m); i++)
2417 38318 : for(j=1; j<lg(m); j++)
2418 32263 : gcoeff(m,i,j) = basistoalg(nf,gcoeff(m,i,j));
2419 1393 : return m;
2420 : }
2421 :
2422 : /* assumes x in algebraic form */
2423 : static GEN
2424 13097 : algalgmultable(GEN al, GEN x)
2425 : {
2426 13097 : switch(alg_type(al))
2427 : {
2428 11998 : case al_CYCLIC: return algalgmultable_cyc(al, x);
2429 1099 : case al_CSA: return algalgmultable_csa(al, x);
2430 : }
2431 : return NULL; /*LCOV_EXCL_LINE*/
2432 : }
2433 :
2434 : /* on the natural basis */
2435 : /* assumes x in algebraic form */
2436 : static GEN
2437 10325 : algZmultable(GEN al, GEN x) {
2438 10325 : pari_sp av = avma;
2439 10325 : return gerepileupto(av, algmtK2Z(al,algalgmultable(al,x)));
2440 : }
2441 :
2442 : /* x integral */
2443 : static GEN
2444 36575 : algbasisrightmultable(GEN al, GEN x)
2445 : {
2446 36575 : long N = alg_get_absdim(al), i,j,k;
2447 36575 : GEN res = zeromatcopy(N,N), c, mt = alg_get_multable(al), p = alg_get_char(al);
2448 36575 : if (gequal0(p)) p = NULL;
2449 330960 : for (i=1; i<=N; i++) {
2450 294385 : c = gel(x,i);
2451 294385 : if (!gequal0(c)) {
2452 892773 : for (j=1; j<=N; j++)
2453 7582134 : for(k=1; k<=N; k++) {
2454 6785842 : if (p) gcoeff(res,k,j) = Fp_add(gcoeff(res,k,j), Fp_mul(c, gcoeff(gel(mt,j),k,i), p), p);
2455 5160974 : else gcoeff(res,k,j) = addii(gcoeff(res,k,j), mulii(c, gcoeff(gel(mt,j),k,i)));
2456 : }
2457 : }
2458 : }
2459 36575 : return res;
2460 : }
2461 :
2462 : /* basis for matrices : 1, E_{i,j} for (i,j)!=(1,1) */
2463 : /* index : ijk = ((i-1)*N+j-1)*n + k */
2464 : /* square matrices only, coefficients in basis form, shallow function */
2465 : static GEN
2466 23961 : algmat2basis(GEN al, GEN M)
2467 : {
2468 23961 : long n = alg_get_absdim(al), N = lg(M)-1, i, j, k, ij, ijk;
2469 : GEN res, x;
2470 23961 : res = zerocol(N*N*n);
2471 75131 : for (i=1; i<=N; i++) {
2472 163310 : for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
2473 112140 : x = gcoeff(M,i,j);
2474 819532 : for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
2475 707392 : gel(res, ijk) = gel(x, k);
2476 707392 : if (i>1 && i==j) gel(res, ijk) = gsub(gel(res,ijk), gel(res,k));
2477 : }
2478 : }
2479 : }
2480 :
2481 23961 : return res;
2482 : }
2483 :
2484 : static GEN
2485 294 : algbasis2mat(GEN al, GEN M, long N)
2486 : {
2487 294 : long n = alg_get_absdim(al), i, j, k, ij, ijk;
2488 : GEN res, x;
2489 294 : res = zeromatcopy(N,N);
2490 882 : for (i=1; i<=N; i++)
2491 1764 : for (j=1; j<=N; j++)
2492 1176 : gcoeff(res,i,j) = zerocol(n);
2493 :
2494 882 : for (i=1; i<=N; i++) {
2495 1764 : for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
2496 1176 : x = gcoeff(res,i,j);
2497 9240 : for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
2498 8064 : gel(x,k) = gel(M,ijk);
2499 8064 : if (i>1 && i==j) gel(x,k) = gadd(gel(x,k), gel(M,k));
2500 : }
2501 : }
2502 : }
2503 :
2504 294 : return res;
2505 : }
2506 :
2507 : static GEN
2508 23884 : algmatbasis_ei(GEN al, long ijk, long N)
2509 : {
2510 23884 : long n = alg_get_absdim(al), i, j, k, ij;
2511 : GEN res;
2512 :
2513 23884 : res = zeromatcopy(N,N);
2514 74900 : for (i=1; i<=N; i++)
2515 162848 : for (j=1; j<=N; j++)
2516 111832 : gcoeff(res,i,j) = zerocol(n);
2517 :
2518 23884 : k = ijk%n;
2519 23884 : if (k==0) k=n;
2520 23884 : ij = (ijk-k)/n+1;
2521 :
2522 23884 : if (ij==1) {
2523 16947 : for (i=1; i<=N; i++)
2524 11410 : gcoeff(res,i,i) = col_ei(n,k);
2525 5537 : return res;
2526 : }
2527 :
2528 18347 : j = ij%N;
2529 18347 : if (j==0) j=N;
2530 18347 : i = (ij-j)/N+1;
2531 :
2532 18347 : gcoeff(res,i,j) = col_ei(n,k);
2533 18347 : return res;
2534 : }
2535 :
2536 : /* FIXME lazy implementation! */
2537 : static GEN
2538 910 : algleftmultable_mat(GEN al, GEN M)
2539 : {
2540 910 : long N = lg(M)-1, n = alg_get_absdim(al), D = N*N*n, j;
2541 : GEN res, x, Mx;
2542 910 : if (N == 0) return cgetg(1, t_MAT);
2543 903 : if (N != nbrows(M)) pari_err_DIM("algleftmultable_mat (nonsquare)");
2544 882 : res = cgetg(D+1, t_MAT);
2545 24766 : for (j=1; j<=D; j++) {
2546 23884 : x = algmatbasis_ei(al, j, N);
2547 23884 : Mx = algmul(al, M, x);
2548 23884 : gel(res, j) = algmat2basis(al, Mx);
2549 : }
2550 882 : return res;
2551 : }
2552 :
2553 : /* left multiplication table on integral basis */
2554 : static GEN
2555 6951 : algleftmultable(GEN al, GEN x)
2556 : {
2557 6951 : pari_sp av = avma;
2558 : long tx;
2559 : GEN res;
2560 :
2561 6951 : checkalg(al);
2562 6951 : tx = alg_model(al,x);
2563 6944 : switch(tx) {
2564 98 : case al_TRIVIAL : res = mkmatcopy(mkcol(gel(x,1))); break;
2565 196 : case al_ALGEBRAIC : x = algalgtobasis(al,x);
2566 6328 : case al_BASIS : res = algbasismultable(al,x); break;
2567 518 : case al_MATRIX : res = algleftmultable_mat(al,x); break;
2568 : default : return NULL; /* LCOV_EXCL_LINE */
2569 : }
2570 6937 : return gerepileupto(av,res);
2571 : }
2572 :
2573 : static GEN
2574 4102 : algbasissplittingmatrix_csa(GEN al, GEN x)
2575 : {
2576 4102 : long d = alg_get_degree(al), i, j;
2577 4102 : GEN rnf = alg_get_splittingfield(al), splba = alg_get_splittingbasis(al), splbainv = alg_get_splittingbasisinv(al), M;
2578 4102 : M = algbasismultable(al,x);
2579 4102 : M = RgM_mul(M, splba); /* TODO best order ? big matrix /Q vs small matrix /nf */
2580 4102 : M = RgM_mul(splbainv, M);
2581 12131 : for (i=1; i<=d; i++)
2582 23912 : for (j=1; j<=d; j++)
2583 15883 : gcoeff(M,i,j) = rnfeltabstorel(rnf, gcoeff(M,i,j));
2584 4102 : return M;
2585 : }
2586 :
2587 : static GEN
2588 728 : algmat_tomatrix(GEN al, GEN x) /* abs = 0 */
2589 : {
2590 : GEN res;
2591 : long i,j;
2592 728 : if (lg(x) == 1) return cgetg(1, t_MAT);
2593 700 : res = zeromatcopy(nbrows(x),lg(x)-1);
2594 2212 : for (j=1; j<lg(x); j++)
2595 4879 : for (i=1; i<lgcols(x); i++)
2596 3367 : gcoeff(res,i,j) = algtomatrix(al,gcoeff(x,i,j),0);
2597 700 : return shallowmatconcat(res);
2598 : }
2599 :
2600 : static GEN
2601 42 : R_tomatrix(GEN x)
2602 : {
2603 42 : long t = H_model(x);
2604 42 : if (t == H_QUATERNION) pari_err_TYPE("R_tomatrix", x);
2605 35 : if (t == H_MATRIX) return x;
2606 21 : return mkmat(mkcol(x));
2607 : }
2608 : static GEN
2609 84 : C_tomatrix(GEN z, long abs)
2610 : {
2611 : GEN x,y;
2612 84 : long t = H_model(z), nrows, ncols;
2613 84 : if (t == H_QUATERNION) pari_err_TYPE("C_tomatrix", z);
2614 77 : if (!abs)
2615 : {
2616 14 : if (t == H_MATRIX) return z;
2617 7 : return mkmat(mkcol(z));
2618 : }
2619 63 : if (t == H_MATRIX)
2620 : {
2621 : /* Warning: this is not the same choice of basis as for other algebras */
2622 : GEN res, a, b;
2623 : long i,j;
2624 56 : RgM_dimensions(z,&nrows,&ncols);
2625 56 : res = zeromatcopy(2*nrows,2*ncols);
2626 168 : for (i=1; i<=nrows; i++)
2627 336 : for (j=1; j<=ncols; j++)
2628 : {
2629 224 : a = real_i(gcoeff(z,i,j));
2630 224 : b = imag_i(gcoeff(z,i,j));
2631 224 : gcoeff(res,2*i-1,2*j-1) = a;
2632 224 : gcoeff(res,2*i,2*j) = a;
2633 224 : gcoeff(res,2*i-1,2*j) = gneg(b);
2634 224 : gcoeff(res,2*i,2*j-1) = b;
2635 : }
2636 56 : return res;
2637 : }
2638 7 : x = real_i(z);
2639 7 : y = imag_i(z);
2640 7 : return mkmat22(x,gneg(y),y,x);
2641 : }
2642 : static GEN
2643 2331 : H_tomatrix(GEN x, long abs)
2644 : {
2645 2331 : long tx = H_model(x);
2646 2324 : GEN a = NULL, b =NULL, c = NULL, d = NULL, md = NULL, M = NULL;
2647 2324 : if (abs) {
2648 287 : if (tx == H_MATRIX) return algleftmultable_mat(NULL,x);
2649 154 : switch(tx)
2650 : {
2651 35 : case H_SCALAR:
2652 35 : a = real_i(x);
2653 35 : b = imag_i(x);
2654 35 : c = gen_0;
2655 35 : d = gen_0;
2656 35 : break;
2657 119 : case H_QUATERNION:
2658 119 : a = gel(x,1);
2659 119 : b = gel(x,2);
2660 119 : c = gel(x,3);
2661 119 : d = gel(x,4);
2662 119 : break;
2663 : }
2664 154 : M = scalarmat(a,4);
2665 154 : gcoeff(M,2,1) = gcoeff(M,4,3) = b;
2666 154 : gcoeff(M,1,2) = gcoeff(M,3,4) = gneg(b);
2667 154 : gcoeff(M,3,1) = gcoeff(M,2,4) = c;
2668 154 : gcoeff(M,4,2) = gcoeff(M,1,3) = gneg(c);
2669 154 : gcoeff(M,4,1) = gcoeff(M,3,2) = d;
2670 154 : gcoeff(M,2,3) = gcoeff(M,1,4) = gneg(d);
2671 : }
2672 : else /* abs == 0 */
2673 : {
2674 2037 : if (tx == H_MATRIX) return algmat_tomatrix(NULL,x);
2675 1778 : switch(tx)
2676 : {
2677 273 : case H_SCALAR:
2678 273 : M = mkmat22(
2679 : x, gen_0,
2680 : gen_0, conj_i(x)
2681 : );
2682 273 : break;
2683 1505 : case H_QUATERNION:
2684 1505 : a = gel(x,1);
2685 1505 : b = gel(x,2);
2686 1505 : c = gel(x,3);
2687 1505 : md = gneg(gel(x,4));
2688 1505 : M = mkmat22(
2689 : mkcomplex(a,b), mkcomplex(gneg(c),md),
2690 : mkcomplex(c,md), mkcomplex(a,gneg(b))
2691 : );
2692 : }
2693 : }
2694 1932 : return M;
2695 : }
2696 :
2697 : GEN
2698 9667 : algtomatrix(GEN al, GEN x, long abs)
2699 : {
2700 9667 : pari_sp av = avma;
2701 9667 : GEN res = NULL;
2702 : long ta, tx;
2703 9667 : checkalg(al);
2704 9667 : ta = alg_type(al);
2705 9667 : if (ta==al_REAL)
2706 : {
2707 2268 : switch(alg_get_absdim(al)) {
2708 42 : case 1: res = R_tomatrix(x); break;
2709 84 : case 2: res = C_tomatrix(x,abs); break;
2710 2135 : case 4: res = H_tomatrix(x,abs); break;
2711 7 : default: pari_err_TYPE("algtomatrix [apply alginit]", al);
2712 : }
2713 2240 : return gerepilecopy(av, res);
2714 : }
2715 7399 : if (abs || ta==al_TABLE) return algleftmultable(al,x);
2716 6622 : tx = alg_model(al,x);
2717 6622 : if (tx == al_MATRIX) res = algmat_tomatrix(al,x);
2718 6153 : else switch (alg_type(al))
2719 : {
2720 2051 : case al_CYCLIC:
2721 2051 : if (tx==al_BASIS) x = algbasistoalg(al,x);
2722 2051 : res = algalgmultable(al,x);
2723 2051 : break;
2724 4102 : case al_CSA:
2725 4102 : if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
2726 4102 : res = algbasissplittingmatrix_csa(al,x);
2727 4102 : break;
2728 : default: return NULL; /*LCOV_EXCL_LINE*/
2729 : }
2730 6622 : return gerepilecopy(av,res);
2731 : }
2732 :
2733 : /* x^(-1)*y, NULL if no solution */
2734 : static GEN
2735 112 : C_divl_i(GEN x, GEN y)
2736 : {
2737 112 : long tx = H_model(x), ty = H_model(y);
2738 112 : if (tx != ty) pari_err_TYPE2("C_divl", x, y);
2739 105 : switch (tx) {
2740 42 : case H_SCALAR:
2741 42 : if (gequal0(x)) return gequal0(y) ? gen_0 : NULL;
2742 14 : else return gdiv(y,x);
2743 56 : case H_MATRIX:
2744 56 : if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
2745 7 : pari_err_DIM("C_divl (nonsquare)");
2746 49 : if (lg(x) != lg(y)) pari_err_DIM("C_divl");
2747 42 : if (lg(y) == 1) return cgetg(1, t_MAT);
2748 42 : return RgM_invimage(x, y);
2749 7 : default: pari_err_TYPE("C_divl", x); return NULL;
2750 : }
2751 : }
2752 : /* H^k -> C^2k */
2753 : static GEN
2754 140 : HC_to_CC(GEN v)
2755 : {
2756 140 : long l = lg(v), i;
2757 140 : GEN w = cgetg(2*l-1, t_COL), a, b, c, d;
2758 420 : for (i=1; i<l; i++)
2759 : {
2760 280 : H_compo(gel(v,i),&a,&b,&c,&d);
2761 280 : gel(w,2*i-1) = mkcomplex(a,b);
2762 280 : gel(w,2*i) = mkcomplex(c,gneg(d));
2763 : }
2764 140 : return w;
2765 : }
2766 : /* C^2k -> H^k */
2767 : static GEN
2768 98 : CC_to_HC(GEN w)
2769 : {
2770 98 : long l = lg(w), i, lv = (l+1)/2;
2771 98 : GEN v = cgetg(lv, t_COL), ab, cd;
2772 294 : for (i=1; i<lv; i++)
2773 : {
2774 196 : ab = gel(w,2*i-1);
2775 196 : cd = gel(w,2*i);
2776 196 : gel(v,i) = mkcol4(real_i(ab),imag_i(ab),real_i(cd),gneg(imag_i(cd)));
2777 : }
2778 98 : return v;
2779 : }
2780 : /* M_{k,n}(H) -> M_{2k,n}(C) */
2781 : static GEN
2782 210 : HM_to_CM(GEN x) pari_APPLY_same(HC_to_CC(gel(x,i)));
2783 : /* M_{2k,n}(C) -> M_{k,n}(H) */
2784 : static GEN
2785 147 : CM_to_HM(GEN x) pari_APPLY_same(CC_to_HC(gel(x,i)));
2786 : /* x^(-1)*y, NULL if no solution */
2787 : static GEN
2788 203 : H_divl_i(GEN x, GEN y)
2789 : {
2790 203 : pari_sp av = avma;
2791 203 : long tx = H_model(x), ty = H_model(y);
2792 189 : if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_divl", x, y);
2793 168 : if (tx==H_MATRIX)
2794 : {
2795 : GEN mx, my, mxdivy;
2796 98 : if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
2797 14 : pari_err_DIM("H_divl (nonsquare)");
2798 84 : if (lg(x) != lg(y)) pari_err_DIM("H_divl");
2799 77 : if (lg(y) == 1) return cgetg(1, t_MAT);
2800 70 : mx = H_tomatrix(x,0);
2801 70 : my = HM_to_CM(y);
2802 70 : mxdivy = RgM_invimage(mx, my);
2803 70 : if (!mxdivy) return gc_NULL(av);
2804 49 : return gerepilecopy(av,CM_to_HM(mxdivy));
2805 : }
2806 70 : if (gequal0(y)) return gen_0;
2807 56 : if (gequal0(x)) return NULL;
2808 42 : return gerepilecopy(av,H_mul(H_inv(x),y));
2809 : }
2810 : /* x^(-1)*y, NULL if no solution */
2811 : static GEN
2812 1715 : algdivl_i(GEN al, GEN x, GEN y, long tx, long ty) {
2813 1715 : pari_sp av = avma;
2814 1715 : GEN res, p = alg_get_char(al), mtx;
2815 1715 : if (tx != ty) {
2816 343 : if (tx==al_ALGEBRAIC) { x = algalgtobasis(al,x); tx=al_BASIS; }
2817 343 : if (ty==al_ALGEBRAIC) { y = algalgtobasis(al,y); ty=al_BASIS; }
2818 : }
2819 1715 : if (ty == al_MATRIX)
2820 : {
2821 77 : if (alg_type(al) != al_TABLE) y = algalgtobasis(al,y);
2822 77 : y = algmat2basis(al,y);
2823 : }
2824 1715 : if (signe(p)) res = FpM_FpC_invimage(algbasismultable(al,x),y,p);
2825 : else
2826 : {
2827 1526 : if (ty==al_ALGEBRAIC) mtx = algalgmultable(al,x);
2828 819 : else mtx = algleftmultable(al,x);
2829 1526 : res = inverseimage(mtx,y);
2830 : }
2831 1715 : if (!res || lg(res)==1) return gc_NULL(av);
2832 1687 : if (tx == al_MATRIX) {
2833 294 : res = algbasis2mat(al, res, lg(x)-1);
2834 294 : return gerepilecopy(av,res);
2835 : }
2836 1393 : return gerepileupto(av,res);
2837 : }
2838 : static GEN
2839 1001 : algdivl_i2(GEN al, GEN x, GEN y)
2840 : {
2841 : long tx, ty;
2842 1001 : checkalg(al);
2843 1001 : if (alg_type(al)==al_REAL) switch(alg_get_absdim(al)) {
2844 112 : case 1: case 2: return C_divl_i(x,y);
2845 147 : case 4: return H_divl_i(x,y);
2846 : }
2847 742 : tx = alg_model(al,x);
2848 735 : ty = alg_model(al,y);
2849 735 : if (tx == al_MATRIX) {
2850 140 : if (ty != al_MATRIX) pari_err_TYPE2("\\", x, y);
2851 133 : if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
2852 28 : pari_err_DIM("algdivl (nonsquare)");
2853 105 : if (lg(x) != lg(y)) pari_err_DIM("algdivl");
2854 84 : if (lg(y) == 1) return cgetg(1, t_MAT);
2855 : }
2856 672 : return algdivl_i(al,x,y,tx,ty);
2857 : }
2858 :
2859 875 : GEN algdivl(GEN al, GEN x, GEN y)
2860 : {
2861 : GEN z;
2862 875 : z = algdivl_i2(al,x,y);
2863 728 : if (!z) pari_err_INV("algdivl", x);
2864 714 : return z;
2865 : }
2866 :
2867 : int
2868 126 : algisdivl(GEN al, GEN x, GEN y, GEN* ptz)
2869 : {
2870 126 : pari_sp av = avma;
2871 126 : GEN z = algdivl_i2(al,x,y);
2872 126 : if (!z) return gc_bool(av,0);
2873 84 : if (ptz != NULL) *ptz = z;
2874 84 : return 1;
2875 : }
2876 :
2877 : static GEN
2878 140 : C_inv(GEN x)
2879 : {
2880 140 : switch (H_model(x))
2881 : {
2882 63 : case H_SCALAR: return gequal0(x) ? NULL : ginv(x);
2883 70 : case H_MATRIX: return RgM_inv(x);
2884 7 : default: pari_err_TYPE("alginv_i", x);
2885 : }
2886 : return NULL; /*LCOV_EXCL_LINE*/
2887 : }
2888 : static GEN
2889 259 : H_inv(GEN x)
2890 : {
2891 259 : pari_sp av = avma;
2892 : GEN nm, xi;
2893 : long i;
2894 259 : switch (H_model(x))
2895 : {
2896 28 : case H_SCALAR:
2897 28 : if (gequal0(x)) return NULL;
2898 14 : return ginv(x);
2899 161 : case H_QUATERNION:
2900 161 : if (gequal0(x)) return NULL;
2901 154 : nm = H_norm(x, 0);
2902 154 : xi = gdiv(x,nm);
2903 616 : for(i=2; i<=4; i++) gel(xi,i) = gneg(gel(xi,i));
2904 154 : return gerepilecopy(av,xi);
2905 63 : case H_MATRIX:
2906 63 : if (lg(x)==1) return cgetg(1,t_MAT);
2907 56 : return H_divl_i(x, matid(lg(x)-1));
2908 : }
2909 : return NULL; /*LCOV_EXCL_LINE*/
2910 : }
2911 : static GEN
2912 1512 : alginv_i(GEN al, GEN x)
2913 : {
2914 1512 : pari_sp av = avma;
2915 1512 : GEN res = NULL, p = alg_get_char(al);
2916 : long tx, n, ta;
2917 1512 : ta = alg_type(al);
2918 1512 : if (ta==al_REAL) switch(alg_get_absdim(al)) {
2919 140 : case 1: case 2: return C_inv(x);
2920 217 : case 4: return H_inv(x);
2921 7 : default: pari_err_TYPE("alginv_i [apply alginit]", al);
2922 : }
2923 1148 : tx = alg_model(al,x);
2924 1127 : switch(tx) {
2925 63 : case al_TRIVIAL :
2926 63 : if (signe(p)) { res = mkcol(Fp_inv(gel(x,1),p)); break; }
2927 49 : else { res = mkcol(ginv(gel(x,1))); break; }
2928 455 : case al_ALGEBRAIC :
2929 : switch(ta) {
2930 350 : case al_CYCLIC: n = alg_get_degree(al); break;
2931 105 : case al_CSA: n = alg_get_dim(al); break;
2932 : default: return NULL; /* LCOV_EXCL_LINE */
2933 : }
2934 455 : res = algdivl_i(al, x, col_ei(n,1), tx, al_ALGEBRAIC); break;
2935 371 : case al_BASIS : res = algdivl_i(al, x, col_ei(alg_get_absdim(al),1), tx,
2936 371 : al_BASIS); break;
2937 238 : case al_MATRIX :
2938 238 : n = lg(x)-1;
2939 238 : if (n==0) return cgetg(1, t_MAT);
2940 224 : if (n != nbrows(x)) pari_err_DIM("alginv_i (nonsquare)");
2941 217 : res = algdivl_i(al, x, col_ei(n*n*alg_get_absdim(al),1), tx, al_BASIS);
2942 : /* cheat on type because wrong dimension */
2943 : }
2944 1106 : if (!res) return gc_NULL(av);
2945 1092 : return gerepilecopy(av,res);
2946 : }
2947 : GEN
2948 1323 : alginv(GEN al, GEN x)
2949 : {
2950 : GEN z;
2951 1323 : checkalg(al);
2952 1323 : z = alginv_i(al,x);
2953 1274 : if (!z) pari_err_INV("alginv", x);
2954 1239 : return z;
2955 : }
2956 :
2957 : int
2958 189 : algisinv(GEN al, GEN x, GEN* ptix)
2959 : {
2960 189 : pari_sp av = avma;
2961 : GEN ix;
2962 189 : if (al) checkalg(al);
2963 189 : ix = alginv_i(al,x);
2964 189 : if (!ix) return gc_bool(av,0);
2965 133 : if (ptix != NULL) *ptix = ix;
2966 133 : return 1;
2967 : }
2968 :
2969 : /* x*y^(-1) */
2970 : GEN
2971 469 : algdivr(GEN al, GEN x, GEN y) { return algmul(al, x, alginv(al, y)); }
2972 :
2973 41315 : static GEN _mul(void* data, GEN x, GEN y) { return algmul((GEN)data,x,y); }
2974 105025 : static GEN _sqr(void* data, GEN x) { return algsqr((GEN)data,x); }
2975 :
2976 : static GEN
2977 21 : algmatid(GEN al, long N)
2978 : {
2979 21 : long n = alg_get_absdim(al), i, j;
2980 : GEN res, one, zero;
2981 :
2982 21 : res = zeromatcopy(N,N);
2983 21 : one = col_ei(n,1);
2984 21 : zero = zerocol(n);
2985 49 : for (i=1; i<=N; i++)
2986 84 : for (j=1; j<=N; j++)
2987 56 : gcoeff(res,i,j) = i==j ? one : zero;
2988 21 : return res;
2989 : }
2990 :
2991 : GEN
2992 14490 : algpow(GEN al, GEN x, GEN n)
2993 : {
2994 14490 : pari_sp av = avma;
2995 : GEN res;
2996 14490 : long s = signe(n);
2997 14490 : checkalg(al);
2998 14490 : if (!s && alg_type(al)==al_REAL)
2999 : {
3000 63 : if (H_model(x) == H_MATRIX) return matid(lg(x)-1);
3001 35 : else return gen_1;
3002 : }
3003 14427 : switch (s) {
3004 28 : case 0:
3005 28 : if (alg_model(al,x) == al_MATRIX)
3006 21 : res = algmatid(al,lg(x)-1);
3007 : else
3008 7 : res = col_ei(alg_get_absdim(al),1);
3009 28 : return res;
3010 14252 : case 1:
3011 14252 : res = gen_pow_i(x, n, (void*)al, _sqr, _mul); break;
3012 147 : default: /* -1 */
3013 147 : res = gen_pow_i(alginv(al,x), gneg(n), (void*)al, _sqr, _mul);
3014 : }
3015 14385 : return gerepilecopy(av,res);
3016 : }
3017 :
3018 : static GEN
3019 378 : algredcharpoly_i(GEN al, GEN x, long v)
3020 : {
3021 378 : GEN rnf = alg_get_splittingfield(al);
3022 378 : GEN cp = charpoly(algtomatrix(al,x,0),v);
3023 371 : long i, m = lg(cp);
3024 1540 : for (i=2; i<m; i++) gel(cp,i) = rnfeltdown(rnf, gel(cp,i));
3025 371 : return cp;
3026 : }
3027 :
3028 : /* assumes al is CSA or CYCLIC */
3029 : static GEN
3030 385 : algredcharpoly(GEN al, GEN x, long v)
3031 : {
3032 385 : pari_sp av = avma;
3033 385 : long w = gvar(rnf_get_pol(alg_get_center(al)));
3034 385 : if (varncmp(v,w)>=0) pari_err_PRIORITY("algredcharpoly",pol_x(v),">=",w);
3035 378 : switch(alg_type(al))
3036 : {
3037 378 : case al_CYCLIC:
3038 : case al_CSA:
3039 378 : return gerepileupto(av, algredcharpoly_i(al, x, v));
3040 : }
3041 : return NULL; /*LCOV_EXCL_LINE*/
3042 : }
3043 :
3044 : static GEN
3045 21399 : algbasischarpoly(GEN al, GEN x, long v)
3046 : {
3047 21399 : pari_sp av = avma;
3048 21399 : GEN p = alg_get_char(al), mx;
3049 21399 : if (alg_model(al,x) == al_MATRIX) mx = algleftmultable_mat(al,x);
3050 21308 : else mx = algbasismultable(al,x);
3051 21392 : if (signe(p)) {
3052 19278 : GEN res = FpM_charpoly(mx,p);
3053 19278 : setvarn(res,v);
3054 19278 : return gerepileupto(av, res);
3055 : }
3056 2114 : return gerepileupto(av, charpoly(mx,v));
3057 : }
3058 :
3059 : static GEN
3060 35 : R_charpoly(GEN x, long v, long abs)
3061 : {
3062 35 : pari_sp av = avma;
3063 35 : GEN res = NULL;
3064 35 : switch (H_model(x))
3065 : {
3066 14 : case H_SCALAR: res = mkpoln(2, gen_1, gneg(x)); break;
3067 14 : case H_MATRIX:
3068 14 : res = charpoly(x,v);
3069 14 : if (abs) res = gpowgs(res,nbrows(x));
3070 14 : break;
3071 7 : default: pari_err_TYPE("R_charpoly", x);
3072 : }
3073 28 : if (v) setvarn(res, v);
3074 28 : return gerepilecopy(av, res);
3075 : }
3076 : static GEN
3077 35 : C_charpoly(GEN x, long v, long abs)
3078 : {
3079 35 : pari_sp av = avma;
3080 35 : GEN res = NULL;
3081 35 : switch (H_model(x))
3082 : {
3083 14 : case H_SCALAR:
3084 14 : if (abs) res = mkpoln(3, gen_1, gneg(gshift(real_i(x),1)), cxnorm(x));
3085 7 : else res = mkpoln(2, gen_1, gneg(x));
3086 14 : break;
3087 14 : case H_MATRIX:
3088 14 : res = charpoly(x,v);
3089 14 : if (abs) res = gpowgs(real_i(gmul(res,gconj(res))),nbrows(x));
3090 14 : break;
3091 7 : default: pari_err_TYPE("C_charpoly", x);
3092 : }
3093 28 : if (v) setvarn(res, v);
3094 28 : return gerepilecopy(av, res);
3095 : }
3096 : static GEN
3097 98 : H_charpoly(GEN x, long v, long abs)
3098 : {
3099 98 : pari_sp av = avma;
3100 : GEN res;
3101 98 : if (H_model(x) == H_MATRIX) return greal(charpoly(H_tomatrix(x,abs),v));
3102 70 : res = mkpoln(3, gen_1, gneg(H_trace(x,0)), H_norm(x,0));
3103 70 : if (v) setvarn(res, v);
3104 70 : if (abs) res = gsqr(res);
3105 70 : return gerepilecopy(av, res);
3106 : }
3107 :
3108 : GEN
3109 21595 : algcharpoly(GEN al, GEN x, long v, long abs)
3110 : {
3111 : long ta;
3112 21595 : if (v<0) v=0;
3113 21595 : checkalg(al);
3114 21595 : ta = alg_type(al);
3115 21595 : if (ta == al_REAL) switch (alg_get_absdim(al)) {
3116 35 : case 1: return R_charpoly(x, v, abs);
3117 35 : case 2: return C_charpoly(x, v, abs);
3118 98 : case 4: return H_charpoly(x, v, abs);
3119 7 : default: pari_err_TYPE("algcharpoly [apply alginit]", al);
3120 : }
3121 :
3122 : /* gneg(x[1]) left on stack */
3123 21420 : if (alg_model(al,x) == al_TRIVIAL) {
3124 56 : GEN p = alg_get_char(al);
3125 56 : if (signe(p)) return deg1pol(gen_1,Fp_neg(gel(x,1),p),v);
3126 42 : return deg1pol(gen_1,gneg(gel(x,1)),v);
3127 : }
3128 :
3129 21357 : switch(ta) {
3130 490 : case al_CYCLIC: case al_CSA:
3131 490 : if (abs)
3132 : {
3133 105 : if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
3134 : }
3135 385 : else return algredcharpoly(al,x,v);
3136 20972 : case al_TABLE: return algbasischarpoly(al,x,v);
3137 : default : return NULL; /* LCOV_EXCL_LINE */
3138 : }
3139 : }
3140 :
3141 : /* assumes x in basis form */
3142 : static GEN
3143 248681 : algabstrace(GEN al, GEN x)
3144 : {
3145 248681 : pari_sp av = avma;
3146 248681 : GEN res = NULL, p = alg_get_char(al);
3147 248681 : if (signe(p)) return FpV_dotproduct(x, alg_get_tracebasis(al), p);
3148 38745 : switch(alg_model(al,x)) {
3149 154 : case al_TRIVIAL: return gcopy(gel(x,1)); break;
3150 38591 : case al_BASIS: res = RgV_dotproduct(x, alg_get_tracebasis(al)); break;
3151 : }
3152 38591 : return gerepileupto(av,res);
3153 : }
3154 :
3155 : static GEN
3156 1372 : algredtrace(GEN al, GEN x)
3157 : {
3158 1372 : pari_sp av = avma;
3159 1372 : GEN res = NULL;
3160 1372 : switch(alg_model(al,x)) {
3161 35 : case al_TRIVIAL: return gcopy(gel(x,1)); break;
3162 490 : case al_BASIS: return algredtrace(al, algbasistoalg(al,x));
3163 : /* TODO precompute too? */
3164 847 : case al_ALGEBRAIC:
3165 847 : switch(alg_type(al))
3166 : {
3167 553 : case al_CYCLIC:
3168 553 : res = rnfelttrace(alg_get_splittingfield(al),gel(x,1));
3169 553 : break;
3170 294 : case al_CSA:
3171 294 : res = gtrace(algalgmultable_csa(al,x));
3172 294 : res = gdiv(res, stoi(alg_get_degree(al)));
3173 294 : break;
3174 : default: return NULL; /* LCOV_EXCL_LINE */
3175 : }
3176 : }
3177 847 : return gerepileupto(av,res);
3178 : }
3179 :
3180 : static GEN
3181 469 : algtrace_mat(GEN al, GEN M, long abs) {
3182 469 : pari_sp av = avma;
3183 469 : long N = lg(M)-1, i;
3184 469 : GEN res, p = alg_get_char(al);
3185 469 : if (N == 0) return gen_0;
3186 448 : if (N != nbrows(M)) pari_err_DIM("algtrace_mat (nonsquare)");
3187 :
3188 434 : if (!signe(p)) p = NULL;
3189 434 : if (alg_type(al) == al_TABLE) abs = 1;
3190 434 : res = algtrace(al, gcoeff(M,1,1), abs);
3191 896 : for (i=2; i<=N; i++) {
3192 462 : if (p) res = Fp_add(res, algtrace(al,gcoeff(M,i,i),abs), p);
3193 455 : else res = gadd(res, algtrace(al,gcoeff(M,i,i),abs));
3194 : }
3195 434 : if (abs) res = gmulgu(res, N); /* absolute trace */
3196 434 : return gerepileupto(av, res);
3197 : }
3198 :
3199 : static GEN
3200 35 : R_trace(GEN x, long abs)
3201 : {
3202 35 : pari_sp av = avma;
3203 35 : GEN res = NULL;
3204 35 : switch (H_model(x))
3205 : {
3206 14 : case H_SCALAR: res = gcopy(x); break;
3207 14 : case H_MATRIX: res = abs? mulrs(gtrace(x),nbrows(x)) : gtrace(x); break;
3208 7 : default: pari_err_TYPE("R_trace", x);
3209 : }
3210 28 : return gerepilecopy(av, res);
3211 : }
3212 : static GEN
3213 35 : C_trace(GEN x, long abs)
3214 : {
3215 35 : pari_sp av = avma;
3216 35 : GEN res = NULL;
3217 35 : switch (H_model(x))
3218 : {
3219 14 : case H_SCALAR: res = abs ? gshift(real_i(x),1) : x; break;
3220 14 : case H_MATRIX:
3221 14 : res = abs ? mulrs(real_i(gtrace(x)),2*nbrows(x)) : gtrace(x); break;
3222 7 : default: pari_err_TYPE("C_trace", x);
3223 : }
3224 28 : return gerepilecopy(av, res);
3225 : }
3226 : static GEN
3227 567 : H_trace(GEN x, long abs)
3228 : {
3229 567 : long s = abs? 2 : 1;
3230 567 : switch (H_model(x))
3231 : {
3232 154 : case H_SCALAR: return gshift(real_i(x),s);
3233 329 : case H_QUATERNION: return gshift(gel(x,1),s);
3234 77 : case H_MATRIX:
3235 77 : return algtrace_mat(NULL, x, abs);
3236 : }
3237 : return NULL; /*LCOV_EXCL_LINE*/
3238 : }
3239 :
3240 : GEN
3241 2632 : algtrace(GEN al, GEN x, long abs)
3242 : {
3243 : long ta;
3244 2632 : checkalg(al);
3245 2632 : ta = alg_type(al);
3246 2632 : if (ta==al_REAL) switch (alg_get_absdim(al)) {
3247 35 : case 1: return R_trace(x,abs);
3248 35 : case 2: return C_trace(x,abs);
3249 497 : case 4: return H_trace(x,abs);
3250 7 : default: pari_err_TYPE("algtrace [apply alginit]", al);
3251 : }
3252 2058 : if (alg_model(al,x) == al_MATRIX) return algtrace_mat(al,x,abs);
3253 1666 : switch(ta) {
3254 1526 : case al_CYCLIC: case al_CSA:
3255 1526 : if (!abs) return algredtrace(al,x);
3256 644 : if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
3257 784 : case al_TABLE: return algabstrace(al,x);
3258 : default : return NULL; /* LCOV_EXCL_LINE */
3259 : }
3260 : }
3261 :
3262 : static GEN
3263 44248 : ZM_trace(GEN x)
3264 : {
3265 44248 : long i, lx = lg(x);
3266 : GEN t;
3267 44248 : if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
3268 43387 : t = gcoeff(x,1,1);
3269 712387 : for (i = 2; i < lx; i++) t = addii(t, gcoeff(x,i,i));
3270 43387 : return t;
3271 : }
3272 : static GEN
3273 131162 : FpM_trace(GEN x, GEN p)
3274 : {
3275 131162 : long i, lx = lg(x);
3276 : GEN t;
3277 131162 : if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
3278 123218 : t = gcoeff(x,1,1);
3279 895546 : for (i = 2; i < lx; i++) t = Fp_add(t, gcoeff(x,i,i), p);
3280 123218 : return t;
3281 : }
3282 :
3283 : static GEN
3284 41141 : algtracebasis(GEN al)
3285 : {
3286 41141 : pari_sp av = avma;
3287 41141 : GEN mt = alg_get_multable(al), p = alg_get_char(al);
3288 41141 : long i, l = lg(mt);
3289 41141 : GEN v = cgetg(l, t_VEC);
3290 172303 : if (signe(p)) for (i=1; i < l; i++) gel(v,i) = FpM_trace(gel(mt,i), p);
3291 50291 : else for (i=1; i < l; i++) gel(v,i) = ZM_trace(gel(mt,i));
3292 41141 : return gerepileupto(av,v);
3293 : }
3294 :
3295 : /* Assume: i > 0, expo := p^i <= absdim, x contained in I_{i-1} given by mult
3296 : * table modulo modu=p^(i+1). Return Tr(x^(p^i)) mod modu */
3297 : static ulong
3298 24902 : algtracei(GEN mt, ulong p, ulong expo, ulong modu)
3299 : {
3300 24902 : pari_sp av = avma;
3301 24902 : long j, l = lg(mt);
3302 24902 : ulong tr = 0;
3303 24902 : mt = Flm_powu(mt,expo,modu);
3304 270185 : for (j=1; j<l; j++) tr += ucoeff(mt,j,j);
3305 24902 : return gc_ulong(av, (tr/expo) % p);
3306 : }
3307 :
3308 : static GEN
3309 42 : R_norm(GEN x, long abs)
3310 : {
3311 42 : pari_sp av = avma;
3312 42 : GEN res = NULL;
3313 42 : switch (H_model(x))
3314 : {
3315 14 : case H_SCALAR: res = gcopy(x); break;
3316 21 : case H_MATRIX: res = abs ? powrs(det(x),nbrows(x)) : det(x); break;
3317 7 : default: pari_err_TYPE("R_norm", x);
3318 : }
3319 35 : return gerepilecopy(av,res);
3320 : }
3321 : static GEN
3322 42 : C_norm(GEN x, long abs)
3323 : {
3324 42 : pari_sp av = avma;
3325 42 : GEN res = NULL;
3326 42 : switch (H_model(x))
3327 : {
3328 14 : case H_SCALAR: res = abs ? cxnorm(x) : x; break;
3329 21 : case H_MATRIX: res = abs ? powrs(cxnorm(det(x)),nbrows(x)) : det(x); break;
3330 7 : default: pari_err_TYPE("C_norm", x);
3331 : }
3332 35 : return gerepilecopy(av,res);
3333 : }
3334 : static GEN
3335 434 : H_norm(GEN x, long abs)
3336 : {
3337 434 : pari_sp av = avma;
3338 434 : switch (H_model(x))
3339 : {
3340 42 : case H_SCALAR:
3341 42 : if (abs) return gerepilecopy(av,gsqr(gnorm(x)));
3342 35 : else return gnorm(x);
3343 322 : case H_QUATERNION:
3344 322 : if (abs) return gerepilecopy(av,gsqr(gnorml2(x)));
3345 294 : else return gnorml2(x);
3346 63 : case H_MATRIX:
3347 63 : return gerepilecopy(av,real_i(det(H_tomatrix(x,abs))));
3348 : }
3349 : return NULL; /*LCOV_EXCL_LINE*/
3350 : }
3351 :
3352 : GEN
3353 1253 : algnorm(GEN al, GEN x, long abs)
3354 : {
3355 1253 : pari_sp av = avma;
3356 : long tx, ta;
3357 : GEN p, rnf, res, mx;
3358 1253 : checkalg(al);
3359 1253 : ta = alg_type(al);
3360 1253 : if (ta==al_REAL) switch (alg_get_absdim(al)) {
3361 42 : case 1: return R_norm(x,abs);
3362 42 : case 2: return C_norm(x,abs);
3363 210 : case 4: return H_norm(x,abs);
3364 7 : default: pari_err_TYPE("algnorm [apply alginit]", al);
3365 : }
3366 952 : p = alg_get_char(al);
3367 952 : tx = alg_model(al,x);
3368 952 : if (signe(p)) {
3369 21 : if (tx == al_MATRIX) mx = algleftmultable_mat(al,x);
3370 14 : else mx = algbasismultable(al,x);
3371 21 : return gerepileupto(av, FpM_det(mx,p));
3372 : }
3373 931 : if (tx == al_TRIVIAL) return gcopy(gel(x,1));
3374 :
3375 889 : switch(ta) {
3376 819 : case al_CYCLIC: case al_CSA:
3377 819 : if (abs)
3378 : {
3379 196 : if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
3380 : }
3381 : else
3382 : {
3383 623 : rnf = alg_get_splittingfield(al);
3384 623 : res = rnfeltdown(rnf, det(algtomatrix(al,x,0)));
3385 616 : break;
3386 : }
3387 : case al_TABLE:
3388 266 : if (tx == al_MATRIX) mx = algleftmultable_mat(al,x);
3389 105 : else mx = algbasismultable(al,x);
3390 259 : res = det(mx);
3391 259 : break;
3392 : default: return NULL; /* LCOV_EXCL_LINE */
3393 : }
3394 875 : return gerepileupto(av, res);
3395 : }
3396 :
3397 : static GEN
3398 50443 : algalgtonat_cyc(GEN al, GEN x)
3399 : {
3400 50443 : pari_sp av = avma;
3401 50443 : GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
3402 50443 : long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
3403 50443 : res = zerocol(N*n);
3404 154570 : for (i=0; i<n; i++) {
3405 104127 : c = gel(x,i+1);
3406 104127 : c = rnfeltreltoabs(rnf,c);
3407 104127 : if (!gequal0(c)) {
3408 78156 : c = algtobasis(nf,c);
3409 412954 : for (i1=1; i1<=N; i1++) gel(res,i*N+i1) = gel(c,i1);
3410 : }
3411 : }
3412 50443 : return gerepilecopy(av, res);
3413 : }
3414 :
3415 : static GEN
3416 11375 : algalgtonat_csa(GEN al, GEN x)
3417 : {
3418 11375 : pari_sp av = avma;
3419 11375 : GEN nf = alg_get_center(al), res, c;
3420 11375 : long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
3421 11375 : res = zerocol(d2*n);
3422 56644 : for (i=0; i<d2; i++) {
3423 45269 : c = gel(x,i+1);
3424 45269 : if (!gequal0(c)) {
3425 31563 : c = algtobasis(nf,c);
3426 95095 : for (i1=1; i1<=n; i1++) gel(res,i*n+i1) = gel(c,i1);
3427 : }
3428 : }
3429 11375 : return gerepilecopy(av, res);
3430 : }
3431 :
3432 : /* assumes al CSA or CYCLIC */
3433 : static GEN
3434 61818 : algalgtonat(GEN al, GEN x)
3435 : {
3436 61818 : switch(alg_type(al))
3437 : {
3438 50443 : case al_CYCLIC: return algalgtonat_cyc(al, x);
3439 11375 : case al_CSA: return algalgtonat_csa(al, x);
3440 : }
3441 : return NULL; /*LCOV_EXCL_LINE*/
3442 : }
3443 :
3444 : static GEN
3445 11669 : algnattoalg_cyc(GEN al, GEN x)
3446 : {
3447 11669 : pari_sp av = avma;
3448 11669 : GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
3449 11669 : long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
3450 11669 : res = zerocol(n);
3451 11669 : c = zerocol(N);
3452 49154 : for (i=0; i<n; i++) {
3453 324527 : for (i1=1; i1<=N; i1++) gel(c,i1) = gel(x,i*N+i1);
3454 37485 : gel(res,i+1) = rnfeltabstorel(rnf,basistoalg(nf,c));
3455 : }
3456 11669 : return gerepilecopy(av, res);
3457 : }
3458 :
3459 : static GEN
3460 1309 : algnattoalg_csa(GEN al, GEN x)
3461 : {
3462 1309 : pari_sp av = avma;
3463 1309 : GEN nf = alg_get_center(al), res, c;
3464 1309 : long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
3465 1309 : res = zerocol(d2);
3466 1309 : c = zerocol(n);
3467 7028 : for (i=0; i<d2; i++) {
3468 19390 : for (i1=1; i1<=n; i1++) gel(c,i1) = gel(x,i*n+i1);
3469 5719 : gel(res,i+1) = basistoalg(nf,c);
3470 : }
3471 1309 : return gerepilecopy(av, res);
3472 : }
3473 :
3474 : /* assumes al CSA or CYCLIC */
3475 : static GEN
3476 12978 : algnattoalg(GEN al, GEN x)
3477 : {
3478 12978 : switch(alg_type(al))
3479 : {
3480 11669 : case al_CYCLIC: return algnattoalg_cyc(al, x);
3481 1309 : case al_CSA: return algnattoalg_csa(al, x);
3482 : }
3483 : return NULL; /*LCOV_EXCL_LINE*/
3484 : }
3485 :
3486 : static GEN
3487 182 : algalgtobasis_mat(GEN al, GEN x) /* componentwise */
3488 : {
3489 182 : pari_sp av = avma;
3490 : long lx, lxj, i, j;
3491 : GEN res;
3492 182 : lx = lg(x);
3493 182 : res = cgetg(lx, t_MAT);
3494 546 : for (j=1; j<lx; j++) {
3495 364 : lxj = lg(gel(x,j));
3496 364 : gel(res,j) = cgetg(lxj, t_COL);
3497 1092 : for (i=1; i<lxj; i++)
3498 728 : gcoeff(res,i,j) = algalgtobasis(al,gcoeff(x,i,j));
3499 : }
3500 182 : return gerepilecopy(av,res);
3501 : }
3502 : GEN
3503 62280 : algalgtobasis(GEN al, GEN x)
3504 : {
3505 : pari_sp av;
3506 : long tx, ta;
3507 62280 : checkalg(al);
3508 62280 : ta = alg_type(al);
3509 62280 : if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algalgtobasis [use alginit]", al);
3510 62259 : tx = alg_model(al,x);
3511 62259 : if (tx==al_BASIS) return gcopy(x);
3512 62000 : if (tx==al_MATRIX) return algalgtobasis_mat(al,x);
3513 61818 : av = avma;
3514 61818 : x = algalgtonat(al,x);
3515 61818 : x = RgM_RgC_mul(alg_get_invbasis(al),x);
3516 61818 : return gerepileupto(av, x);
3517 : }
3518 :
3519 : static GEN
3520 119 : algbasistoalg_mat(GEN al, GEN x) /* componentwise */
3521 : {
3522 119 : long j, lx = lg(x);
3523 119 : GEN res = cgetg(lx, t_MAT);
3524 357 : for (j=1; j<lx; j++) {
3525 238 : long i, lxj = lg(gel(x,j));
3526 238 : gel(res,j) = cgetg(lxj, t_COL);
3527 714 : for (i=1; i<lxj; i++) gcoeff(res,i,j) = algbasistoalg(al,gcoeff(x,i,j));
3528 : }
3529 119 : return res;
3530 : }
3531 : GEN
3532 2926 : algbasistoalg(GEN al, GEN x)
3533 : {
3534 : pari_sp av;
3535 : long tx, ta;
3536 2926 : checkalg(al);
3537 2926 : ta = alg_type(al);
3538 2926 : if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algbasistoalg [use alginit]", al);
3539 2905 : tx = alg_model(al,x);
3540 2905 : if (tx==al_ALGEBRAIC) return gcopy(x);
3541 2772 : if (tx==al_MATRIX) return algbasistoalg_mat(al,x);
3542 2653 : av = avma;
3543 2653 : x = RgM_RgC_mul(alg_get_basis(al),x);
3544 2653 : x = algnattoalg(al,x);
3545 2653 : return gerepileupto(av, x);
3546 : }
3547 :
3548 : static GEN
3549 4466 : R_random(GEN b)
3550 : {
3551 4466 : pari_sp av = avma;
3552 4466 : long prec = realprec(b);
3553 4466 : GEN z = randomr(prec); shiftr_inplace(z, 1);
3554 4466 : return gerepileuptoleaf(av, mulrr(b,addsr(-1, z)));
3555 : }
3556 : static GEN
3557 182 : C_random(GEN b)
3558 : {
3559 182 : retmkcomplex(R_random(b), R_random(b));
3560 : }
3561 : static GEN
3562 980 : H_random(GEN b)
3563 : {
3564 980 : GEN res = cgetg(5, t_COL);
3565 : long i;
3566 4900 : for (i=1; i<=4; i++) gel(res,i) = R_random(b);
3567 980 : return res;
3568 : }
3569 : GEN
3570 19698 : algrandom(GEN al, GEN b)
3571 : {
3572 19698 : GEN res = NULL, p, N;
3573 : long i, n;
3574 19698 : checkalg(al);
3575 19684 : if (alg_type(al)==al_REAL)
3576 : {
3577 1365 : if (typ(b) != t_REAL) pari_err_TYPE("algrandom",b);
3578 1358 : if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
3579 1351 : switch(alg_get_absdim(al))
3580 : {
3581 182 : case 1: res = R_random(b); break;
3582 182 : case 2: res = C_random(b); break;
3583 980 : case 4: res = H_random(b); break;
3584 7 : default: pari_err_TYPE("algrandom [apply alginit]", al);
3585 : }
3586 1344 : return res;
3587 : }
3588 18319 : if (typ(b) != t_INT) pari_err_TYPE("algrandom",b);
3589 18312 : if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
3590 18305 : n = alg_get_absdim(al);
3591 18305 : N = addiu(shifti(b,1), 1); /* left on stack */
3592 18305 : p = alg_get_char(al); if (!signe(p)) p = NULL;
3593 18305 : res = cgetg(n+1,t_COL);
3594 164353 : for (i = 1; i <= n; i++)
3595 : {
3596 146048 : pari_sp av = avma;
3597 146048 : GEN t = subii(randomi(N),b);
3598 146048 : if (p) t = modii(t, p);
3599 146048 : gel(res,i) = gerepileuptoint(av, t);
3600 : }
3601 18305 : return res;
3602 : }
3603 :
3604 : static GEN
3605 77 : H_poleval(GEN pol, GEN x)
3606 : {
3607 77 : pari_sp av = avma;
3608 : GEN res;
3609 : long i;
3610 77 : switch (H_model(x))
3611 : {
3612 21 : case H_SCALAR: return RgX_cxeval(pol, x, NULL);
3613 42 : case H_QUATERNION: break;
3614 7 : default: pari_err_TYPE("H_poleval", x);
3615 : }
3616 :
3617 42 : res = zerocol(4);
3618 189 : for (i=lg(pol)-1; i>1; i--)
3619 : {
3620 147 : gel(res,1) = gadd(gel(res,1), gel(pol,i));
3621 147 : if (i>2) res = H_mul(x, res);
3622 : }
3623 :
3624 42 : return gerepilecopy(av,res);
3625 : }
3626 :
3627 : /* Assumes pol has coefficients in the same ring as the COL x; x either
3628 : * in basis or algebraic form or [x,mx] where mx is the mult. table of x.
3629 : TODO more general version: pol with coeffs in center and x in basis form */
3630 : GEN
3631 17429 : algpoleval(GEN al, GEN pol, GEN x)
3632 : {
3633 17429 : pari_sp av = avma;
3634 17429 : GEN p, mx = NULL, res;
3635 : long i;
3636 17429 : if (typ(pol) != t_POL) pari_err_TYPE("algpoleval", pol);
3637 17415 : checkalg(al);
3638 17415 : if (alg_type(al)==al_REAL) return H_poleval(pol,x);
3639 17338 : p = alg_get_char(al);
3640 17338 : if (typ(x) == t_VEC)
3641 : {
3642 6097 : if (lg(x) != 3) pari_err_TYPE("algpoleval [vector must be of length 2]", x);
3643 6090 : mx = gel(x,2);
3644 6090 : x = gel(x,1);
3645 6090 : if (typ(mx)!=t_MAT || !gequal(x,gel(mx,1)))
3646 21 : pari_err_TYPE("algpoleval [mx must be the multiplication table of x]", mx);
3647 : }
3648 : else
3649 : {
3650 11241 : switch(alg_model(al,x))
3651 : {
3652 14 : case al_ALGEBRAIC: mx = algalgmultable(al,x); break;
3653 11199 : case al_BASIS: if (!RgX_is_QX(pol))
3654 7 : pari_err_IMPL("algpoleval with x in basis form and pol not in Q[x]");
3655 11206 : case al_TRIVIAL: mx = algbasismultable(al,x); break;
3656 7 : default: pari_err_TYPE("algpoleval", x);
3657 : }
3658 : }
3659 17289 : res = zerocol(lg(mx)-1);
3660 17289 : if (signe(p)) {
3661 64486 : for (i=lg(pol)-1; i>1; i--)
3662 : {
3663 48100 : gel(res,1) = Fp_add(gel(res,1), gel(pol,i), p);
3664 48100 : if (i>2) res = FpM_FpC_mul(mx, res, p);
3665 : }
3666 : }
3667 : else {
3668 5670 : for (i=lg(pol)-1; i>1; i--)
3669 : {
3670 4767 : gel(res,1) = gadd(gel(res,1), gel(pol,i));
3671 4767 : if (i>2) res = RgM_RgC_mul(mx, res);
3672 : }
3673 : }
3674 17289 : return gerepileupto(av, res);
3675 : }
3676 :
3677 : /** GRUNWALD-WANG **/
3678 : /*
3679 : Song Wang's PhD thesis (pdf pages)
3680 : p.25 definition of chi_b. K^Ker(chi_b) = K(b^(1/m))
3681 : p.26 bound on the conductor (also Cohen adv. GTM 193 p.166)
3682 : p.21 & p.34 description special case, also on wikipedia:
3683 : http://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem#Special_fields
3684 : p.77 Kummer case
3685 : */
3686 :
3687 : /* n > 0. Is n = 2^k ? */
3688 : static int
3689 329 : uispow2(ulong n) { return !(n &(n-1)); }
3690 :
3691 : static GEN
3692 378 : get_phi0(GEN bnr, GEN Lpr, GEN Ld, GEN pl, long *pr, long *pn)
3693 : {
3694 378 : const long NTRY = 10; /* FIXME: magic constant */
3695 378 : const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
3696 378 : GEN S = bnr_get_cyc(bnr);
3697 : GEN Sst, G, globGmod, loc, X, Rglob, Rloc, H, U, Lconj;
3698 : long i, j, r, nbfrob, nbloc, nz, t;
3699 :
3700 378 : *pn = n;
3701 378 : *pr = r = lg(S)-1;
3702 378 : if (!r) return NULL;
3703 329 : Sst = cgetg(r+1, t_VECSMALL); /* Z/n-dual */
3704 1589 : for (i=1; i<=r; i++) Sst[i] = ugcdiu(gel(S,i), n);
3705 329 : if (Sst[1] != n) return NULL;
3706 329 : Lconj = NULL;
3707 329 : nbloc = nbfrob = lg(Lpr)-1;
3708 329 : if (uispow2(n))
3709 : {
3710 259 : long l = lg(pl), k = 0;
3711 259 : GEN real = cgetg(l, t_VECSMALL);
3712 973 : for (i = 1; i < l; i++)
3713 714 : if (pl[i] == -1) real[++k] = i;
3714 259 : if (k)
3715 : {
3716 259 : GEN nf = bnr_get_nf(bnr), I = bid_get_fact(bnr_get_bid(bnr));
3717 259 : GEN v, y, C = idealchineseinit(bnr, I);
3718 259 : long r1 = nf_get_r1(nf), n = nbrows(I);
3719 259 : nbloc += k;
3720 259 : Lconj = cgetg(k+1, t_VEC);
3721 259 : v = const_vecsmall(r1, 1);
3722 259 : y = const_vec(n, gen_1);
3723 707 : for (i = 1; i <= k; i++)
3724 : {
3725 448 : v[real[i]] = -1; gel(Lconj,i) = idealchinese(nf, mkvec2(C,v), y);
3726 448 : v[real[i]] = 1;
3727 : }
3728 : }
3729 : }
3730 329 : globGmod = cgetg(r+1,t_MAT);
3731 329 : G = cgetg(r+1,t_VECSMALL);
3732 1589 : for (i = 1; i <= r; i++)
3733 : {
3734 1260 : G[i] = n / Sst[i]; /* pairing between S and Sst */
3735 1260 : gel(globGmod,i) = cgetg(nbloc+1,t_VECSMALL);
3736 : }
3737 :
3738 : /* compute images of Frobenius elements (and complex conjugation) */
3739 329 : loc = cgetg(nbloc+1,t_VECSMALL);
3740 700 : for (i = 1; i <= nbloc; i++)
3741 : {
3742 : long L;
3743 539 : if (i <= nbfrob)
3744 : {
3745 224 : X = gel(Lpr, i);
3746 224 : L = Ld[i];
3747 : }
3748 : else
3749 : { /* X = 1 (mod f), sigma_i(x) < 0, positive at all other real places */
3750 315 : X = gel(Lconj, i-nbfrob);
3751 315 : L = 2;
3752 : }
3753 539 : X = ZV_to_Flv(isprincipalray(bnr,X), n);
3754 2275 : for (nz=0,j=1; j<=r; j++)
3755 : {
3756 1736 : ulong c = (X[j] * G[j]) % L;
3757 1736 : ucoeff(globGmod,i,j) = c;
3758 1736 : if (c) nz = 1;
3759 : }
3760 539 : if (!nz) return NULL;
3761 371 : loc[i] = L;
3762 : }
3763 :
3764 : /* try some random elements in the dual */
3765 161 : Rglob = cgetg(r+1,t_VECSMALL);
3766 443 : for (t=0; t<NTRY; t++) {
3767 1656 : for (j = 1; j <= r; j++) Rglob[j] = random_Fl(Sst[j]);
3768 436 : Rloc = zm_zc_mul(globGmod,Rglob);
3769 968 : for (i = 1; i <= nbloc; i++)
3770 814 : if (Rloc[i] % loc[i] == 0) break;
3771 436 : if (i > nbloc) return zv_to_ZV(Rglob);
3772 : }
3773 :
3774 : /* try to realize some random elements of the product of the local duals */
3775 7 : H = ZM_hnfall_i(shallowconcat(zm_to_ZM(globGmod),
3776 : diagonal_shallow(zv_to_ZV(loc))), &U, 2);
3777 : /* H,U nbloc x nbloc */
3778 7 : Rloc = cgetg(nbloc+1,t_COL);
3779 77 : for (t = 0; t < NTRY; t++)
3780 : { /* nonzero random coordinate */ /* TODO add special case ? */
3781 560 : for (i = 1; i <= nbloc; i++) gel(Rloc,i) = stoi(1 + random_Fl(loc[i]-1));
3782 70 : Rglob = hnf_invimage(H, Rloc);
3783 70 : if (Rglob)
3784 : {
3785 0 : Rglob = ZM_ZC_mul(U,Rglob);
3786 0 : return vecslice(Rglob,1,r);
3787 : }
3788 : }
3789 7 : return NULL;
3790 : }
3791 :
3792 : static GEN
3793 378 : bnrgwsearch(GEN bnr, GEN Lpr, GEN Ld, GEN pl)
3794 : {
3795 378 : pari_sp av = avma;
3796 : long n, r;
3797 378 : GEN phi0 = get_phi0(bnr,Lpr,Ld,pl, &r,&n), gn, v, H,U;
3798 378 : if (!phi0) return gc_const(av, gen_0);
3799 154 : gn = stoi(n);
3800 : /* compute kernel of phi0 */
3801 154 : v = ZV_extgcd(vec_append(phi0, gn));
3802 154 : U = vecslice(gel(v,2), 1,r);
3803 154 : H = ZM_hnfmodid(rowslice(U, 1,r), gn);
3804 154 : return gerepileupto(av, H);
3805 : }
3806 :
3807 : GEN
3808 154 : bnfgwgeneric(GEN bnf, GEN Lpr, GEN Ld, GEN pl, long var)
3809 : {
3810 154 : pari_sp av = avma;
3811 154 : const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
3812 : forprime_t S;
3813 154 : GEN bnr = NULL, ideal = gen_1, nf, dec, H = gen_0, finf, pol;
3814 : ulong ell, p;
3815 : long deg, i, degell;
3816 154 : (void)uisprimepower(n, &ell);
3817 154 : nf = bnf_get_nf(bnf);
3818 154 : deg = nf_get_degree(nf);
3819 154 : degell = ugcd(deg,ell-1);
3820 154 : finf = cgetg(lg(pl),t_VEC);
3821 427 : for (i=1; i<lg(pl); i++) gel(finf,i) = pl[i]==-1 ? gen_1 : gen_0;
3822 :
3823 154 : u_forprime_init(&S, 2, ULONG_MAX);
3824 679 : while ((p = u_forprime_next(&S))) {
3825 679 : if (Fl_powu(p % ell, degell, ell) != 1) continue; /* ell | p^deg-1 ? */
3826 364 : dec = idealprimedec(nf, utoipos(p));
3827 700 : for (i=1; i<lg(dec); i++) {
3828 490 : GEN pp = gel(dec,i);
3829 490 : if (RgV_isin(Lpr,pp)) continue;
3830 : /* TODO also accept the prime ideals at which there is a condition
3831 : * (use local Artin)? */
3832 434 : if (smodis(idealnorm(nf,pp),ell) != 1) continue; /* ell | N(pp)-1 ? */
3833 378 : ideal = idealmul(bnf,ideal,pp);
3834 : /* TODO: give factorization ? */
3835 378 : bnr = Buchray(bnf, mkvec2(ideal,finf), nf_INIT);
3836 378 : H = bnrgwsearch(bnr,Lpr,Ld,pl);
3837 378 : if (H != gen_0)
3838 : {
3839 154 : pol = rnfkummer(bnr,H,nf_get_prec(nf));
3840 154 : setvarn(pol, var);
3841 154 : return gerepileupto(av,pol);
3842 : }
3843 : }
3844 : }
3845 : pari_err_BUG("bnfgwgeneric (no suitable p)"); /*LCOV_EXCL_LINE*/
3846 : return NULL;/*LCOV_EXCL_LINE*/
3847 : }
3848 :
3849 : /* pr.p != ell */
3850 : static GEN
3851 1554 : localextdeg(GEN nf, GEN pr, long d, ulong ell, long n)
3852 : {
3853 : GEN modpr, T, p, gen, k;
3854 1554 : if (d == 1) return gen_1;
3855 1540 : k = powuu(ell, Z_lval(subiu(pr_norm(pr),1), ell));
3856 1540 : k = divis(k, n / d);
3857 1540 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
3858 1540 : (void)Fq_sqrtn(gen_1, k, T, p, &gen);
3859 1540 : return Fq_to_nf(gen, modpr);
3860 : }
3861 : /* pr.p = ell */
3862 : static GEN
3863 133 : localextdegell(GEN nf, GEN pr, GEN E, long d, long n)
3864 : {
3865 : GEN x;
3866 133 : if (d == 1) return gen_1;
3867 126 : x = nfadd(nf, gen_1, pr_get_gen(pr));
3868 126 : return nfpowmodideal(nf, x, stoi(n / d), idealpow(nf, pr, E));
3869 : }
3870 :
3871 : /* Ld[i] must be nontrivial powers of the same prime ell */
3872 : /* pl : -1 at real places at which the extension must ramify, 0 elsewhere */
3873 : GEN
3874 210 : nfgwkummer(GEN nf, GEN Lpr, GEN Ld, GEN pl, long var)
3875 : {
3876 210 : const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
3877 : ulong ell;
3878 210 : long i, l = lg(Lpr), v = uisprimepower(n, &ell);
3879 210 : GEN E = cgetg(l, t_COL), y = cgetg(l, t_VEC), fa;
3880 :
3881 1897 : for (i = 1; i < l; i++)
3882 : {
3883 1687 : GEN pr = gel(Lpr,i), p = pr_get_p(pr);
3884 1687 : if (!absequalui(ell, p))
3885 : {
3886 1554 : gel(E, i) = gen_1;
3887 1554 : gel(y, i) = localextdeg(nf, pr, Ld[i], ell, n);
3888 : }
3889 : else
3890 : {
3891 133 : long e = pr_get_e(pr);
3892 133 : gel(E, i) = addui(1 + v*e, divsi(e, subiu(p,1)));
3893 133 : gel(y, i) = localextdegell(nf, pr, gel(E,i), Ld[i], n);
3894 : }
3895 : }
3896 210 : y = factoredextchinese(nf, mkmat2(shallowtrans(Lpr),E), y, pl, &fa);
3897 210 : return gsub(gpowgs(pol_x(var),n), basistoalg(nf, y));
3898 : }
3899 :
3900 : static GEN
3901 833 : get_vecsmall(GEN v)
3902 : {
3903 833 : switch(typ(v))
3904 : {
3905 707 : case t_VECSMALL: return v;
3906 119 : case t_VEC: if (RgV_is_ZV(v)) return ZV_to_zv(v);
3907 : }
3908 7 : pari_err_TYPE("nfgrunwaldwang",v);
3909 : return NULL;/*LCOV_EXCL_LINE*/
3910 : }
3911 : GEN
3912 462 : nfgrunwaldwang(GEN nf0, GEN Lpr, GEN Ld, GEN pl, long var)
3913 : {
3914 : ulong n, ell, ell2;
3915 462 : pari_sp av = avma;
3916 : GEN nf, bnf;
3917 : long t, w, i, vnf;
3918 :
3919 462 : if (var < 0) var = 0;
3920 462 : nf = get_nf(nf0,&t);
3921 462 : if (!nf) pari_err_TYPE("nfgrunwaldwang",nf0);
3922 462 : vnf = nf_get_varn(nf);
3923 462 : if (varncmp(var, vnf) >= 0)
3924 7 : pari_err_PRIORITY("nfgrunwaldwang", pol_x(var), ">=", vnf);
3925 455 : if (typ(Lpr) != t_VEC) pari_err_TYPE("nfgrunwaldwang",Lpr);
3926 441 : if (lg(Lpr) != lg(Ld)) pari_err_DIM("nfgrunwaldwang [#Lpr != #Ld]");
3927 434 : if (nf_get_degree(nf)==1) Lpr = shallowcopy(Lpr);
3928 2359 : for (i=1; i<lg(Lpr); i++) {
3929 1932 : GEN pr = gel(Lpr,i);
3930 1932 : if (nf_get_degree(nf)==1 && typ(pr)==t_INT)
3931 77 : gel(Lpr,i) = gel(idealprimedec(nf,pr), 1);
3932 1855 : else checkprid(pr);
3933 : }
3934 427 : if (lg(pl)-1 != nf_get_r1(nf))
3935 7 : pari_err_DOMAIN("nfgrunwaldwang [pl should have r1 components]", "#pl",
3936 7 : "!=", stoi(nf_get_r1(nf)), stoi(lg(pl)-1));
3937 :
3938 420 : Ld = get_vecsmall(Ld);
3939 413 : pl = get_vecsmall(pl);
3940 413 : bnf = get_bnf(nf0,&t);
3941 413 : n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
3942 :
3943 413 : if (!uisprimepower(n, &ell))
3944 7 : pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (a)");
3945 2296 : for (i=1; i<lg(Ld); i++)
3946 1897 : if (Ld[i]!=1 && (!uisprimepower(Ld[i],&ell2) || ell2!=ell))
3947 7 : pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (b)");
3948 1043 : for (i=1; i<lg(pl); i++)
3949 651 : if (pl[i]==-1 && ell%2)
3950 7 : pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (c)");
3951 :
3952 392 : w = bnf? bnf_get_tuN(bnf): itos(gel(nfrootsof1(nf),1));
3953 :
3954 : /* TODO choice between kummer and generic ? Let user choose between speed
3955 : * and size */
3956 392 : if (w%n==0 && lg(Ld)>1)
3957 210 : return gerepileupto(av, nfgwkummer(nf,Lpr,Ld,pl,var));
3958 182 : if (ell==n)
3959 : {
3960 154 : if (!bnf) bnf = Buchall(nf, nf_FORCE, 0);
3961 154 : return gerepileupto(av, bnfgwgeneric(bnf,Lpr,Ld,pl,var));
3962 : }
3963 28 : pari_err_IMPL("nfgrunwaldwang for nonprime degree");
3964 : return NULL; /*LCOV_EXCL_LINE*/
3965 : }
3966 :
3967 : /** HASSE INVARIANTS **/
3968 :
3969 : /* TODO long -> ulong + uel */
3970 : static GEN
3971 1064 : hasseconvert(GEN H, long n)
3972 : {
3973 : GEN h, c;
3974 : long i, l;
3975 1064 : switch(typ(H)) {
3976 994 : case t_VEC:
3977 994 : l = lg(H); h = cgetg(l,t_VECSMALL);
3978 994 : if (l == 1) return h;
3979 882 : c = gel(H,1);
3980 882 : if (typ(c) == t_VEC && l == 3)
3981 336 : return mkvec2(gel(H,1),hasseconvert(gel(H,2),n));
3982 2891 : for (i=1; i<l; i++)
3983 : {
3984 2373 : c = gel(H,i);
3985 2373 : switch(typ(c)) {
3986 728 : case t_INT: break;
3987 7 : case t_INTMOD:
3988 7 : c = gel(c,2); break;
3989 1617 : case t_FRAC :
3990 1617 : c = gmulgs(c,n);
3991 1617 : if (typ(c) == t_INT) break;
3992 7 : pari_err_DOMAIN("hasseconvert [degree should be a denominator of the invariant]", "denom(h)", "ndiv", stoi(n), Q_denom(gel(H,i)));
3993 21 : default : pari_err_TYPE("Hasse invariant", c);
3994 : }
3995 2345 : h[i] = smodis(c,n);
3996 : }
3997 518 : return h;
3998 63 : case t_VECSMALL: return H;
3999 : }
4000 7 : pari_err_TYPE("Hasse invariant", H);
4001 : return NULL;/*LCOV_EXCL_LINE*/
4002 : }
4003 :
4004 : /* assume f >= 2 */
4005 : static long
4006 385 : cyclicrelfrob0(GEN nf, GEN aut, GEN pr, GEN q, long f, long g)
4007 : {
4008 385 : GEN T, p, a, b, modpr = nf_to_Fq_init(nf,&pr,&T,&p);
4009 : long s;
4010 :
4011 385 : a = pol_x(nf_get_varn(nf));
4012 385 : b = galoisapply(nf, aut, modpr_genFq(modpr));
4013 385 : b = nf_to_Fq(nf, b, modpr);
4014 1169 : for (s = 0; !ZX_equal(a, b); s++) a = Fq_pow(a, q, T, p);
4015 385 : return g * Fl_inv(s, f); /* < n */
4016 : }
4017 :
4018 : static long
4019 2471 : cyclicrelfrob(GEN rnf, GEN auts, GEN pr)
4020 : {
4021 2471 : pari_sp av = avma;
4022 2471 : long f,g,frob, n = rnf_get_degree(rnf);
4023 2471 : GEN P = rnfidealprimedec(rnf, pr);
4024 :
4025 2471 : if (pr_get_e(gel(P,1)) > pr_get_e(pr))
4026 0 : pari_err_DOMAIN("cyclicrelfrob","e(PR/pr)",">",gen_1,pr);
4027 2471 : g = lg(P) - 1;
4028 2471 : f = n / g;
4029 :
4030 2471 : if (f <= 2) frob = g % n;
4031 : else {
4032 385 : GEN nf2, PR = gel(P,1);
4033 385 : GEN autabs = rnfeltreltoabs(rnf,gel(auts,g));
4034 385 : nf2 = obj_check(rnf,rnf_NFABS);
4035 385 : autabs = nfadd(nf2, autabs, gmul(rnf_get_k(rnf), rnf_get_alpha(rnf)));
4036 385 : frob = cyclicrelfrob0(nf2, autabs, PR, pr_norm(pr), f, g);
4037 : }
4038 2471 : return gc_long(av, frob);
4039 : }
4040 :
4041 : static long
4042 623 : localhasse(GEN rnf, GEN cnd, GEN pl, GEN auts, GEN b, long k)
4043 : {
4044 623 : pari_sp av = avma;
4045 : long v, m, h, lfa, frob, n, i;
4046 : GEN previous, y, pr, nf, q, fa;
4047 623 : nf = rnf_get_nf(rnf);
4048 623 : n = rnf_get_degree(rnf);
4049 623 : pr = gcoeff(cnd,k,1);
4050 623 : v = nfval(nf, b, pr);
4051 623 : m = lg(cnd)>1 ? nbrows(cnd) : 0;
4052 :
4053 : /* add the valuation of b to the conductor... */
4054 623 : previous = gcoeff(cnd,k,2);
4055 623 : gcoeff(cnd,k,2) = addis(previous, v);
4056 :
4057 623 : y = const_vec(m, gen_1);
4058 623 : gel(y,k) = b;
4059 : /* find a factored element y congruent to b mod pr^(vpr(b)+vpr(cnd)) and to 1 mod the conductor. */
4060 623 : y = factoredextchinese(nf, cnd, y, pl, &fa);
4061 623 : h = 0;
4062 623 : lfa = nbrows(fa);
4063 : /* sum of all Hasse invariants of (rnf/nf,aut,y) is 0, Hasse invariants at q!=pr are easy, Hasse invariant at pr is the same as for al=(rnf/nf,aut,b). */
4064 1225 : for (i=1; i<=lfa; i++) {
4065 602 : q = gcoeff(fa,i,1);
4066 602 : if (cmp_prime_ideal(pr,q)) {
4067 567 : frob = cyclicrelfrob(rnf, auts, q);
4068 567 : frob = Fl_mul(frob,umodiu(gcoeff(fa,i,2),n),n);
4069 567 : h = Fl_add(h,frob,n);
4070 : }
4071 : }
4072 : /* ...then restore it. */
4073 623 : gcoeff(cnd,k,2) = previous;
4074 623 : return gc_long(av, Fl_neg(h,n));
4075 : }
4076 :
4077 : static GEN
4078 819 : allauts(GEN rnf, GEN aut)
4079 : {
4080 819 : long n = rnf_get_degree(rnf), i;
4081 819 : GEN pol = rnf_get_pol(rnf), vaut;
4082 819 : if (n==1) n=2;
4083 819 : vaut = cgetg(n,t_VEC);
4084 819 : aut = lift_shallow(rnfbasistoalg(rnf,aut));
4085 819 : if (typ(aut) != t_POL || varn(pol) != varn(aut))
4086 0 : pari_err_TYPE("alg_cyclic", aut);
4087 819 : gel(vaut,1) = aut;
4088 1141 : for (i=1; i<n-1; i++)
4089 322 : gel(vaut,i+1) = RgX_rem(poleval(gel(vaut,i), aut), pol);
4090 819 : return vaut;
4091 : }
4092 :
4093 : static GEN
4094 273 : clean_factor(GEN fa)
4095 : {
4096 273 : GEN P2,E2, P = gel(fa,1), E = gel(fa,2);
4097 273 : long l = lg(P), i, j = 1;
4098 273 : P2 = cgetg(l, t_COL);
4099 273 : E2 = cgetg(l, t_COL);
4100 2339 : for (i = 1;i < l; i++)
4101 2066 : if (signe(gel(E,i))) {
4102 526 : gel(P2,j) = gel(P,i);
4103 526 : gel(E2,j) = gel(E,i); j++;
4104 : }
4105 273 : setlg(P2,j);
4106 273 : setlg(E2,j); return mkmat2(P2,E2);
4107 : }
4108 :
4109 : /* shallow concat x[1],...x[nx],y[1], ... y[ny], returning a t_COL. To be
4110 : * used when we do not know whether x,y are t_VEC or t_COL */
4111 : static GEN
4112 546 : colconcat(GEN x, GEN y)
4113 : {
4114 546 : long i, lx = lg(x), ly = lg(y);
4115 546 : GEN z=cgetg(lx+ly-1, t_COL);
4116 3696 : for (i=1; i<lx; i++) z[i] = x[i];
4117 1528 : for (i=1; i<ly; i++) z[lx+i-1]= y[i];
4118 546 : return z;
4119 : }
4120 :
4121 : /* return v(x) at all primes in listpr, replace x by cofactor */
4122 : static GEN
4123 1092 : nfmakecoprime(GEN nf, GEN *px, GEN listpr)
4124 : {
4125 1092 : long j, l = lg(listpr);
4126 1092 : GEN x1, x = *px, L = cgetg(l, t_COL);
4127 :
4128 1092 : if (typ(x) != t_MAT)
4129 : { /* scalar, divide at the end (fast valuation) */
4130 945 : x1 = NULL;
4131 4943 : for (j=1; j<l; j++)
4132 : {
4133 3998 : GEN pr = gel(listpr,j), e;
4134 3998 : long v = nfval(nf, x, pr);
4135 3998 : e = stoi(v); gel(L,j) = e;
4136 5587 : if (v) x1 = x1? idealmulpowprime(nf, x1, pr, e)
4137 1589 : : idealpow(nf, pr, e);
4138 : }
4139 945 : if (x1) x = idealdivexact(nf, idealhnf(nf,x), x1);
4140 : }
4141 : else
4142 : { /* HNF, divide as we proceed (reduce size) */
4143 273 : for (j=1; j<l; j++)
4144 : {
4145 126 : GEN pr = gel(listpr,j);
4146 126 : long v = idealval(nf, x, pr);
4147 126 : gel(L,j) = stoi(v);
4148 126 : if (v) x = idealmulpowprime(nf, x, pr, stoi(-v));
4149 : }
4150 : }
4151 1092 : *px = x; return L;
4152 : }
4153 :
4154 : /* Caveat: factorizations are not sorted wrt cmp_prime_ideal: Lpr comes first */
4155 : static GEN
4156 273 : computecnd(GEN rnf, GEN Lpr)
4157 : {
4158 : GEN id, nf, fa, Le, P,E;
4159 273 : long n = rnf_get_degree(rnf);
4160 :
4161 273 : nf = rnf_get_nf(rnf);
4162 273 : id = rnf_get_idealdisc(rnf);
4163 273 : Le = nfmakecoprime(nf, &id, Lpr);
4164 273 : fa = idealfactor(nf, id); /* part of D_{L/K} coprime with Lpr */
4165 273 : P = colconcat(Lpr,gel(fa,1));
4166 273 : E = colconcat(Le, gel(fa,2));
4167 273 : fa = mkmat2(P, gdiventgs(E, eulerphiu(n)));
4168 273 : return mkvec2(fa, clean_factor(fa));
4169 : }
4170 :
4171 : /* h >= 0 */
4172 : static void
4173 35 : nextgen(GEN gene, long h, GEN* gens, GEN* hgens, long* ngens, long* curgcd) {
4174 35 : long nextgcd = ugcd(h,*curgcd);
4175 35 : if (nextgcd == *curgcd) return;
4176 35 : (*ngens)++;
4177 35 : gel(*gens,*ngens) = gene;
4178 35 : gel(*hgens,*ngens) = utoi(h);
4179 35 : *curgcd = nextgcd;
4180 35 : return;
4181 : }
4182 :
4183 : static int
4184 49 : dividesmod(long d, long h, long n) { return !(h%cgcd(d,n)); }
4185 :
4186 : /* ramified prime with nontrivial Hasse invariant */
4187 : static GEN
4188 35 : localcomplete(GEN rnf, GEN pl, GEN cnd, GEN auts, long j, long n, long h, long* v)
4189 : {
4190 : GEN nf, gens, hgens, pr, modpr, T, p, sol, U, b, gene, randg, pu;
4191 : long ngens, i, d, np, d1, d2, hg, dnf, vcnd, curgcd;
4192 35 : nf = rnf_get_nf(rnf);
4193 35 : pr = gcoeff(cnd,j,1);
4194 35 : np = umodiu(pr_norm(pr), n);
4195 35 : dnf = nf_get_degree(nf);
4196 35 : vcnd = itos(gcoeff(cnd,j,2));
4197 35 : ngens = 13+dnf;
4198 35 : gens = zerovec(ngens);
4199 35 : hgens = zerovec(ngens);
4200 35 : *v = 0;
4201 35 : curgcd = 0;
4202 35 : ngens = 0;
4203 :
4204 35 : if (!uisprime(n)) {
4205 0 : gene = pr_get_gen(pr);
4206 0 : hg = localhasse(rnf, cnd, pl, auts, gene, j);
4207 0 : nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
4208 : }
4209 :
4210 35 : if (ugcd(np,n) != 1) { /* GCD(Np,n) != 1 */
4211 35 : pu = idealprincipalunits(nf,pr,vcnd);
4212 35 : pu = abgrp_get_gen(pu);
4213 70 : for (i=1; i<lg(pu) && !dividesmod(curgcd,h,n); i++) {
4214 35 : gene = gel(pu,i);
4215 35 : hg = localhasse(rnf, cnd, pl, auts, gene, j);
4216 35 : nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
4217 : }
4218 : }
4219 :
4220 35 : d = ugcd(np-1,n);
4221 35 : if (d != 1) { /* GCD(Np-1,n) != 1 */
4222 7 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
4223 7 : while (!dividesmod(curgcd,h,n)) { /* TODO gener_FpXQ_local */
4224 0 : if (T==NULL) randg = randomi(p);
4225 0 : else randg = random_FpX(degpol(T), varn(T),p);
4226 :
4227 0 : if (!gequal0(randg) && !gequal1(randg)) {
4228 0 : gene = Fq_to_nf(randg, modpr);
4229 0 : hg = localhasse(rnf, cnd, pl, auts, gene, j);
4230 0 : nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
4231 : }
4232 : }
4233 : }
4234 :
4235 35 : setlg(gens,ngens+1);
4236 35 : setlg(hgens,ngens+1);
4237 :
4238 35 : sol = ZV_extgcd(hgens);
4239 35 : U = ZV_to_Flv(gmael(sol,2,ngens), n);
4240 35 : d = itou(gel(sol,1));
4241 35 : d1 = ugcd(d, n);
4242 35 : d2 = d / d1;
4243 35 : d = Fl_mul(h / d1, Fl_inv(d2,n), n);
4244 35 : if (d != 1) U = Flv_Fl_mul(U, d, n);
4245 70 : for (i = 1, b = gen_1; i <= ngens; i++)
4246 35 : if (U[i]) b = nfmul(nf, b, nfpow_u(nf, gel(gens,i), U[i]));
4247 35 : *v = U[1]; return b;
4248 : }
4249 :
4250 : static int
4251 828 : testsplits(GEN data, GEN fa)
4252 : {
4253 828 : GEN rnf = gel(data,1), forbid = gel(data,2), P = gel(fa,1), E = gel(fa,2);
4254 828 : long i, n, l = lg(P);
4255 :
4256 1167 : for (i = 1; i < l; i++)
4257 : {
4258 823 : GEN pr = gel(P,i);
4259 823 : if (tablesearch(forbid, pr, &cmp_prime_ideal)) return 0;
4260 : }
4261 344 : n = rnf_get_degree(rnf);
4262 510 : for (i = 1; i < l; i++)
4263 : {
4264 237 : long e = itos(gel(E,i)) % n;
4265 237 : if (e)
4266 : {
4267 223 : GEN L = rnfidealprimedec(rnf, gel(P,i));
4268 223 : long g = lg(L) - 1;
4269 223 : if ((e * g) % n) return 0;
4270 : }
4271 : }
4272 273 : return 1;
4273 : }
4274 :
4275 : /* remove entries with Hasse invariant 0 */
4276 : static GEN
4277 574 : hassereduce(GEN hf)
4278 : {
4279 574 : GEN pr,h, PR = gel(hf,1), H = gel(hf,2);
4280 574 : long i, j, l = lg(PR);
4281 :
4282 574 : pr= cgetg(l, t_VEC);
4283 574 : h = cgetg(l, t_VECSMALL);
4284 4081 : for (i = j = 1; i < l; i++)
4285 3507 : if (H[i]) {
4286 3178 : gel(pr,j) = gel(PR,i);
4287 3178 : h[j] = H[i]; j++;
4288 : }
4289 574 : setlg(pr,j);
4290 574 : setlg(h,j); return mkvec2(pr,h);
4291 : }
4292 :
4293 : /* rnf complete */
4294 : static GEN
4295 273 : alg_complete0(GEN rnf, GEN aut, GEN hf, GEN hi, long flag)
4296 : {
4297 273 : pari_sp av = avma;
4298 : GEN nf, pl, pl2, cnd, prcnd, cnds, y, Lpr, auts, b, fa, data, hfe;
4299 : GEN forbid, al, ind;
4300 : long D, n, d, i, j, l;
4301 273 : nf = rnf_get_nf(rnf);
4302 273 : n = rnf_get_degree(rnf);
4303 273 : d = nf_get_degree(nf);
4304 273 : D = d*n*n;
4305 273 : checkhasse(nf,hf,hi,n);
4306 273 : hf = hassereduce(hf);
4307 273 : Lpr = gel(hf,1);
4308 273 : hfe = gel(hf,2);
4309 :
4310 273 : auts = allauts(rnf,aut);
4311 :
4312 273 : pl = leafcopy(hi); /* conditions on the final b */
4313 273 : pl2 = leafcopy(hi); /* conditions for computing local Hasse invariants */
4314 273 : l = lg(pl); ind = cgetg(l, t_VECSMALL);
4315 686 : for (i = j = 1; i < l; i++)
4316 413 : if (hi[i]) { pl[i] = -1; pl2[i] = 1; } else ind[j++] = i;
4317 273 : setlg(ind, j);
4318 273 : y = nfpolsturm(nf, rnf_get_pol(rnf), ind);
4319 511 : for (i = 1; i < j; i++)
4320 238 : if (!signe(gel(y,i))) { pl[ind[i]] = 1; pl2[ind[i]] = 1; }
4321 :
4322 273 : cnds = computecnd(rnf,Lpr);
4323 273 : prcnd = gel(cnds,1);
4324 273 : cnd = gel(cnds,2);
4325 273 : y = cgetg(lgcols(prcnd),t_VEC);
4326 273 : forbid = vectrunc_init(lg(Lpr));
4327 1848 : for (i=j=1; i<lg(Lpr); i++)
4328 : {
4329 1575 : GEN pr = gcoeff(prcnd,i,1), yi;
4330 1575 : long v, e = itou( gcoeff(prcnd,i,2) );
4331 1575 : if (!e) {
4332 1540 : long frob = cyclicrelfrob(rnf,auts,pr), f1 = ugcd(frob,n);
4333 1540 : vectrunc_append(forbid, pr);
4334 1540 : yi = gen_0;
4335 1540 : v = ((hfe[i]/f1) * Fl_inv(frob/f1,n)) % n;
4336 : }
4337 : else
4338 35 : yi = localcomplete(rnf, pl2, cnd, auts, j++, n, hfe[i], &v);
4339 1575 : gel(y,i) = yi;
4340 1575 : gcoeff(prcnd,i,2) = stoi(e + v);
4341 : }
4342 764 : for (; i<lgcols(prcnd); i++) gel(y,i) = gen_1;
4343 273 : gen_sort_inplace(forbid, (void*)&cmp_prime_ideal, &cmp_nodata, NULL);
4344 273 : data = mkvec2(rnf,forbid);
4345 273 : b = factoredextchinesetest(nf,prcnd,y,pl,&fa,data,testsplits);
4346 :
4347 273 : al = cgetg(12, t_VEC);
4348 273 : gel(al,10)= gen_0; /* must be set first */
4349 273 : gel(al,1) = rnf;
4350 273 : gel(al,2) = auts;
4351 273 : gel(al,3) = basistoalg(nf,b);
4352 273 : gel(al,4) = hi;
4353 : /* add primes | disc or b with trivial Hasse invariant to hf */
4354 273 : Lpr = gel(prcnd,1); y = b;
4355 273 : (void)nfmakecoprime(nf, &y, Lpr);
4356 273 : Lpr = shallowconcat(Lpr, gel(idealfactor(nf,y), 1));
4357 273 : settyp(Lpr,t_VEC);
4358 273 : hf = mkvec2(Lpr, shallowconcat(hfe, const_vecsmall(lg(Lpr)-lg(hfe), 0)));
4359 273 : gel(al,5) = hf;
4360 273 : gel(al,6) = gen_0;
4361 273 : gel(al,7) = matid(D);
4362 273 : gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
4363 273 : gel(al,9) = algnatmultable(al,D);
4364 273 : gel(al,11)= algtracebasis(al);
4365 273 : if (flag & al_MAXORD) al = alg_maximal_primes(al, prV_primes(Lpr));
4366 273 : return gerepilecopy(av, al);
4367 : }
4368 :
4369 : GEN
4370 0 : alg_complete(GEN rnf, GEN aut, GEN hf, GEN hi, long flag)
4371 : {
4372 0 : long n = rnf_get_degree(rnf);
4373 0 : rnfcomplete(rnf);
4374 0 : return alg_complete0(rnf, aut, hasseconvert(hf,n), hasseconvert(hi,n), flag);
4375 : }
4376 :
4377 : void
4378 1421 : checkhasse(GEN nf, GEN hf, GEN hi, long n)
4379 : {
4380 : GEN Lpr, Lh;
4381 : long i, sum;
4382 1421 : if (typ(hf) != t_VEC || lg(hf) != 3) pari_err_TYPE("checkhasse [hf]", hf);
4383 1414 : Lpr = gel(hf,1);
4384 1414 : Lh = gel(hf,2);
4385 1414 : if (typ(Lpr) != t_VEC) pari_err_TYPE("checkhasse [Lpr]", Lpr);
4386 1414 : if (typ(Lh) != t_VECSMALL) pari_err_TYPE("checkhasse [Lh]", Lh);
4387 1414 : if (typ(hi) != t_VECSMALL) pari_err_TYPE("checkhasse [hi]", hi);
4388 1414 : if ((nf && lg(hi) != nf_get_r1(nf)+1))
4389 7 : pari_err_DOMAIN("checkhasse [hi should have r1 components]","#hi","!=",stoi(nf_get_r1(nf)),stoi(lg(hi)-1));
4390 1407 : if (lg(Lpr) != lg(Lh))
4391 7 : pari_err_DIM("checkhasse [Lpr and Lh should have same length]");
4392 7455 : for (i=1; i<lg(Lpr); i++) checkprid(gel(Lpr,i));
4393 1400 : if (lg(gen_sort_uniq(Lpr, (void*)cmp_prime_ideal, cmp_nodata)) < lg(Lpr))
4394 7 : pari_err(e_MISC, "error in checkhasse [duplicate prime ideal]");
4395 1393 : sum = 0;
4396 7434 : for (i=1; i<lg(Lh); i++) sum = (sum+Lh[i])%n;
4397 3234 : for (i=1; i<lg(hi); i++) {
4398 1855 : if (hi[i] && 2*hi[i] != n) pari_err_DOMAIN("checkhasse", "Hasse invariant at real place [must be 0 or 1/2]", "!=", n%2? gen_0 : stoi(n/2), stoi(hi[i]));
4399 1841 : sum = (sum+hi[i])%n;
4400 : }
4401 1379 : if (sum<0) sum = n+sum;
4402 1379 : if (sum != 0)
4403 7 : pari_err_DOMAIN("checkhasse","sum(Hasse invariants)","!=",gen_0,Lh);
4404 1372 : }
4405 :
4406 : static GEN
4407 371 : hassecoprime(GEN hf, GEN hi, long n)
4408 : {
4409 371 : pari_sp av = avma;
4410 : long l, i, j, lk, inv;
4411 : GEN fa, P,E, res, hil, hfl;
4412 371 : hi = hasseconvert(hi, n);
4413 357 : hf = hasseconvert(hf, n);
4414 336 : checkhasse(NULL,hf,hi,n);
4415 294 : fa = factoru(n);
4416 294 : P = gel(fa,1); l = lg(P);
4417 294 : E = gel(fa,2);
4418 294 : res = cgetg(l,t_VEC);
4419 595 : for (i=1; i<l; i++) {
4420 301 : lk = upowuu(P[i],E[i]);
4421 301 : inv = Fl_invsafe((n/lk)%lk, lk);
4422 301 : hil = gcopy(hi);
4423 301 : hfl = gcopy(hf);
4424 :
4425 301 : if (P[i] == 2)
4426 651 : for (j=1; j<lg(hil); j++) hil[j] = hi[j]==0 ? 0 : lk/2;
4427 : else
4428 98 : for (j=1; j<lg(hil); j++) hil[j] = 0;
4429 2233 : for (j=1; j<lgcols(hfl); j++) gel(hfl,2)[j] = (gel(hf,2)[j]*inv)%lk;
4430 301 : hfl = hassereduce(hfl);
4431 301 : gel(res,i) = mkvec3(hfl,hil,utoi(lk));
4432 : }
4433 :
4434 294 : return gerepilecopy(av, res);
4435 : }
4436 :
4437 : /* no garbage collection */
4438 : static GEN
4439 77 : genefrob(GEN nf, GEN gal, GEN r)
4440 : {
4441 : long i;
4442 77 : GEN g = identity_perm(nf_get_degree(nf)), fa = Z_factor(r), p, pr, frob;
4443 126 : for (i=1; i<lgcols(fa); i++) {
4444 49 : p = gcoeff(fa,i,1);
4445 49 : pr = idealprimedec(nf, p);
4446 49 : pr = gel(pr,1);
4447 49 : frob = idealfrobenius(nf, gal, pr);
4448 49 : g = perm_mul(g, perm_pow(frob, gcoeff(fa,i,2)));
4449 : }
4450 77 : return g;
4451 : }
4452 :
4453 : static GEN
4454 273 : rnfcycaut(GEN rnf)
4455 : {
4456 273 : GEN nf2 = obj_check(rnf, rnf_NFABS);
4457 : GEN L, alpha, pol, salpha, s, sj, polabs, k, X, pol0, nf;
4458 : long i, d, j;
4459 273 : d = rnf_get_degree(rnf);
4460 273 : L = galoisconj(nf2,NULL);
4461 273 : alpha = lift_shallow(rnf_get_alpha(rnf));
4462 273 : pol = rnf_get_pol(rnf);
4463 273 : k = rnf_get_k(rnf);
4464 273 : polabs = rnf_get_polabs(rnf);
4465 273 : nf = rnf_get_nf(rnf);
4466 273 : pol0 = nf_get_pol(nf);
4467 273 : X = RgX_rem(pol_x(varn(pol0)), pol0);
4468 :
4469 : /* TODO check mod prime of degree 1 */
4470 386 : for (i=1; i<lg(L); i++) {
4471 386 : s = gel(L,i);
4472 386 : salpha = RgX_RgXQ_eval(alpha,s,polabs);
4473 386 : if (!gequal(alpha,salpha)) continue;
4474 :
4475 336 : s = lift_shallow(rnfeltabstorel(rnf,s));
4476 336 : sj = s = gsub(s, gmul(k,X));
4477 651 : for (j=1; !gequal0(gsub(sj,pol_x(varn(s)))); j++)
4478 315 : sj = RgX_RgXQ_eval(sj,s,pol);
4479 336 : if (j<d) continue;
4480 273 : return s;
4481 : }
4482 : return NULL; /*LCOV_EXCL_LINE*/
4483 : }
4484 :
4485 : /* returns the smallest prime not in P */
4486 : static GEN
4487 84 : extraprime(GEN P)
4488 : {
4489 : forprime_t T;
4490 : GEN p;
4491 84 : forprime_init(&T, gen_2, NULL);
4492 98 : while ((p = forprime_next(&T))) if (!ZV_search(P, p)) break;
4493 84 : return p;
4494 : }
4495 :
4496 : /* true nf */
4497 : GEN
4498 385 : alg_hasse(GEN nf, long n, GEN hf, GEN hi, long var, long flag)
4499 : {
4500 385 : pari_sp av = avma;
4501 385 : GEN primary, al = gen_0, al2, rnf, hil, hfl, Ld, pl, pol, Lpr, aut, Lpr2, Ld2;
4502 : long i, lk, j, maxdeg;
4503 385 : dbg_printf(1)("alg_hasse\n");
4504 385 : if (n<=1) pari_err_DOMAIN("alg_hasse", "degree", "<=", gen_1, stoi(n));
4505 371 : primary = hassecoprime(hf, hi, n);
4506 574 : for (i=1; i<lg(primary); i++) {
4507 301 : lk = itos(gmael(primary,i,3));
4508 301 : hfl = gmael(primary,i,1);
4509 301 : hil = gmael(primary,i,2);
4510 301 : checkhasse(nf, hfl, hil, lk);
4511 294 : dbg_printf(1)("alg_hasse: i=%d hf=%Ps hi=%Ps lk=%d\n", i, hfl, hil, lk);
4512 :
4513 294 : if (lg(gel(hfl,1))>1 || lk%2==0) {
4514 287 : maxdeg = 1;
4515 287 : Lpr = gel(hfl,1);
4516 287 : Ld = gcopy(gel(hfl,2));
4517 1876 : for (j=1; j<lg(Ld); j++)
4518 : {
4519 1589 : Ld[j] = lk/ugcd(lk,Ld[j]);
4520 1589 : maxdeg = maxss(Ld[j],maxdeg);
4521 : }
4522 287 : pl = leafcopy(hil);
4523 714 : for (j=1; j<lg(pl); j++) if(pl[j])
4524 : {
4525 175 : pl[j] = -1;
4526 175 : maxdeg = maxss(maxdeg,2);
4527 : }
4528 :
4529 287 : Lpr2 = Lpr;
4530 287 : Ld2 = Ld;
4531 287 : if (maxdeg<lk)
4532 : {
4533 154 : if (maxdeg==1 && lk==2 && lg(pl)>1) pl[1] = -1;
4534 : else
4535 : {
4536 84 : GEN p = extraprime(prV_primes(Lpr));
4537 84 : Lpr2 = vec_append(Lpr2, idealprimedec_galois(nf, p));
4538 84 : Ld2 = vecsmall_append(Ld2, lk);
4539 : }
4540 : }
4541 :
4542 287 : dbg_printf(2)("alg_hasse: calling nfgrunwaldwang Lpr=%Ps Pd=%Ps pl=%Ps\n",
4543 : Lpr, Ld, pl);
4544 287 : pol = nfgrunwaldwang(nf, Lpr2, Ld2, pl, var);
4545 273 : dbg_printf(2)("alg_hasse: calling rnfinit(%Ps)\n", pol);
4546 273 : rnf = rnfinit0(nf,pol,1);
4547 273 : dbg_printf(2)("alg_hasse: computing automorphism\n");
4548 273 : aut = rnfcycaut(rnf);
4549 273 : dbg_printf(2)("alg_hasse: calling alg_complete\n");
4550 273 : al2 = alg_complete0(rnf, aut, hfl, hil, flag);
4551 : }
4552 7 : else al2 = alg_matrix(nf, lk, var, flag);
4553 :
4554 280 : if (i==1) al = al2;
4555 7 : else al = algtensor(al,al2,flag);
4556 : }
4557 273 : return gerepilecopy(av,al);
4558 : }
4559 :
4560 : /** CYCLIC ALGEBRA WITH GIVEN HASSE INVARIANTS **/
4561 :
4562 : /* no garbage collection */
4563 : static GEN
4564 77 : subcycloindep(GEN nf, long n, long v, GEN *pr)
4565 : {
4566 : pari_sp av;
4567 : forprime_t S;
4568 : ulong p;
4569 77 : u_forprime_arith_init(&S, 1, ULONG_MAX, 1, n);
4570 77 : av = avma;
4571 84 : while ((p = u_forprime_next(&S)))
4572 : {
4573 84 : ulong r = pgener_Fl(p);
4574 84 : GEN pol = galoissubcyclo(utoipos(p), utoipos(Fl_powu(r,n,p)), 0, v);
4575 84 : GEN fa = nffactor(nf, pol);
4576 84 : if (lgcols(fa) == 2) { *pr = utoipos(r); return pol; }
4577 7 : set_avma(av);
4578 : }
4579 : pari_err_BUG("subcycloindep (no suitable prime = 1(mod n))"); /*LCOV_EXCL_LINE*/
4580 : *pr = NULL; return NULL; /*LCOV_EXCL_LINE*/
4581 : }
4582 :
4583 : GEN
4584 84 : alg_matrix(GEN nf, long n, long v, long flag)
4585 : {
4586 84 : pari_sp av = avma;
4587 : GEN pol, gal, rnf, cyclo, g, r, aut;
4588 84 : dbg_printf(1)("alg_matrix\n");
4589 84 : if (n<=0) pari_err_DOMAIN("alg_matrix", "n", "<=", gen_0, stoi(n));
4590 77 : pol = subcycloindep(nf, n, v, &r);
4591 77 : rnf = rnfinit(nf, pol);
4592 77 : cyclo = nfinit(pol, nf_get_prec(nf));
4593 77 : gal = galoisinit(cyclo, NULL);
4594 77 : g = genefrob(cyclo,gal,r);
4595 77 : aut = galoispermtopol(gal,g);
4596 77 : return gerepileupto(av, alg_cyclic(rnf, aut, gen_1, flag));
4597 : }
4598 :
4599 : GEN
4600 329 : alg_hilbert(GEN nf, GEN a, GEN b, long v, long flag)
4601 : {
4602 329 : pari_sp av = avma;
4603 : GEN rnf, aut, rnfpol;
4604 329 : dbg_printf(1)("alg_hilbert\n");
4605 329 : if (!isint1(Q_denom(a)))
4606 7 : pari_err_DOMAIN("alg_hilbert", "denominator(a)", "!=", gen_1,a);
4607 322 : if (!isint1(Q_denom(b)))
4608 7 : pari_err_DOMAIN("alg_hilbert", "denominator(b)", "!=", gen_1,b);
4609 :
4610 315 : if (v < 0) v = 0;
4611 315 : rnfpol = deg2pol_shallow(gen_1, gen_0, gneg(a), v);
4612 315 : if (!(flag & al_FACTOR)) rnfpol = mkvec2(rnfpol, stoi(1<<20));
4613 315 : rnf = rnfinit(nf, rnfpol);
4614 308 : aut = gneg(pol_x(v));
4615 308 : return gerepileupto(av, alg_cyclic(rnf, aut, b, flag));
4616 : }
4617 :
4618 : /* return a structure representing the algebra of real numbers */
4619 : static GEN
4620 14 : mk_R()
4621 : {
4622 14 : pari_sp av = avma;
4623 : GEN al;
4624 14 : al = zerovec(11);
4625 14 : gel(al,1) = stor(1,3);
4626 14 : gel(al,2) = mkvec(gel(al,1));
4627 14 : gel(al,3) = gen_1;
4628 14 : gel(al,4) = mkvecsmall(0);
4629 14 : gel(al,8) = gel(al,7) = matid(1);
4630 14 : gel(al,9) = mkvec(matid(1));
4631 14 : return gerepilecopy(av,al);
4632 : }
4633 : /* return a structure representing the algebra of complex numbers */
4634 : static GEN
4635 14 : mk_C()
4636 : {
4637 14 : pari_sp av = avma;
4638 : GEN al, I;
4639 14 : al = zerovec(11);
4640 14 : I = gen_I();
4641 14 : gel(al,1) = I;
4642 14 : gel(al,2) = mkvec(I);
4643 14 : gel(al,3) = gen_1;
4644 14 : gel(al,4) = cgetg(1,t_VECSMALL);
4645 14 : gel(al,8) = gel(al,7) = matid(2);
4646 14 : gel(al,9) = mkvec2(
4647 : matid(2),
4648 : mkmat22(gen_0,gen_m1,gen_1,gen_0)
4649 : );
4650 14 : return gerepilecopy(av,al);
4651 : }
4652 : /* return a structure representing the Hamilton quaternion algebra */
4653 : static GEN
4654 14 : mk_H()
4655 : {
4656 14 : pari_sp av = avma;
4657 : GEN al, I;
4658 14 : al = zerovec(11);
4659 14 : I = gen_I();
4660 14 : gel(al,1) = I;
4661 14 : gel(al,2) = mkvec(gconj(I));
4662 14 : gel(al,3) = gen_m1;
4663 14 : gel(al,4) = mkvecsmall(1);
4664 14 : gel(al,8) = gel(al,7) = matid(4);
4665 14 : gel(al,9) = mkvec4(
4666 : matid(4),
4667 : H_tomatrix(I,1),
4668 : H_tomatrix(mkcol4(gen_0,gen_0,gen_1,gen_0),1),
4669 : H_tomatrix(mkcol4(gen_0,gen_0,gen_0,gen_1),1)
4670 : );
4671 14 : return gerepilecopy(av,al);
4672 : }
4673 :
4674 : GEN
4675 1239 : alginit(GEN A, GEN B, long v, long flag)
4676 : {
4677 : long w;
4678 1239 : if (typ(A) == t_COMPLEX) return mk_C();
4679 1225 : if (typ(A) == t_REAL)
4680 : {
4681 35 : if (is_scalar_t(typ(B)) && gequal0(B)) return mk_R();
4682 21 : if (typ(B) == t_FRAC && gequal(B, mkfrac(gen_1,gen_2))) return mk_H();
4683 7 : pari_err_DOMAIN("alginit", "real Hasse invariant [must be 0 or 1/2]", "", NULL, B);
4684 : }
4685 1190 : switch(nftyp(A))
4686 : {
4687 1001 : case typ_NF:
4688 1001 : if (v<0) v=0;
4689 1001 : w = gvar(nf_get_pol(A));
4690 1001 : if (varncmp(v,w)>=0) pari_err_PRIORITY("alginit", pol_x(v), ">=", w);
4691 987 : switch(typ(B))
4692 : {
4693 : long nB;
4694 77 : case t_INT: return alg_matrix(A, itos(B), v, flag);
4695 903 : case t_VEC:
4696 903 : nB = lg(B)-1;
4697 903 : if (nB && typ(gel(B,1)) == t_MAT) return alg_csa_table(A,B,v,flag);
4698 : switch(nB)
4699 : {
4700 329 : case 2: return alg_hilbert(A, gel(B,1), gel(B,2), v, flag);
4701 392 : case 3:
4702 392 : if (typ(gel(B,1))!=t_INT)
4703 7 : pari_err_TYPE("alginit [degree should be an integer]", gel(B,1));
4704 385 : return alg_hasse(A, itos(gel(B,1)), gel(B,2), gel(B,3), v,
4705 : flag);
4706 : }
4707 : }
4708 14 : pari_err_TYPE("alginit", B); break;
4709 :
4710 175 : case typ_RNF:
4711 175 : if (typ(B) != t_VEC || lg(B) != 3) pari_err_TYPE("alginit", B);
4712 161 : return alg_cyclic(A, gel(B,1), gel(B,2), flag);
4713 : }
4714 14 : pari_err_TYPE("alginit", A);
4715 : return NULL;/*LCOV_EXCL_LINE*/
4716 : }
4717 :
4718 : /* assumes al CSA or CYCLIC */
4719 : static GEN
4720 966 : algnatmultable(GEN al, long D)
4721 : {
4722 : GEN res, x;
4723 : long i;
4724 966 : res = cgetg(D+1,t_VEC);
4725 11291 : for (i=1; i<=D; i++) {
4726 10325 : x = algnattoalg(al,col_ei(D,i));
4727 10325 : gel(res,i) = algZmultable(al,x);
4728 : }
4729 966 : return res;
4730 : }
4731 :
4732 140 : static int normfact_is_partial(GEN nf, GEN x, GEN fax)
4733 : {
4734 : long i;
4735 : GEN nfx;
4736 140 : nfx = RgM_shallowcopy(fax);
4737 385 : for (i=1; i<lg(gel(nfx,1)); i++)
4738 245 : gcoeff(nfx,i,1) = idealnorm(nf, gcoeff(nfx,i,1));
4739 140 : nfx = factorback(nfx);
4740 140 : return !gequal(idealnorm(nf, x), nfx);
4741 : }
4742 : /* no garbage collection */
4743 : static void
4744 546 : algcomputehasse(GEN al, long flag)
4745 : {
4746 : int partialfact;
4747 : long r1, k, n, m, m1, m2, m3, i, m23, m123;
4748 : GEN rnf, nf, b, fab, disc2, cnd, fad, auts, pr, pl, perm, y, hi, PH, H, L;
4749 :
4750 546 : rnf = alg_get_splittingfield(al);
4751 546 : n = rnf_get_degree(rnf);
4752 546 : nf = rnf_get_nf(rnf);
4753 546 : b = alg_get_b(al);
4754 546 : r1 = nf_get_r1(nf);
4755 546 : auts = alg_get_auts(al);
4756 546 : (void)alg_get_abssplitting(al);
4757 :
4758 546 : y = nfpolsturm(nf, rnf_get_pol(rnf), NULL);
4759 546 : pl = cgetg(r1+1, t_VECSMALL);
4760 : /* real places where rnf/nf ramifies */
4761 1134 : for (k = 1; k <= r1; k++) pl[k] = !signe(gel(y,k));
4762 :
4763 : /* infinite Hasse invariants */
4764 546 : if (odd(n)) hi = const_vecsmall(r1, 0);
4765 : else
4766 : {
4767 462 : GEN s = nfsign(nf, b);
4768 462 : hi = cgetg(r1+1, t_VECSMALL);
4769 994 : for (k = 1; k<=r1; k++) hi[k] = (s[k] && pl[k]) ? (n/2) : 0;
4770 : }
4771 546 : gel(al,4) = hi;
4772 :
4773 546 : partialfact = 0;
4774 546 : if (flag & al_FACTOR)
4775 462 : fab = idealfactor(nf, b);
4776 : else {
4777 84 : fab = idealfactor_limit(nf, b, 1<<20);
4778 : /* does not report whether factorisation was partial; check it */
4779 84 : partialfact = normfact_is_partial(nf, b, fab);
4780 : }
4781 :
4782 546 : disc2 = rnf_get_idealdisc(rnf);
4783 546 : L = nfmakecoprime(nf, &disc2, gel(fab,1));
4784 546 : m = lg(L)-1;
4785 : /* m1 = #{pr|b: pr \nmid disc}, m3 = #{pr|b: pr | disc} */
4786 546 : perm = cgetg(m+1, t_VECSMALL);
4787 1029 : for (i=1, m1=m, k=1; k<=m; k++)
4788 483 : if (signe(gel(L,k))) perm[m1--] = k; else perm[i++] = k;
4789 546 : m3 = m - m1;
4790 :
4791 : /* disc2 : factor of disc coprime to b */
4792 546 : if (flag & al_FACTOR)
4793 462 : fad = idealfactor(nf, disc2);
4794 : else {
4795 84 : fad = idealfactor_limit(nf, disc2, 1<<20);
4796 84 : partialfact = partialfact || normfact_is_partial(nf, disc2, fad);
4797 : }
4798 :
4799 : /* if factorisation is partial, do not compute Hasse invariants */
4800 : /* we could compute their sum at composite factors */
4801 546 : if (partialfact)
4802 : {
4803 35 : if (!(flag & al_MAXORD))
4804 : {
4805 28 : gel(al,5) = gen_0;
4806 35 : return;
4807 : }
4808 : /* but transmit list of factors found for computation of maximal order */
4809 7 : PH = prV_primes(shallowconcat(gel(fab,1), gel(fad,1)));
4810 7 : gel(al,5) = mkvec2(PH, gen_0);;
4811 7 : return;
4812 : }
4813 :
4814 : /* m2 : number of prime factors of disc not dividing b */
4815 511 : m2 = nbrows(fad);
4816 511 : m23 = m2+m3;
4817 511 : m123 = m1+m2+m3;
4818 :
4819 : /* initialize the possibly ramified primes (hasse) and the factored conductor of rnf/nf (cnd) */
4820 511 : cnd = zeromatcopy(m23,2);
4821 511 : PH = cgetg(m123+1, t_VEC); /* ramified primes */
4822 511 : H = cgetg(m123+1, t_VECSMALL); /* Hasse invariant */
4823 : /* compute Hasse invariant at primes that are unramified in rnf/nf */
4824 875 : for (k=1; k<=m1; k++) {/* pr | b, pr \nmid disc */
4825 364 : long frob, e, j = perm[k];
4826 364 : pr = gcoeff(fab,j,1);
4827 364 : e = itos(gcoeff(fab,j,2));
4828 364 : frob = cyclicrelfrob(rnf, auts, pr);
4829 364 : gel(PH,k) = pr;
4830 364 : H[k] = Fl_mul(frob, e, n);
4831 : }
4832 : /* compute Hasse invariant at primes that are ramified in rnf/nf */
4833 1064 : for (k=1; k<=m2; k++) {/* pr \nmid b, pr | disc */
4834 553 : pr = gcoeff(fad,k,1);
4835 553 : gel(PH,k+m1) = pr;
4836 553 : gcoeff(cnd,k,1) = pr;
4837 553 : gcoeff(cnd,k,2) = gcoeff(fad,k,2);
4838 : }
4839 546 : for (k=1; k<=m3; k++) { /* pr | (b, disc) */
4840 35 : long j = perm[k+m1];
4841 35 : pr = gcoeff(fab,j,1);
4842 35 : gel(PH,k+m1+m2) = pr;
4843 35 : gcoeff(cnd,k+m2,1) = pr;
4844 35 : gcoeff(cnd,k+m2,2) = gel(L,j);
4845 : }
4846 511 : gel(cnd,2) = gdiventgs(gel(cnd,2), eulerphiu(n));
4847 1099 : for (k=1; k<=m23; k++) H[k+m1] = localhasse(rnf, cnd, pl, auts, b, k);
4848 511 : perm = gen_indexsort(PH, (void*)&cmp_prime_ideal, &cmp_nodata);
4849 511 : gel(al,5) = mkvec2(vecpermute(PH,perm),vecsmallpermute(H,perm));
4850 511 : checkhasse(nf, alg_get_hasse_f(al), alg_get_hasse_i(al), n);
4851 : }
4852 :
4853 : static GEN
4854 805 : alg_maximal_primes(GEN al, GEN P)
4855 : {
4856 805 : pari_sp av = avma;
4857 805 : long l = lg(P), i;
4858 2855 : for (i=1; i<l; i++)
4859 : {
4860 2050 : if (i != 1) al = gerepilecopy(av, al);
4861 2050 : al = alg_pmaximal(al,gel(P,i));
4862 : }
4863 805 : return al;
4864 : }
4865 :
4866 : GEN
4867 560 : alg_cyclic(GEN rnf, GEN aut, GEN b, long flag)
4868 : {
4869 560 : pari_sp av = avma;
4870 : GEN al, nf;
4871 : long D, n, d;
4872 560 : dbg_printf(1)("alg_cyclic\n");
4873 560 : checkrnf(rnf); nf = rnf_get_nf(rnf);
4874 560 : b = nf_to_scalar_or_basis(nf, b);
4875 553 : if (typ(b) == t_FRAC || (typ(b) == t_COL && !RgV_is_ZV(b)))
4876 7 : pari_err_DOMAIN("alg_cyclic", "denominator(b)", "!=", gen_1,b);
4877 :
4878 546 : n = rnf_get_degree(rnf);
4879 546 : d = nf_get_degree(nf);
4880 546 : D = d*n*n;
4881 :
4882 546 : al = cgetg(12,t_VEC);
4883 546 : gel(al,10)= gen_0; /* must be set first */
4884 546 : gel(al,1) = rnf;
4885 546 : gel(al,2) = allauts(rnf, aut);
4886 546 : gel(al,3) = basistoalg(nf,b);
4887 546 : rnf_build_nfabs(rnf, nf_get_prec(nf));
4888 546 : gel(al,6) = gen_0;
4889 546 : gel(al,7) = matid(D);
4890 546 : gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
4891 546 : gel(al,9) = algnatmultable(al,D);
4892 546 : gel(al,11)= algtracebasis(al);
4893 :
4894 546 : algcomputehasse(al, flag);
4895 :
4896 546 : if (flag & al_MAXORD) {
4897 448 : GEN hf = alg_get_hasse_f(al), pr = gel(hf,1);
4898 448 : if (typ(gel(hf,2)) == t_INT) /* factorisation was partial */
4899 7 : gel(al,5) = gen_0;
4900 441 : else pr = prV_primes(pr);
4901 448 : al = alg_maximal_primes(al, pr);
4902 : }
4903 546 : return gerepilecopy(av, al);
4904 : }
4905 :
4906 : static int
4907 427 : ismaximalsubfield(GEN al, GEN x, GEN d, long v, GEN *pt_minpol)
4908 : {
4909 427 : GEN cp = algbasischarpoly(al, x, v), lead;
4910 427 : if (!ispower(cp, d, pt_minpol)) return 0;
4911 427 : lead = leading_coeff(*pt_minpol);
4912 427 : if (isintm1(lead)) *pt_minpol = gneg(*pt_minpol);
4913 427 : return ZX_is_irred(*pt_minpol);
4914 : }
4915 :
4916 : static GEN
4917 147 : findmaximalsubfield(GEN al, GEN d, long v)
4918 : {
4919 147 : long count, nb=2, i, N = alg_get_absdim(al), n = nf_get_degree(alg_get_center(al));
4920 147 : GEN x, minpol, maxc = gen_1;
4921 :
4922 238 : for (i=n+1; i<=N; i+=n) {
4923 399 : for (count=0; count<2 && i+count<=N; count++) {
4924 308 : x = col_ei(N,i+count);
4925 308 : if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
4926 : }
4927 : }
4928 :
4929 : while(1) {
4930 119 : x = zerocol(N);
4931 504 : for (count=0; count<nb; count++)
4932 : {
4933 385 : i = random_Fl(N)+1;
4934 385 : gel(x,i) = addiu(randomi(maxc),1);
4935 385 : if (random_bits(1)) gel(x,i) = negi(gel(x,i));
4936 : }
4937 119 : if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
4938 56 : if (!random_bits(3)) maxc = addiu(maxc,1);
4939 56 : if (nb<N) nb++;
4940 : }
4941 :
4942 : return NULL; /* LCOV_EXCL_LINE */
4943 : }
4944 :
4945 : static GEN
4946 147 : frobeniusform(GEN al, GEN x)
4947 : {
4948 : GEN M, FP, P, Pi;
4949 :
4950 : /* /!\ has to be the *right* multiplication table */
4951 147 : M = algbasisrightmultable(al, x);
4952 :
4953 147 : FP = matfrobenius(M,2,0); /* M = P^(-1)*F*P */
4954 147 : P = gel(FP,2);
4955 147 : Pi = RgM_inv(P);
4956 147 : return mkvec2(P, Pi);
4957 : }
4958 :
4959 : static void
4960 147 : computesplitting(GEN al, long d, long v, long flag)
4961 : {
4962 147 : GEN subf, x, pol, polabs, basis, P, Pi, nf = alg_get_center(al), rnf, Lbasis, Lbasisinv, Q, pows;
4963 147 : long i, n = nf_get_degree(nf), nd = n*d, N = alg_get_absdim(al), j, j2;
4964 :
4965 147 : subf = findmaximalsubfield(al, utoipos(d), v);
4966 147 : x = gel(subf, 1);
4967 147 : polabs = gel(subf, 2);
4968 :
4969 : /* Frobenius form to obtain L-vector space structure */
4970 147 : basis = frobeniusform(al, x);
4971 147 : P = gel(basis, 1);
4972 147 : Pi = gel(basis, 2);
4973 :
4974 : /* construct rnf of splitting field */
4975 147 : pol = nffactor(nf,polabs);
4976 147 : pol = gcoeff(pol,1,1);
4977 147 : if (!(flag & al_FACTOR)) pol = mkvec2(pol, stoi(1<<20));
4978 147 : gel(al,1) = rnf = rnfinit(nf, pol);
4979 : /* since pol is irreducible over Q, we have k=0 in rnf. */
4980 147 : if (!gequal0(rnf_get_k(rnf)))
4981 : pari_err_BUG("computesplitting (k!=0)"); /*LCOV_EXCL_LINE*/
4982 147 : gel(al,6) = gen_0;
4983 147 : rnf_build_nfabs(rnf, nf_get_prec(nf));
4984 :
4985 : /* construct splitting data */
4986 147 : Lbasis = cgetg(d+1, t_MAT);
4987 399 : for (j=j2=1; j<=d; j++, j2+=nd)
4988 252 : gel(Lbasis,j) = gel(Pi,j2);
4989 :
4990 147 : Q = zeromatcopy(d,N);
4991 147 : pows = pol_x_powers(nd,v);
4992 399 : for (i=j=1; j<=N; j+=nd, i++)
4993 1197 : for (j2=0; j2<nd; j2++)
4994 945 : gcoeff(Q,i,j+j2) = mkpolmod(gel(pows,j2+1),polabs);
4995 147 : Lbasisinv = RgM_mul(Q,P);
4996 :
4997 147 : gel(al,3) = mkvec3(x,Lbasis,Lbasisinv);
4998 147 : }
4999 :
5000 : /* assumes that mt defines a central simple algebra over nf */
5001 : GEN
5002 175 : alg_csa_table(GEN nf, GEN mt0, long v, long flag)
5003 : {
5004 175 : pari_sp av = avma;
5005 : GEN al, mt;
5006 175 : long n, D, d2 = lg(mt0)-1, d = usqrt(d2);
5007 175 : dbg_printf(1)("alg_csa_table\n");
5008 :
5009 175 : mt = check_relmt(nf,mt0);
5010 161 : if (!mt) pari_err_TYPE("alg_csa_table", mt0);
5011 154 : n = nf_get_degree(nf);
5012 154 : D = n*d2;
5013 154 : if (d*d != d2)
5014 7 : pari_err_DOMAIN("alg_csa_table","(nonsquare) dimension","!=",stoi(d*d),mt);
5015 :
5016 147 : al = cgetg(12, t_VEC);
5017 147 : gel(al,10) = gen_0; /* must be set first */
5018 147 : gel(al,1) = zerovec(12); gmael(al,1,10) = nf;
5019 147 : gmael(al,1,1) = gpowgs(pol_x(0), d); /* placeholder before splitting field */
5020 147 : gel(al,2) = mt;
5021 147 : gel(al,3) = gen_0; /* placeholder */
5022 147 : gel(al,4) = gel(al,5) = gen_0; /* TODO Hasse invariants if flag&al_FACTOR */
5023 147 : gel(al,5) = gel(al,6) = gen_0; /* placeholder */
5024 147 : gel(al,7) = matid(D);
5025 147 : gel(al,8) = matid(D);
5026 147 : gel(al,9) = algnatmultable(al,D);
5027 147 : gel(al,11)= algtracebasis(al);
5028 147 : if (flag & al_MAXORD) al = alg_maximal(al);
5029 147 : computesplitting(al, d, v, flag);
5030 147 : return gerepilecopy(av, al);
5031 : }
5032 :
5033 : static GEN
5034 38374 : algtableinit_i(GEN mt0, GEN p)
5035 : {
5036 : GEN al, mt;
5037 : long i, n;
5038 :
5039 38374 : if (p && !signe(p)) p = NULL;
5040 38374 : mt = check_mt(mt0,p);
5041 38374 : if (!mt) pari_err_TYPE("algtableinit", mt0);
5042 38367 : if (!p && !isint1(Q_denom(mt0)))
5043 7 : pari_err_DOMAIN("algtableinit", "denominator(mt)", "!=", gen_1, mt0);
5044 38360 : n = lg(mt)-1;
5045 38360 : al = cgetg(12, t_VEC);
5046 268520 : for (i=1; i<=6; i++) gel(al,i) = gen_0;
5047 38360 : gel(al,7) = matid(n);
5048 38360 : gel(al,8) = matid(n);
5049 38360 : gel(al,9) = mt;
5050 38360 : gel(al,10) = p? p: gen_0;
5051 38360 : gel(al,11)= algtracebasis(al);
5052 38360 : return al;
5053 : }
5054 : GEN
5055 4200 : algtableinit(GEN mt0, GEN p)
5056 : {
5057 4200 : pari_sp av = avma;
5058 4200 : if (p)
5059 : {
5060 4074 : if (typ(p) != t_INT) pari_err_TYPE("algtableinit",p);
5061 4067 : if (signe(p) && !BPSW_psp(p)) pari_err_PRIME("algtableinit",p);
5062 : }
5063 4179 : return gerepilecopy(av, algtableinit_i(mt0, p));
5064 : }
5065 :
5066 : /** REPRESENTATIONS OF GROUPS **/
5067 :
5068 : static GEN
5069 294 : list_to_regular_rep(GEN elts, long n)
5070 : {
5071 : GEN reg, elts2, g;
5072 : long i,j;
5073 294 : elts = shallowcopy(elts);
5074 294 : gen_sort_inplace(elts, (void*)&vecsmall_lexcmp, &cmp_nodata, NULL);
5075 294 : reg = cgetg(n+1, t_VEC);
5076 294 : gel(reg,1) = identity_perm(n);
5077 3857 : for (i=2; i<=n; i++) {
5078 3563 : g = perm_inv(gel(elts,i));
5079 3563 : elts2 = cgetg(n+1, t_VEC);
5080 74543 : for (j=1; j<=n; j++) gel(elts2,j) = perm_mul(g,gel(elts,j));
5081 3563 : gen_sort_inplace(elts2, (void*)&vecsmall_lexcmp, &cmp_nodata, &gel(reg,i));
5082 : }
5083 294 : return reg;
5084 : }
5085 :
5086 : static GEN
5087 3857 : matrix_perm(GEN perm, long n)
5088 : {
5089 : GEN m;
5090 : long j;
5091 3857 : m = cgetg(n+1, t_MAT);
5092 78694 : for (j=1; j<=n; j++) {
5093 74837 : gel(m,j) = col_ei(n,perm[j]);
5094 : }
5095 3857 : return m;
5096 : }
5097 :
5098 : GEN
5099 847 : conjclasses_algcenter(GEN cc, GEN p)
5100 : {
5101 847 : GEN mt, elts = gel(cc,1), conjclass = gel(cc,2), rep = gel(cc,3), card;
5102 847 : long i, nbcl = lg(rep)-1, n = lg(elts)-1;
5103 : pari_sp av;
5104 :
5105 847 : card = zero_Flv(nbcl);
5106 14819 : for (i=1; i<=n; i++) card[conjclass[i]]++;
5107 :
5108 : /* multiplication table of the center of Z[G] (class functions) */
5109 847 : mt = cgetg(nbcl+1,t_VEC);
5110 7217 : for (i=1;i<=nbcl;i++) gel(mt,i) = zero_Flm_copy(nbcl,nbcl);
5111 847 : av = avma;
5112 7217 : for (i=1;i<=nbcl;i++)
5113 : {
5114 6370 : GEN xi = gel(elts,rep[i]), mi = gel(mt,i);
5115 : long j,k;
5116 132244 : for (j=1;j<=n;j++)
5117 : {
5118 125874 : GEN xj = gel(elts,j);
5119 125874 : k = vecsearch(elts, perm_mul(xi,xj), NULL);
5120 125874 : ucoeff(mi, conjclass[k], conjclass[j])++;
5121 : }
5122 70238 : for (k=1; k<=nbcl; k++)
5123 852362 : for (j=1; j<=nbcl; j++)
5124 : {
5125 788494 : ucoeff(mi,k,j) *= card[i];
5126 788494 : ucoeff(mi,k,j) /= card[k];
5127 : }
5128 6370 : set_avma(av);
5129 : }
5130 7217 : for (i=1;i<=nbcl;i++) gel(mt,i) = Flm_to_ZM(gel(mt,i));
5131 847 : return algtableinit_i(mt,p);
5132 : }
5133 :
5134 : GEN
5135 329 : alggroupcenter(GEN G, GEN p, GEN *pcc)
5136 : {
5137 329 : pari_sp av = avma;
5138 329 : GEN cc = group_to_cc(G), al = conjclasses_algcenter(cc, p);
5139 315 : if (!pcc) return gerepilecopy(av,al);
5140 7 : *pcc = cc; return gc_all(av, 2, &al, pcc);
5141 : }
5142 :
5143 : static GEN
5144 294 : groupelts_algebra(GEN elts, GEN p)
5145 : {
5146 294 : pari_sp av = avma;
5147 : GEN mt;
5148 294 : long i, n = lg(elts)-1;
5149 294 : elts = list_to_regular_rep(elts,n);
5150 294 : mt = cgetg(n+1, t_VEC);
5151 4151 : for (i=1; i<=n; i++) gel(mt,i) = matrix_perm(gel(elts,i),n);
5152 294 : return gerepilecopy(av, algtableinit_i(mt,p));
5153 : }
5154 :
5155 : GEN
5156 329 : alggroup(GEN gal, GEN p)
5157 : {
5158 329 : GEN elts = checkgroupelts(gal);
5159 294 : return groupelts_algebra(elts, p);
5160 : }
5161 :
5162 : /** MAXIMAL ORDER **/
5163 :
5164 : static GEN
5165 51461 : mattocol(GEN M, long n)
5166 : {
5167 51461 : GEN C = cgetg(n*n+1, t_COL);
5168 : long i,j,ic;
5169 51461 : ic = 1;
5170 899534 : for (i=1; i<=n; i++)
5171 19807952 : for (j=1; j<=n; j++, ic++) gel(C,ic) = gcoeff(M,i,j);
5172 51461 : return C;
5173 : }
5174 :
5175 : /* Ip is a lift of a left O/pO-ideal where O is the integral basis of al */
5176 : static GEN
5177 4775 : algleftordermodp(GEN al, GEN Ip, GEN p)
5178 : {
5179 4775 : pari_sp av = avma;
5180 : GEN I, Ii, M, mt, K, imi, p2;
5181 : long n, i;
5182 4775 : n = alg_get_absdim(al);
5183 4775 : mt = alg_get_multable(al);
5184 4775 : p2 = sqri(p);
5185 :
5186 4775 : I = ZM_hnfmodid(Ip, p);
5187 4775 : Ii = ZM_inv(I,NULL);
5188 :
5189 4775 : M = cgetg(n+1, t_MAT);
5190 56236 : for (i=1; i<=n; i++) {
5191 51461 : imi = FpM_mul(Ii, FpM_mul(gel(mt,i), I, p2), p2);
5192 51461 : imi = ZM_Z_divexact(imi, p);
5193 51461 : gel(M,i) = mattocol(imi, n);
5194 : }
5195 4775 : K = FpM_ker(M, p);
5196 4775 : if (lg(K)==1) { set_avma(av); return matid(n); }
5197 1815 : K = ZM_hnfmodid(K,p);
5198 :
5199 1815 : return gerepileupto(av, ZM_Z_div(K,p));
5200 : }
5201 :
5202 : static GEN
5203 6796 : alg_ordermodp(GEN al, GEN p)
5204 : {
5205 : GEN alp;
5206 6796 : long i, N = alg_get_absdim(al);
5207 6796 : alp = cgetg(12, t_VEC);
5208 61164 : for (i=1; i<=8; i++) gel(alp,i) = gen_0;
5209 6796 : gel(alp,9) = cgetg(N+1, t_VEC);
5210 69764 : for (i=1; i<=N; i++) gmael(alp,9,i) = FpM_red(gmael(al,9,i), p);
5211 6796 : gel(alp,10) = p;
5212 6796 : gel(alp,11) = cgetg(N+1, t_VEC);
5213 69764 : for (i=1; i<=N; i++) gmael(alp,11,i) = Fp_red(gmael(al,11,i), p);
5214 :
5215 6796 : return alp;
5216 : }
5217 :
5218 : static GEN
5219 3865 : algpradical_i(GEN al, GEN p, GEN zprad, GEN projs)
5220 : {
5221 3865 : pari_sp av = avma;
5222 3865 : GEN alp = alg_ordermodp(al, p), liftrad, projrad, alq, alrad, res, Lalp, radq;
5223 : long i;
5224 3865 : if (lg(zprad)==1) {
5225 2854 : liftrad = NULL;
5226 2854 : projrad = NULL;
5227 : }
5228 : else {
5229 1011 : alq = alg_quotient(alp, zprad, 1);
5230 1011 : alp = gel(alq,1);
5231 1011 : projrad = gel(alq,2);
5232 1011 : liftrad = gel(alq,3);
5233 : }
5234 :
5235 3865 : if (projs) {
5236 572 : if (projrad) {
5237 28 : projs = gcopy(projs);
5238 84 : for (i=1; i<lg(projs); i++)
5239 56 : gel(projs,i) = FpM_FpC_mul(projrad, gel(projs,i), p);
5240 : }
5241 572 : Lalp = alg_centralproj(alp, projs, 1);
5242 :
5243 572 : alrad = cgetg(lg(Lalp),t_VEC);
5244 2088 : for (i=1; i<lg(Lalp); i++) {
5245 1516 : alq = gel(Lalp,i);
5246 1516 : radq = algradical(gel(alq,1));
5247 1516 : if (gequal0(radq))
5248 880 : gel(alrad,i) = cgetg(1,t_MAT);
5249 : else {
5250 636 : radq = FpM_mul(gel(alq,3),radq,p);
5251 636 : gel(alrad,i) = radq;
5252 : }
5253 : }
5254 572 : alrad = shallowmatconcat(alrad);
5255 572 : alrad = FpM_image(alrad,p);
5256 : }
5257 3293 : else alrad = algradical(alp);
5258 :
5259 3865 : if (!gequal0(alrad)) {
5260 3061 : if (liftrad) alrad = FpM_mul(liftrad, alrad, p);
5261 3061 : res = shallowmatconcat(mkvec2(alrad, zprad));
5262 3061 : res = FpM_image(res,p);
5263 : }
5264 804 : else res = lg(zprad)==1 ? gen_0 : zprad;
5265 3865 : return gerepilecopy(av, res);
5266 : }
5267 :
5268 : static GEN
5269 2931 : algpdecompose0(GEN al, GEN prad, GEN p, GEN projs)
5270 : {
5271 2931 : pari_sp av = avma;
5272 2931 : GEN alp, quo, ss, liftm = NULL, projm = NULL, dec, res, I, Lss, deci;
5273 : long i, j;
5274 :
5275 2931 : alp = alg_ordermodp(al, p);
5276 2931 : if (!gequal0(prad)) {
5277 2400 : quo = alg_quotient(alp, prad, 1);
5278 2400 : ss = gel(quo,1);
5279 2400 : projm = gel(quo,2);
5280 2400 : liftm = gel(quo,3);
5281 : }
5282 531 : else ss = alp;
5283 :
5284 2931 : if (projs) {
5285 502 : if (projm) {
5286 1263 : for (i=1; i<lg(projs); i++)
5287 914 : gel(projs,i) = FpM_FpC_mul(projm, gel(projs,i), p);
5288 : }
5289 502 : Lss = alg_centralproj(ss, projs, 1);
5290 :
5291 502 : dec = cgetg(lg(Lss),t_VEC);
5292 1857 : for (i=1; i<lg(Lss); i++) {
5293 1355 : gel(dec,i) = algsimpledec_ss(gmael(Lss,i,1), 1);
5294 1355 : deci = gel(dec,i);
5295 3066 : for (j=1; j<lg(deci); j++)
5296 1711 : gmael(deci,j,3) = FpM_mul(gmael(Lss,i,3), gmael(deci,j,3), p);
5297 : }
5298 502 : dec = shallowconcat1(dec);
5299 : }
5300 2429 : else dec = algsimpledec_ss(ss,1);
5301 :
5302 2931 : res = cgetg(lg(dec),t_VEC);
5303 7834 : for (i=1; i<lg(dec); i++) {
5304 4903 : I = gmael(dec,i,3);
5305 4903 : if (liftm) I = FpM_mul(liftm,I,p);
5306 4903 : I = shallowmatconcat(mkvec2(I,prad));
5307 4903 : gel(res,i) = I;
5308 : }
5309 :
5310 2931 : return gerepilecopy(av, res);
5311 : }
5312 :
5313 : /* finds a nontrivial ideal of O/prad or gen_0 if there is none. */
5314 : static GEN
5315 881 : algpdecompose_i(GEN al, GEN p, GEN zprad, GEN projs)
5316 : {
5317 881 : pari_sp av = avma;
5318 881 : GEN prad = algpradical_i(al,p,zprad,projs);
5319 881 : return gerepileupto(av, algpdecompose0(al, prad, p, projs));
5320 : }
5321 :
5322 : /* ord is assumed to be in hnf wrt the integral basis of al. */
5323 : /* assumes that alg_get_invbasis(al) is integral. */
5324 : static GEN
5325 1815 : alg_change_overorder_shallow(GEN al, GEN ord)
5326 : {
5327 : GEN al2, mt, iord, mtx, den, den2, div;
5328 : long i, n;
5329 1815 : n = alg_get_absdim(al);
5330 :
5331 1815 : iord = QM_inv(ord);
5332 1815 : al2 = shallowcopy(al);
5333 1815 : ord = Q_remove_denom(ord,&den);
5334 :
5335 1815 : gel(al2,7) = Q_remove_denom(gel(al,7), &den2);
5336 1815 : if (den2) div = mulii(den,den2);
5337 693 : else div = den;
5338 1815 : gel(al2,7) = ZM_Z_div(ZM_mul(gel(al2,7), ord), div);
5339 :
5340 1815 : gel(al2,8) = ZM_mul(iord, gel(al,8));
5341 :
5342 1815 : mt = cgetg(n+1,t_VEC);
5343 1815 : gel(mt,1) = matid(n);
5344 1815 : div = sqri(den);
5345 19846 : for (i=2; i<=n; i++) {
5346 18031 : mtx = algbasismultable(al,gel(ord,i));
5347 18031 : gel(mt,i) = ZM_mul(iord, ZM_mul(mtx, ord));
5348 18031 : gel(mt,i) = ZM_Z_divexact(gel(mt,i), div);
5349 : }
5350 1815 : gel(al2,9) = mt;
5351 :
5352 1815 : gel(al2,11) = algtracebasis(al2);
5353 :
5354 1815 : return al2;
5355 : }
5356 :
5357 : static GEN
5358 12209 : algfromcenter(GEN al, GEN x)
5359 : {
5360 12209 : GEN nf = alg_get_center(al);
5361 : long n;
5362 12209 : switch(alg_type(al)) {
5363 11075 : case al_CYCLIC:
5364 11075 : n = alg_get_degree(al);
5365 11075 : break;
5366 1134 : case al_CSA:
5367 1134 : n = alg_get_dim(al);
5368 1134 : break;
5369 : default: return NULL; /*LCOV_EXCL_LINE*/
5370 : }
5371 12209 : return algalgtobasis(al, scalarcol(basistoalg(nf, x), n));
5372 : }
5373 :
5374 : /* x is an ideal of the center in hnf form */
5375 : static GEN
5376 3865 : algfromcenterhnf(GEN al, GEN x)
5377 : {
5378 : GEN res;
5379 : long i;
5380 3865 : res = cgetg(lg(x), t_MAT);
5381 11265 : for (i=1; i<lg(x); i++) gel(res,i) = algfromcenter(al, gel(x,i));
5382 3865 : return res;
5383 : }
5384 :
5385 : /* assumes al is CSA or CYCLIC */
5386 : static GEN
5387 2050 : algcenter_precompute(GEN al, GEN p)
5388 : {
5389 2050 : GEN fa, pdec, nfprad, projs, nf = alg_get_center(al);
5390 : long i, np;
5391 :
5392 2050 : pdec = idealprimedec(nf, p);
5393 2050 : settyp(pdec, t_COL);
5394 2050 : np = lg(pdec)-1;
5395 2050 : fa = mkmat2(pdec, const_col(np, gen_1));
5396 2050 : if (dvdii(nf_get_disc(nf), p))
5397 336 : nfprad = idealprodprime(nf, pdec);
5398 : else
5399 1714 : nfprad = scalarmat_shallow(p, nf_get_degree(nf));
5400 2050 : fa = idealchineseinit(nf, fa);
5401 2050 : projs = cgetg(np+1, t_VEC);
5402 4528 : for (i=1; i<=np; i++) gel(projs, i) = idealchinese(nf, fa, vec_ei(np,i));
5403 2050 : return mkvec2(nfprad, projs);
5404 : }
5405 :
5406 : static GEN
5407 3865 : algcenter_prad(GEN al, GEN p, GEN pre)
5408 : {
5409 : GEN nfprad, zprad, mtprad;
5410 : long i;
5411 3865 : nfprad = gel(pre,1);
5412 3865 : zprad = algfromcenterhnf(al, nfprad);
5413 3865 : zprad = FpM_image(zprad, p);
5414 3865 : mtprad = cgetg(lg(zprad), t_VEC);
5415 5409 : for (i=1; i<lg(zprad); i++) gel(mtprad, i) = algbasismultable(al, gel(zprad,i));
5416 3865 : mtprad = shallowmatconcat(mtprad);
5417 3865 : zprad = FpM_image(mtprad, p);
5418 3865 : return zprad;
5419 : }
5420 :
5421 : static GEN
5422 3865 : algcenter_p_projs(GEN al, GEN p, GEN pre)
5423 : {
5424 : GEN projs, zprojs;
5425 : long i;
5426 3865 : projs = gel(pre,2);
5427 3865 : zprojs = cgetg(lg(projs), t_VEC);
5428 8674 : for (i=1; i<lg(projs); i++) gel(zprojs,i) = FpC_red(algfromcenter(al, gel(projs,i)),p);
5429 3865 : return zprojs;
5430 : }
5431 :
5432 : /* al is assumed to be simple */
5433 : static GEN
5434 2050 : alg_pmaximal(GEN al, GEN p)
5435 : {
5436 : pari_sp av;
5437 2050 : long n = alg_get_absdim(al);
5438 2050 : GEN id = matid(n), al2 = al, prad, lord = gen_0, dec, zprad, projs, pre;
5439 :
5440 2050 : dbg_printf(0)("Round 2 (noncommutative) at p=%Ps, dim=%d\n", p, n);
5441 2050 : pre = algcenter_precompute(al,p); av = avma;
5442 : while (1) {
5443 2984 : zprad = algcenter_prad(al2, p, pre);
5444 2984 : projs = algcenter_p_projs(al2, p, pre);
5445 2984 : if (lg(projs) == 2) projs = NULL;
5446 2984 : prad = algpradical_i(al2,p,zprad,projs);
5447 2984 : if (typ(prad) == t_INT) break;
5448 2956 : lord = algleftordermodp(al2,prad,p);
5449 2956 : if (!cmp_universal(lord,id)) break;
5450 934 : al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
5451 : }
5452 :
5453 2050 : dec = algpdecompose0(al2,prad,p,projs); av = avma;
5454 2931 : while (lg(dec) > 2) {
5455 : long i;
5456 2098 : for (i = 1; i < lg(dec); i++) {
5457 1819 : GEN I = gel(dec,i);
5458 1819 : lord = algleftordermodp(al2,I,p);
5459 1819 : if (cmp_universal(lord,id)) break;
5460 : }
5461 1160 : if (i==lg(dec)) break;
5462 881 : al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
5463 881 : zprad = algcenter_prad(al2, p, pre);
5464 881 : projs = algcenter_p_projs(al2, p, pre);
5465 881 : if (lg(projs) == 2) projs = NULL;
5466 881 : dec = algpdecompose_i(al2,p,zprad,projs);
5467 : }
5468 2050 : return al2;
5469 : }
5470 :
5471 : static GEN
5472 6419 : algtracematrix(GEN al)
5473 : {
5474 : GEN M, mt;
5475 : long n, i, j;
5476 6419 : n = alg_get_absdim(al);
5477 6419 : mt = alg_get_multable(al);
5478 6419 : M = cgetg(n+1, t_MAT);
5479 48414 : for (i=1; i<=n; i++)
5480 : {
5481 41995 : gel(M,i) = cgetg(n+1,t_MAT);
5482 289892 : for (j=1; j<=i; j++)
5483 247897 : gcoeff(M,j,i) = gcoeff(M,i,j) = algabstrace(al,gmael(mt,i,j));
5484 : }
5485 6419 : return M;
5486 : }
5487 : static GEN
5488 161 : algdisc_i(GEN al) { return ZM_det(algtracematrix(al)); }
5489 : GEN
5490 49 : algdisc(GEN al)
5491 : {
5492 49 : pari_sp av = avma;
5493 49 : checkalg(al);
5494 49 : if (alg_type(al) == al_REAL) pari_err_TYPE("algdisc [real algebra]", al);
5495 28 : return gerepileuptoint(av, algdisc_i(al));
5496 : }
5497 : static GEN
5498 133 : alg_maximal(GEN al)
5499 : {
5500 133 : GEN fa = absZ_factor(algdisc_i(al));
5501 133 : return alg_maximal_primes(al, gel(fa,1));
5502 : }
5503 :
5504 : /** LATTICES **/
5505 :
5506 : /*
5507 : Convention: lattice = [I,t] representing t*I, where
5508 : - I integral nonsingular upper-triangular matrix representing a lattice over
5509 : the integral basis of the algebra, and
5510 : - t>0 either an integer or a rational number.
5511 :
5512 : Recommended and returned by the functions below:
5513 : - I HNF and primitive
5514 : */
5515 :
5516 : /* TODO use hnfmodid whenever possible using a*O <= I <= O
5517 : * for instance a = ZM_det_triangular(I) */
5518 :
5519 : static GEN
5520 63343 : primlat(GEN lat)
5521 : {
5522 : GEN m, t, c;
5523 63343 : m = alglat_get_primbasis(lat);
5524 63343 : t = alglat_get_scalar(lat);
5525 63343 : m = Q_primitive_part(m,&c);
5526 63343 : if (c) return mkvec2(m,gmul(t,c));
5527 53809 : return lat;
5528 : }
5529 :
5530 : /* assumes the lattice contains d * integral basis, d=0 allowed */
5531 : GEN
5532 51072 : alglathnf(GEN al, GEN m, GEN d)
5533 : {
5534 51072 : pari_sp av = avma;
5535 : long N,i,j;
5536 : GEN m2, c;
5537 51072 : checkalg(al);
5538 51072 : if (alg_type(al) == al_REAL) pari_err_TYPE("alglathnf [real algebra]", al);
5539 51065 : N = alg_get_absdim(al);
5540 51065 : if (!d) d = gen_0;
5541 51065 : if (typ(m) == t_VEC) m = matconcat(m);
5542 51065 : if (typ(m) == t_COL) m = algleftmultable(al,m);
5543 51065 : if (typ(m) != t_MAT) pari_err_TYPE("alglathnf",m);
5544 51058 : if (typ(d) != t_FRAC && typ(d) != t_INT) pari_err_TYPE("alglathnf",d);
5545 51058 : if (lg(m)-1 < N || lg(gel(m,1))-1 != N) pari_err_DIM("alglathnf");
5546 459242 : for (i=1; i<=N; i++)
5547 6820758 : for (j=1; j<lg(m); j++)
5548 6412546 : if (typ(gcoeff(m,i,j)) != t_FRAC && typ(gcoeff(m,i,j)) != t_INT)
5549 7 : pari_err_TYPE("alglathnf", gcoeff(m,i,j));
5550 51023 : m2 = Q_primitive_part(m,&c);
5551 51023 : if (!c) c = gen_1;
5552 51023 : if (!signe(d)) d = detint(m2);
5553 45593 : else d = gdiv(d,c); /* should be an integer */
5554 51023 : if (!signe(d)) pari_err_INV("alglathnf [m does not have full rank]", m2);
5555 51009 : m2 = ZM_hnfmodid(m2,d);
5556 51009 : return gerepilecopy(av, mkvec2(m2,c));
5557 : }
5558 :
5559 : static GEN
5560 10689 : prepare_multipliers(GEN *a, GEN *b)
5561 : {
5562 : GEN na, nb, da, db, d;
5563 10689 : na = numer_i(*a); da = denom_i(*a);
5564 10689 : nb = numer_i(*b); db = denom_i(*b);
5565 10689 : na = mulii(na,db);
5566 10689 : nb = mulii(nb,da);
5567 10689 : d = gcdii(na,nb);
5568 10689 : *a = diviiexact(na,d);
5569 10689 : *b = diviiexact(nb,d);
5570 10689 : return gdiv(d, mulii(da,db));
5571 : }
5572 :
5573 : static GEN
5574 10689 : prepare_lat(GEN m1, GEN t1, GEN m2, GEN t2)
5575 : {
5576 10689 : GEN d = prepare_multipliers(&t1, &t2);
5577 10689 : m1 = ZM_Z_mul(m1,t1);
5578 10689 : m2 = ZM_Z_mul(m2,t2);
5579 10689 : return mkvec3(m1,m2,d);
5580 : }
5581 :
5582 : static GEN
5583 10703 : alglataddinter(GEN al, GEN lat1, GEN lat2, GEN *sum, GEN *inter)
5584 : {
5585 : GEN d, m1, m2, t1, t2, M, prep, d1, d2, ds, di, K;
5586 10703 : checkalg(al);
5587 10703 : if (alg_type(al) == al_REAL)
5588 14 : pari_err_TYPE("alglataddinter [real algebra]", al);
5589 10689 : checklat(al,lat1);
5590 10689 : checklat(al,lat2);
5591 :
5592 10689 : m1 = alglat_get_primbasis(lat1);
5593 10689 : t1 = alglat_get_scalar(lat1);
5594 10689 : m2 = alglat_get_primbasis(lat2);
5595 10689 : t2 = alglat_get_scalar(lat2);
5596 10689 : prep = prepare_lat(m1, t1, m2, t2);
5597 10689 : m1 = gel(prep,1);
5598 10689 : m2 = gel(prep,2);
5599 10689 : d = gel(prep,3);
5600 10689 : M = matconcat(mkvec2(m1,m2));
5601 10689 : d1 = ZM_det_triangular(m1);
5602 10689 : d2 = ZM_det_triangular(m2);
5603 10689 : ds = gcdii(d1,d2);
5604 10689 : if (inter)
5605 : {
5606 7112 : di = diviiexact(mulii(d1,d2),ds);
5607 7112 : K = matkermod(M,di,sum);
5608 7112 : K = rowslice(K,1,lg(m1));
5609 7112 : *inter = hnfmodid(FpM_mul(m1,K,di),di);
5610 7112 : if (sum) *sum = hnfmodid(*sum,ds);
5611 : }
5612 3577 : else *sum = hnfmodid(M,ds);
5613 10689 : return d;
5614 : }
5615 :
5616 : GEN
5617 3605 : alglatinter(GEN al, GEN lat1, GEN lat2, GEN* psum)
5618 : {
5619 3605 : pari_sp av = avma;
5620 : GEN inter, d;
5621 3605 : d = alglataddinter(al, lat1, lat2, psum, &inter);
5622 3598 : inter = primlat(mkvec2(inter, d));
5623 3598 : if (!psum) return gerepilecopy(av, inter);
5624 14 : *psum = primlat(mkvec2(*psum,d));
5625 14 : return gc_all(av, 2, &inter, psum);
5626 : }
5627 :
5628 : GEN
5629 7098 : alglatadd(GEN al, GEN lat1, GEN lat2, GEN* pinter)
5630 : {
5631 7098 : pari_sp av = avma;
5632 : GEN sum, d;
5633 7098 : d = alglataddinter(al, lat1, lat2, &sum, pinter);
5634 7091 : sum = primlat(mkvec2(sum, d));
5635 7091 : if (!pinter) return gerepilecopy(av, sum);
5636 3514 : *pinter = primlat(mkvec2(*pinter,d));
5637 3514 : return gc_all(av, 2, &sum, pinter);
5638 : }
5639 :
5640 : /* TODO version that returns the quotient as abelian group? */
5641 : /* return matrices to convert coordinates from one to other? */
5642 : int
5643 31556 : alglatsubset(GEN al, GEN lat1, GEN lat2, GEN* pindex)
5644 : {
5645 31556 : pari_sp av = avma;
5646 : int res;
5647 : GEN m1, m2, m2i, m, t;
5648 31556 : checkalg(al);
5649 31556 : if (alg_type(al) == al_REAL) pari_err_TYPE("alglatsubset [real algebra]", al);
5650 31549 : checklat(al,lat1);
5651 31549 : checklat(al,lat2);
5652 31549 : m1 = alglat_get_primbasis(lat1);
5653 31549 : m2 = alglat_get_primbasis(lat2);
5654 31549 : m2i = RgM_inv_upper(m2);
5655 31549 : t = gdiv(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
5656 31549 : m = RgM_Rg_mul(RgM_mul(m2i,m1), t);
5657 31549 : res = RgM_is_ZM(m);
5658 31549 : if (!res || !pindex) return gc_int(av, res);
5659 1757 : *pindex = gerepileuptoint(av, mpabs(ZM_det_triangular(m)));
5660 1757 : return 1;
5661 : }
5662 :
5663 : GEN
5664 5271 : alglatindex(GEN al, GEN lat1, GEN lat2)
5665 : {
5666 5271 : pari_sp av = avma;
5667 : long N;
5668 : GEN res;
5669 5271 : checkalg(al);
5670 5271 : if (alg_type(al) == al_REAL) pari_err_TYPE("alglatindex [real algebra]", al);
5671 5264 : checklat(al,lat1);
5672 5264 : checklat(al,lat2);
5673 5264 : N = alg_get_absdim(al);
5674 5264 : res = alglat_get_scalar(lat1);
5675 5264 : res = gdiv(res, alglat_get_scalar(lat2));
5676 5264 : res = gpowgs(res, N);
5677 5264 : res = gmul(res,RgM_det_triangular(alglat_get_primbasis(lat1)));
5678 5264 : res = gdiv(res, RgM_det_triangular(alglat_get_primbasis(lat2)));
5679 5264 : res = gabs(res,0);
5680 5264 : return gerepilecopy(av, res);
5681 : }
5682 :
5683 : GEN
5684 45612 : alglatmul(GEN al, GEN lat1, GEN lat2)
5685 : {
5686 45612 : pari_sp av = avma;
5687 : long N,i;
5688 : GEN m1, m2, m, V, lat, t, d, dp;
5689 45612 : checkalg(al);
5690 45612 : if (alg_type(al) == al_REAL) pari_err_TYPE("alglatmul [real algebra]", al);
5691 45605 : if (typ(lat1)==t_COL)
5692 : {
5693 19292 : if (typ(lat2)==t_COL)
5694 7 : pari_err_TYPE("alglatmul [one of lat1, lat2 has to be a lattice]", lat2);
5695 19285 : checklat(al,lat2);
5696 19285 : lat1 = Q_remove_denom(lat1,&d);
5697 19285 : m = algbasismultable(al,lat1);
5698 19285 : m2 = alglat_get_primbasis(lat2);
5699 19285 : dp = mulii(detint(m),ZM_det_triangular(m2));
5700 19285 : m = ZM_mul(m,m2);
5701 19285 : t = alglat_get_scalar(lat2);
5702 19285 : if (d) t = gdiv(t,d);
5703 : }
5704 : else /* typ(lat1)!=t_COL */
5705 : {
5706 26313 : checklat(al,lat1);
5707 26313 : if (typ(lat2)==t_COL)
5708 : {
5709 19285 : lat2 = Q_remove_denom(lat2,&d);
5710 19285 : m = algbasisrightmultable(al,lat2);
5711 19285 : m1 = alglat_get_primbasis(lat1);
5712 19285 : dp = mulii(detint(m),ZM_det_triangular(m1));
5713 19285 : m = ZM_mul(m,m1);
5714 19285 : t = alglat_get_scalar(lat1);
5715 19285 : if (d) t = gdiv(t,d);
5716 : }
5717 : else /* typ(lat2)!=t_COL */
5718 : {
5719 7028 : checklat(al,lat2);
5720 7021 : N = alg_get_absdim(al);
5721 7021 : m1 = alglat_get_primbasis(lat1);
5722 7021 : m2 = alglat_get_primbasis(lat2);
5723 7021 : dp = mulii(ZM_det_triangular(m1), ZM_det_triangular(m2));
5724 7021 : V = cgetg(N+1,t_VEC);
5725 63189 : for (i=1; i<=N; i++) {
5726 56168 : gel(V,i) = algbasismultable(al,gel(m1,i));
5727 56168 : gel(V,i) = ZM_mul(gel(V,i),m2);
5728 : }
5729 7021 : m = matconcat(V);
5730 7021 : t = gmul(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
5731 : }
5732 : }
5733 :
5734 45591 : lat = alglathnf(al,m,dp);
5735 45591 : gel(lat,2) = gmul(alglat_get_scalar(lat), t);
5736 45591 : lat = primlat(lat);
5737 45591 : return gerepilecopy(av, lat);
5738 : }
5739 :
5740 : int
5741 17528 : alglatcontains(GEN al, GEN lat, GEN x, GEN *ptc)
5742 : {
5743 17528 : pari_sp av = avma;
5744 : GEN m, t, sol;
5745 17528 : checkalg(al);
5746 17528 : if (alg_type(al) == al_REAL)
5747 7 : pari_err_TYPE("alglatcontains [real algebra]", al);
5748 17521 : checklat(al,lat);
5749 17521 : m = alglat_get_primbasis(lat);
5750 17521 : t = alglat_get_scalar(lat);
5751 17521 : x = RgC_Rg_div(x,t);
5752 17521 : if (!RgV_is_ZV(x)) return gc_bool(av,0);
5753 17521 : sol = hnf_solve(m,x);
5754 17521 : if (!sol) return gc_bool(av,0);
5755 8771 : if (!ptc) return gc_bool(av,1);
5756 8764 : *ptc = gerepilecopy(av, sol); return 1;
5757 : }
5758 :
5759 : GEN
5760 8778 : alglatelement(GEN al, GEN lat, GEN c)
5761 : {
5762 8778 : pari_sp av = avma;
5763 : GEN res;
5764 8778 : checkalg(al);
5765 8778 : if (alg_type(al) == al_REAL)
5766 7 : pari_err_TYPE("alglatelement [real algebra]", al);
5767 8771 : checklat(al,lat);
5768 8771 : if (typ(c)!=t_COL) pari_err_TYPE("alglatelement", c);
5769 8764 : res = ZM_ZC_mul(alglat_get_primbasis(lat),c);
5770 8764 : res = RgC_Rg_mul(res, alglat_get_scalar(lat));
5771 8764 : return gerepilecopy(av,res);
5772 : }
5773 :
5774 : /* idem QM_invimZ, knowing result is contained in 1/c*Z^n */
5775 : static GEN
5776 3535 : QM_invimZ_mod(GEN m, GEN c)
5777 : {
5778 : GEN d, m0, K;
5779 3535 : m0 = Q_remove_denom(m, &d);
5780 3535 : if (d) d = mulii(d,c);
5781 35 : else d = c;
5782 3535 : K = matkermod(m0, d, NULL);
5783 3535 : if (lg(K)==1) K = scalarmat(d, lg(m)-1);
5784 3493 : else K = hnfmodid(K, d);
5785 3535 : return RgM_Rg_div(K,c);
5786 : }
5787 :
5788 : /* If m is injective, computes a Z-basis of the submodule of elements whose
5789 : * image under m is integral */
5790 : static GEN
5791 14 : QM_invimZ(GEN m)
5792 : {
5793 14 : return RgM_invimage(m, QM_ImQ_hnf(m));
5794 : }
5795 :
5796 : /* An isomorphism of R-modules M_{m,n}(R) -> R^{m*n} */
5797 : static GEN
5798 28322 : mat2col(GEN M, long m, long n)
5799 : {
5800 : long i,j,k,p;
5801 : GEN C;
5802 28322 : p = m*n;
5803 28322 : C = cgetg(p+1,t_COL);
5804 254702 : for (i=1,k=1;i<=m;i++)
5805 2036804 : for (j=1;j<=n;j++,k++)
5806 1810424 : gel(C,k) = gcoeff(M,i,j);
5807 28322 : return C;
5808 : }
5809 :
5810 : static GEN
5811 3535 : alglattransporter_i(GEN al, GEN lat1, GEN lat2, long right)
5812 : {
5813 : GEN m1, m2, m2i, M, MT, mt, t1, t2, T, c;
5814 : long N, i;
5815 3535 : N = alg_get_absdim(al);
5816 3535 : m1 = alglat_get_primbasis(lat1);
5817 3535 : m2 = alglat_get_primbasis(lat2);
5818 3535 : m2i = RgM_inv_upper(m2);
5819 3535 : c = detint(m1);
5820 3535 : t1 = alglat_get_scalar(lat1);
5821 3535 : m1 = RgM_Rg_mul(m1,t1);
5822 3535 : t2 = alglat_get_scalar(lat2);
5823 3535 : m2i = RgM_Rg_div(m2i,t2);
5824 :
5825 3535 : MT = right? NULL: alg_get_multable(al);
5826 3535 : M = cgetg(N+1, t_MAT);
5827 31815 : for (i=1; i<=N; i++) {
5828 28280 : if (right) mt = algbasisrightmultable(al, vec_ei(N,i));
5829 14168 : else mt = gel(MT,i);
5830 28280 : mt = RgM_mul(m2i,mt);
5831 28280 : mt = RgM_mul(mt,m1);
5832 28280 : gel(M,i) = mat2col(mt, N, N);
5833 : }
5834 :
5835 3535 : c = gdiv(t2,gmul(c,t1));
5836 3535 : c = denom_i(c);
5837 3535 : T = QM_invimZ_mod(M,c);
5838 3535 : return primlat(mkvec2(T,gen_1));
5839 : }
5840 :
5841 : /*
5842 : { x in al | x*lat1 subset lat2}
5843 : */
5844 : GEN
5845 1778 : alglatlefttransporter(GEN al, GEN lat1, GEN lat2)
5846 : {
5847 1778 : pari_sp av = avma;
5848 1778 : checkalg(al);
5849 1778 : if (alg_type(al) == al_REAL)
5850 7 : pari_err_TYPE("alglatlefttransporter [real algebra]", al);
5851 1771 : checklat(al,lat1);
5852 1771 : checklat(al,lat2);
5853 1771 : return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,0));
5854 : }
5855 :
5856 : /*
5857 : { x in al | lat1*x subset lat2}
5858 : */
5859 : GEN
5860 1771 : alglatrighttransporter(GEN al, GEN lat1, GEN lat2)
5861 : {
5862 1771 : pari_sp av = avma;
5863 1771 : checkalg(al);
5864 1771 : if (alg_type(al) == al_REAL)
5865 7 : pari_err_TYPE("alglatrighttransporter [real algebra]", al);
5866 1764 : checklat(al,lat1);
5867 1764 : checklat(al,lat2);
5868 1764 : return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,1));
5869 : }
5870 :
5871 : GEN
5872 42 : algmakeintegral(GEN mt0, long maps)
5873 : {
5874 42 : pari_sp av = avma;
5875 : long n,i;
5876 : GEN m,P,Pi,mt2,mt;
5877 42 : n = lg(mt0)-1;
5878 42 : mt = check_mt(mt0,NULL);
5879 42 : if (!mt) pari_err_TYPE("algmakeintegral", mt0);
5880 21 : if (isint1(Q_denom(mt0))) {
5881 7 : if (maps) mt = mkvec3(mt,matid(n),matid(n));
5882 7 : return gerepilecopy(av,mt);
5883 : }
5884 14 : dbg_printf(2)(" algmakeintegral: dim=%d, denom=%Ps\n", n, Q_denom(mt0));
5885 14 : m = cgetg(n+1,t_MAT);
5886 56 : for (i=1;i<=n;i++)
5887 42 : gel(m,i) = mat2col(gel(mt,i),n,n);
5888 14 : dbg_printf(2)(" computing order, dims m = %d x %d...\n", nbrows(m), lg(m)-1);
5889 14 : P = QM_invimZ(m);
5890 14 : dbg_printf(2)(" ...done.\n");
5891 14 : P = shallowmatconcat(mkvec2(col_ei(n,1),P));
5892 14 : P = hnf(P);
5893 14 : Pi = RgM_inv(P);
5894 14 : mt2 = change_Rgmultable(mt,P,Pi);
5895 14 : if (maps) mt2 = mkvec3(mt2,Pi,P); /* mt2, mt->mt2, mt2->mt */
5896 14 : return gerepilecopy(av,mt2);
5897 : }
5898 :
5899 : /** ORDERS **/
5900 :
5901 : /** IDEALS **/
5902 :
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