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 :
15 : /*******************************************************************/
16 : /* */
17 : /* BASIC NF OPERATIONS */
18 : /* (continued) */
19 : /* */
20 : /*******************************************************************/
21 : #include "pari.h"
22 : #include "paripriv.h"
23 :
24 : #define DEBUGLEVEL DEBUGLEVEL_nf
25 :
26 : /*******************************************************************/
27 : /* */
28 : /* IDEAL OPERATIONS */
29 : /* */
30 : /*******************************************************************/
31 :
32 : /* A valid ideal is either principal (valid nf_element), or prime, or a matrix
33 : * on the integer basis in HNF.
34 : * A prime ideal is of the form [p,a,e,f,b], where the ideal is p.Z_K+a.Z_K,
35 : * p is a rational prime, a belongs to Z_K, e=e(P/p), f=f(P/p), and b
36 : * is Lenstra's constant, such that p.P^(-1)= p Z_K + b Z_K.
37 : *
38 : * An extended ideal is a couple [I,F] where I is an ideal and F is either an
39 : * algebraic number, or a factorization matrix attached to an algebraic number.
40 : * All routines work with either extended ideals or ideals (an omitted F is
41 : * assumed to be factor(1)). All ideals are output in HNF form. */
42 :
43 : /* types and conversions */
44 :
45 : long
46 16552244 : idealtyp(GEN *ideal, GEN *arch)
47 : {
48 16552244 : GEN x = *ideal;
49 16552244 : long t,lx,tx = typ(x);
50 :
51 16552244 : if (tx!=t_VEC || lg(x)!=3) { if (arch) *arch = NULL; }
52 : else
53 : {
54 1240396 : GEN a = gel(x,2);
55 1240396 : if (typ(a) == t_MAT && lg(a) != 3)
56 : { /* allow [;] */
57 14 : if (lg(a) != 1) pari_err_TYPE("idealtyp [extended ideal]",x);
58 7 : if (arch) *arch = trivial_fact();
59 : }
60 : else
61 1240382 : if (arch) *arch = a;
62 1240389 : x = gel(x,1); tx = typ(x);
63 : }
64 16552237 : switch(tx)
65 : {
66 12539442 : case t_MAT: lx = lg(x);
67 12539442 : if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
68 12539267 : if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
69 12539239 : t = id_MAT;
70 12539239 : break;
71 :
72 3050761 : case t_VEC:
73 3050761 : if (!checkprid_i(x)) pari_err_TYPE("idealtyp [fake prime ideal]",x);
74 3050713 : t = id_PRIME; break;
75 :
76 962396 : case t_POL: case t_POLMOD: case t_COL:
77 : case t_INT: case t_FRAC:
78 962396 : t = id_PRINCIPAL; break;
79 6 : default:
80 6 : pari_err_TYPE("idealtyp",x);
81 : return 0; /*LCOV_EXCL_LINE*/
82 : }
83 16552523 : *ideal = x; return t;
84 : }
85 :
86 : /* true nf; v = [a,x,...], a in Z. Return (a,x) */
87 : GEN
88 803500 : idealhnf_two(GEN nf, GEN v)
89 : {
90 803500 : GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
91 803501 : if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
92 730120 : return ZM_hnfmodid(m, p);
93 : }
94 : /* true nf */
95 : GEN
96 3802931 : pr_hnf(GEN nf, GEN pr)
97 : {
98 3802931 : GEN p = pr_get_p(pr), m;
99 3802910 : if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
100 3366131 : m = zk_scalar_or_multable(nf, pr_get_gen(pr));
101 3365804 : return ZM_hnfmodprime(m, p);
102 : }
103 :
104 : GEN
105 1393275 : idealhnf_principal(GEN nf, GEN x)
106 : {
107 : GEN cx;
108 1393275 : x = nf_to_scalar_or_basis(nf, x);
109 1393272 : switch(typ(x))
110 : {
111 1132051 : case t_COL: break;
112 242977 : case t_INT: if (!signe(x)) return cgetg(1,t_MAT);
113 241311 : return scalarmat(absi_shallow(x), nf_get_degree(nf));
114 18245 : case t_FRAC:
115 18245 : return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
116 0 : default: pari_err_TYPE("idealhnf",x);
117 : }
118 1132051 : x = Q_primitive_part(x, &cx);
119 1132051 : RgV_check_ZV(x, "idealhnf");
120 1132050 : x = zk_multable(nf, x);
121 1132046 : x = ZM_hnfmodid(x, zkmultable_capZ(x));
122 1132054 : return cx? ZM_Q_mul(x,cx): x;
123 : }
124 :
125 : /* true nf; x integral Z_K-module as t_MAT generated by its columns.
126 : * Return square hnf representation */
127 : static GEN
128 455 : vec_mulid(GEN nf, GEN x)
129 : {
130 455 : long i, j, l = lg(x);
131 455 : GEN D = NULL, v;
132 455 : v = cgetg(l, t_VEC);
133 777 : for (i = j = 1; i < l; i++)
134 : {
135 : GEN m, d;
136 630 : if (D && ZV_Z_dvd(gel(x,i), D)) l--;
137 630 : gel(v,j++) = m = zk_multable(nf, gel(x,i));
138 630 : d = zkmultable_capZ(m);
139 630 : D = D? gcdii(D, d): d;
140 630 : if (is_pm1(D)) return matid(lg(m)-1);
141 : }
142 147 : setlg(v, l); if (l == 1) return cgetg(1, t_MAT);
143 147 : return ZM_hnfmodid(shallowconcat1(v), D);
144 : }
145 :
146 : static GEN
147 448 : nfV_to_ZM(GEN nf, GEN x)
148 2254 : { pari_APPLY_type(t_MAT, algtobasis(nf, gel(x,i))) }
149 :
150 :
151 : GEN
152 448 : nfV_idealhnf(GEN nf, GEN v, GEN *pden)
153 : {
154 448 : GEN H = ZM_hnf(Q_remove_denom(nfV_to_ZM(nf, v), pden));
155 448 : return vec_mulid(nf, H);
156 : }
157 :
158 : GEN
159 42 : idealfromgens(GEN nf, GEN v)
160 : {
161 42 : pari_sp av = avma;
162 : GEN H, den;
163 42 : long t = typ(v);
164 42 : if (t==t_MAT) { v = shallowcopy(v); settyp(v, t_VEC); }
165 35 : else if (!is_vec_t(t)) pari_err_TYPE("idealfromgens", v);
166 42 : nf = checknf(nf);
167 42 : H = nfV_idealhnf(nf, v, &den);
168 42 : return gc_upto(av, den ? gdiv(H, den): H);
169 : }
170 :
171 : /* true nf */
172 : GEN
173 1857374 : idealhnf_shallow(GEN nf, GEN x)
174 : {
175 1857374 : long tx = typ(x), lx = lg(x), N;
176 :
177 : /* cannot use idealtyp because here we allow nonsquare matrices */
178 1857374 : if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
179 1857374 : if (tx == t_VEC && lx == 6)
180 : {
181 538657 : if (!checkprid_i(x)) pari_err_TYPE("idealhnf [fake prime ideal]",x);
182 538645 : return pr_hnf(nf,x); /* PRIME */
183 : }
184 1318717 : switch(tx)
185 : {
186 91929 : case t_MAT:
187 : {
188 : GEN cx;
189 91929 : long nx = lx-1;
190 91929 : N = nf_get_degree(nf);
191 91929 : if (nx == 0) return cgetg(1, t_MAT);
192 91908 : if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
193 91901 : if (nx == 1) return idealhnf_principal(nf, gel(x,1));
194 :
195 75018 : if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
196 46382 : x = Q_primitive_part(x, &cx);
197 46382 : if (nx < N)
198 7 : x = vec_mulid(nf, x); /* build ZK-module generated from cols */
199 : else
200 46375 : x = ZM_hnfmod(x, ZM_detmult(x)); /* assume Z-span cols is ZK-module */
201 46382 : return cx? ZM_Q_mul(x,cx): x;
202 : }
203 14 : case t_QFB:
204 : {
205 14 : pari_sp av = avma;
206 14 : GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
207 14 : GEN A = gel(x,1), B = gel(x,2);
208 14 : N = nf_get_degree(nf);
209 14 : if (N != 2)
210 0 : pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
211 14 : if (!equalii(qfb_disc(x), D))
212 7 : pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
213 : /* x -> A Z + (-B + sqrt(D)) / 2 Z
214 : K = Q[t]/T(t), t^2 + ut + v = 0, u^2 - 4v = Df^2
215 : => t = (-u + sqrt(D) f)/2
216 : => sqrt(D)/2 = (t + u/2)/f */
217 7 : u = gel(T,3);
218 7 : B = deg1pol_shallow(ginv(f),
219 : gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
220 7 : varn(T));
221 7 : return gc_upto(av, idealhnf_two(nf, mkvec2(A,B)));
222 : }
223 1226774 : default: return idealhnf_principal(nf, x); /* PRINCIPAL */
224 : }
225 : }
226 : /* true nf */
227 : GEN
228 665 : idealhnf(GEN nf, GEN x)
229 : {
230 665 : pari_sp av = avma;
231 665 : GEN y = idealhnf_shallow(nf, x);
232 651 : return (avma == av)? gcopy(y): gc_upto(av, y);
233 : }
234 :
235 : static GEN
236 84 : nfV_eltembed(GEN nf, GEN x, long prec)
237 518 : { pari_APPLY_type(t_VEC, nfeltembed(nf, gel(x,i), NULL, prec)) }
238 :
239 : /* true nf */
240 : static GEN
241 84 : nfweilheight_i(GEN nf, GEN v, long prec)
242 : {
243 84 : long i, j, r1, r2, u, N, l = lg(v);
244 84 : GEN den, h = gen_1, id = nfV_idealhnf(nf, v, &den);
245 84 : GEN V = nfV_eltembed(nf, v, prec);
246 :
247 84 : nf_get_sign(nf, &r1, &r2); u = r1 + r2; N = u + r2;
248 259 : for (i = 1; i <= r1; i++)
249 1029 : for (j = 1; j < l; j++) gmael(V,j,i) = gabs(gmael(V,j,i), prec);
250 343 : for ( ; i <= u; i++)
251 1771 : for (j = 1; j < l; j++) gmael(V,j,i) = gnorm(gmael(V,j,i));
252 518 : for (i = 1; i <= u; i++)
253 : {
254 434 : long j0 = 1;
255 2366 : for (j = 2; j < l; j++)
256 1932 : if (gcmp(gmael(V,j,i), gmael(V,j0,i)) > 0) j0 = j;
257 434 : h = gmul(h, gmael(V,j0,i));
258 : }
259 84 : if (den) h = gmul(h, powiu(den, N));
260 84 : return divru(glog(gdiv(h, idealnorm(nf, id)), prec), N);
261 : }
262 :
263 : GEN
264 84 : nfweilheight(GEN nf, GEN v, long prec)
265 : {
266 84 : pari_sp av = avma;
267 84 : nf = checknf(nf);
268 84 : if (!is_vec_t(typ(v)) || lg(v) < 2) pari_err_TYPE("nfweilheight",v);
269 84 : return gc_upto(av, nfweilheight_i(nf, v, prec));
270 : }
271 :
272 : /* GP functions */
273 :
274 : GEN
275 2485 : idealtwoelt0(GEN nf, GEN x, GEN a)
276 : {
277 2485 : if (!a) return idealtwoelt(nf,x);
278 42 : return idealtwoelt2(nf,x,a);
279 : }
280 :
281 : GEN
282 2499 : idealpow0(GEN nf, GEN x, GEN n, long flag)
283 : {
284 2499 : if (flag) return idealpowred(nf,x,n);
285 2492 : return idealpow(nf,x,n);
286 : }
287 :
288 : GEN
289 70 : idealmul0(GEN nf, GEN x, GEN y, long flag)
290 : {
291 70 : if (flag) return idealmulred(nf,x,y);
292 63 : return idealmul(nf,x,y);
293 : }
294 :
295 : GEN
296 56 : idealdiv0(GEN nf, GEN x, GEN y, long flag)
297 : {
298 56 : switch(flag)
299 : {
300 28 : case 0: return idealdiv(nf,x,y);
301 28 : case 1: return idealdivexact(nf,x,y);
302 0 : default: pari_err_FLAG("idealdiv");
303 : }
304 : return NULL; /* LCOV_EXCL_LINE */
305 : }
306 :
307 : GEN
308 70 : idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
309 : {
310 70 : if (!arg2) return idealaddmultoone(nf,arg1);
311 35 : return idealaddtoone(nf,arg1,arg2);
312 : }
313 :
314 : /* b not a scalar */
315 : static GEN
316 77 : hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
317 : /* b not a scalar */
318 : static GEN
319 70 : hnf_Z_QC(GEN nf, GEN a, GEN b)
320 : {
321 : GEN db;
322 70 : b = Q_remove_denom(b, &db);
323 70 : if (db) a = mulii(a, db);
324 70 : b = hnf_Z_ZC(nf,a,b);
325 70 : return db? RgM_Rg_div(b, db): b;
326 : }
327 : /* b not a scalar (not point in trying to optimize for this case) */
328 : static GEN
329 77 : hnf_Q_QC(GEN nf, GEN a, GEN b)
330 : {
331 : GEN da, db;
332 77 : if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
333 7 : da = gel(a,2);
334 7 : a = gel(a,1);
335 7 : b = Q_remove_denom(b, &db);
336 : /* write da = d*A, db = d*B, gcd(A,B) = 1
337 : * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
338 7 : if (db)
339 : {
340 7 : GEN d = gcdii(da,db);
341 7 : if (!is_pm1(d)) db = diviiexact(db,d); /* B */
342 7 : if (!is_pm1(db))
343 : {
344 7 : a = mulii(a, db); /* a B */
345 7 : da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
346 : }
347 : }
348 7 : return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
349 : }
350 : static GEN
351 7 : hnf_QC_QC(GEN nf, GEN a, GEN b)
352 : {
353 : GEN da, db, d, x;
354 7 : a = Q_remove_denom(a, &da);
355 7 : b = Q_remove_denom(b, &db);
356 7 : if (da) b = ZC_Z_mul(b, da);
357 7 : if (db) a = ZC_Z_mul(a, db);
358 7 : d = mul_denom(da, db);
359 7 : a = zk_multable(nf,a); da = zkmultable_capZ(a);
360 7 : b = zk_multable(nf,b); db = zkmultable_capZ(b);
361 7 : x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
362 7 : return d? RgM_Rg_div(x, d): x;
363 : }
364 : static GEN
365 21 : hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
366 : GEN
367 413 : idealhnf0(GEN nf, GEN a, GEN b)
368 : {
369 : long ta, tb;
370 : pari_sp av;
371 : GEN x;
372 413 : nf = checknf(nf);
373 413 : if (!b) return idealhnf(nf,a);
374 :
375 : /* HNF of aZ_K+bZ_K */
376 112 : av = avma;
377 112 : a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
378 112 : b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
379 105 : if (ta == t_COL)
380 14 : x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
381 : else
382 91 : x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
383 105 : return gc_upto(av, x);
384 : }
385 :
386 : /*******************************************************************/
387 : /* */
388 : /* TWO-ELEMENT FORM */
389 : /* */
390 : /*******************************************************************/
391 : static GEN idealapprfact_i(GEN nf, GEN x, int nored);
392 :
393 : static int
394 226198 : ok_elt(GEN x, GEN xZ, GEN y)
395 : {
396 226198 : pari_sp av = avma;
397 226198 : return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
398 : }
399 :
400 : /* a + s * b, a and b ZM, s integer */
401 : static GEN
402 66617 : addmul_mat(GEN a, GEN s, GEN b)
403 : {
404 66617 : if (!signe(s)) return a;
405 57955 : if (!equali1(s)) b = ZM_Z_mul(b, s);
406 57955 : return a? ZM_add(a, b): b;
407 : }
408 :
409 : static GEN
410 118264 : get_random_a(GEN nf, GEN x, GEN xZ)
411 : {
412 : pari_sp av;
413 118264 : long i, lm, l = lg(x);
414 : GEN z, beta, mul;
415 :
416 118264 : beta= cgetg(l, t_MAT);
417 118264 : mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
418 : /* look for a in x such that a O/xZ = x O/xZ */
419 251663 : for (i = 2; i < l; i++)
420 : {
421 241430 : GEN xi = gel(x,i);
422 241430 : GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
423 241426 : if (gequal0(t)) continue;
424 197913 : if (ok_elt(x,xZ, t)) return xi;
425 89885 : gel(beta,lm) = xi;
426 : /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
427 89885 : gel(mul,lm) = t; lm++;
428 : }
429 10233 : setlg(mul, lm);
430 10233 : setlg(beta,lm); z = cgetg(lm, t_VEC);
431 30142 : for(av = avma;; set_avma(av))
432 19909 : {
433 30142 : GEN a = NULL;
434 96759 : for (i = 1; i < lm; i++)
435 : {
436 66617 : gel(z,i) = randomi(xZ);
437 66617 : a = addmul_mat(a, gel(z,i), gel(mul,i));
438 : }
439 : /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
440 30142 : if (a && ok_elt(x,xZ, a)) break;
441 : }
442 10233 : return ZM_ZC_mul(beta, z);
443 : }
444 :
445 : /* x square matrix, assume it is HNF */
446 : static GEN
447 251101 : mat_ideal_two_elt(GEN nf, GEN x)
448 : {
449 : GEN y, a, cx, xZ;
450 251101 : long N = nf_get_degree(nf);
451 : pari_sp av, tetpil;
452 :
453 251101 : if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
454 251087 : if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
455 :
456 136846 : y = cgetg(3,t_VEC); av = avma;
457 136845 : cx = Q_content(x);
458 136846 : xZ = gcoeff(x,1,1);
459 136846 : if (gequal(xZ, cx)) /* x = (cx) */
460 : {
461 7566 : gel(y,1) = cx;
462 7566 : gel(y,2) = gen_0; return y;
463 : }
464 129280 : if (equali1(cx)) cx = NULL;
465 : else
466 : {
467 1026 : x = Q_div_to_int(x, cx);
468 1026 : xZ = gcoeff(x,1,1);
469 : }
470 129280 : if (N < 6)
471 109813 : a = get_random_a(nf, x, xZ);
472 : else
473 : {
474 19467 : const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
475 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
476 : };
477 19467 : GEN P, E, a1 = Z_lsmoothen(xZ, (GEN)FB, &P, &E);
478 19467 : if (!a1) /* factors completely */
479 11016 : a = idealapprfact_i(nf, idealfactor(nf,x), 1);
480 8451 : else if (lg(P) == 1) /* no small factors */
481 2604 : a = get_random_a(nf, x, xZ);
482 : else /* general case */
483 : {
484 : GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
485 5847 : a0 = diviiexact(xZ, a1);
486 5847 : A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
487 5847 : A1 = ZM_hnfmodid(x, a1); /* cofactor */
488 5847 : pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
489 5847 : pi1 = get_random_a(nf, A1, a1);
490 5847 : (void)bezout(a0, a1, &v0,&v1);
491 5847 : u0 = mulii(a0, v0);
492 5847 : u1 = mulii(a1, v1);
493 5847 : if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
494 : else
495 5847 : { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
496 5847 : u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
497 5847 : a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
498 : }
499 : }
500 129280 : if (cx)
501 : {
502 1026 : a = centermod(a, xZ);
503 1026 : tetpil = avma;
504 1026 : if (typ(cx) == t_INT)
505 : {
506 91 : gel(y,1) = mulii(xZ, cx);
507 91 : gel(y,2) = ZC_Z_mul(a, cx);
508 : }
509 : else
510 : {
511 935 : gel(y,1) = gmul(xZ, cx);
512 935 : gel(y,2) = RgC_Rg_mul(a, cx);
513 : }
514 : }
515 : else
516 : {
517 128254 : tetpil = avma;
518 128254 : gel(y,1) = icopy(xZ);
519 128254 : gel(y,2) = centermod(a, xZ);
520 : }
521 129280 : gc_slice_unsafe(av,tetpil,y+1,2); return y;
522 : }
523 :
524 : /* Given an ideal x, returns [a,alpha] such that a is in Q,
525 : * x = a Z_K + alpha Z_K, alpha in K^*
526 : * a = 0 or alpha = 0 are possible, but do not try to determine whether
527 : * x is principal. */
528 : GEN
529 97948 : idealtwoelt(GEN nf, GEN x)
530 : {
531 : pari_sp av;
532 97948 : long tx = idealtyp(&x, NULL);
533 97940 : nf = checknf(nf);
534 97940 : if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
535 735 : if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
536 : /* id_PRINCIPAL */
537 714 : av = avma; x = nf_to_scalar_or_basis(nf, x);
538 1232 : return gc_GEN(av, typ(x)==t_COL? mkvec2(gen_0,x):
539 609 : mkvec2(Q_abs_shallow(x),gen_0));
540 : }
541 :
542 : /*******************************************************************/
543 : /* */
544 : /* FACTORIZATION */
545 : /* */
546 : /*******************************************************************/
547 : /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
548 : static long
549 4247787 : idealHNF_norm_pval(GEN x, GEN p, long Zval)
550 : {
551 4247787 : long i, v = Zval, l = lg(x);
552 32401857 : for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
553 4247816 : return v;
554 : }
555 :
556 : /* x integral in HNF, f0 = partial factorization of a multiple of
557 : * x[1,1] = x\cap Z */
558 : GEN
559 293983 : idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
560 : {
561 293983 : GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
562 : long i, l;
563 294002 : P = gel(f,1); l = lg(P);
564 294002 : E = gel(f,2);
565 294002 : *pvN = vN = cgetg(l, t_VECSMALL);
566 294009 : *pvZ = vZ = cgetg(l, t_VECSMALL);
567 734746 : for (i = 1; i < l; i++)
568 : {
569 440736 : GEN p = gel(P,i);
570 440736 : vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
571 440735 : vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
572 : }
573 294010 : return P;
574 : }
575 : /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
576 : * x integral in HNF */
577 : GEN
578 0 : idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
579 0 : { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
580 :
581 : /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
582 : * Return v_P(A) */
583 : static long
584 4385032 : idealHNF_val(GEN A, GEN P, long Nval, long Zval)
585 : {
586 4385032 : long f = pr_get_f(P), vmax, v, e, i, j, k, l;
587 : GEN mul, B, a, y, r, p, pk, cx, vals;
588 : pari_sp av;
589 :
590 4385032 : if (Nval < f) return 0;
591 4381862 : p = pr_get_p(P);
592 4381866 : e = pr_get_e(P);
593 : /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
594 4381869 : vmax = minss(Zval * e, Nval / f);
595 4381866 : mul = pr_get_tau(P);
596 4381864 : l = lg(mul);
597 4381864 : B = cgetg(l,t_MAT);
598 : /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
599 4387739 : gel(B,1) = gen_0; /* dummy */
600 23790019 : for (j = 2; j < l; j++)
601 : {
602 21318912 : GEN x = gel(A,j);
603 21318912 : gel(B,j) = y = cgetg(l, t_COL);
604 229569838 : for (i = 1; i < l; i++)
605 : { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
606 210167558 : a = mulii(gel(x,1), gcoeff(mul,i,1));
607 1470715622 : for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
608 : /* p | a ? */
609 210160056 : gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
610 : }
611 : }
612 2471107 : vals = cgetg(l, t_VECSMALL);
613 : /* vals[1] not needed */
614 16968903 : for (j = 2; j < l; j++)
615 : {
616 14497617 : gel(B,j) = Q_primitive_part(gel(B,j), &cx);
617 14497674 : vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
618 : }
619 2471286 : pk = powiu(p, ceildivuu(vmax, e));
620 2471229 : av = avma; y = cgetg(l,t_COL);
621 : /* can compute mod p^ceil((vmax-v)/e) */
622 4160506 : for (v = 1; v < vmax; v++)
623 : { /* we know v_pr(Bj) >= v for all j */
624 1719572 : if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
625 8525845 : for (j = 2; j < l; j++)
626 : {
627 6836587 : GEN x = gel(B,j); if (v < vals[j]) continue;
628 44680269 : for (i = 1; i < l; i++)
629 : {
630 40068324 : pari_sp av2 = avma;
631 40068324 : a = mulii(gel(x,1), gcoeff(mul,i,1));
632 521979173 : for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
633 : /* a = (x.t_0)_i; p | a ? */
634 40067684 : a = dvmdii(a,p,&r); if (signe(r)) return v;
635 40037225 : if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
636 40037221 : gel(y,i) = gc_INT(av2, a);
637 : }
638 4611945 : gel(B,j) = y; y = x;
639 4611945 : if (gc_needed(av,3))
640 : {
641 0 : if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
642 0 : (void)gc_all(av,3, &y,&B,&pk);
643 : }
644 : }
645 : }
646 2440934 : return v;
647 : }
648 : /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
649 : * FA integer factorization matrix or NULL. Return partial factorization of
650 : * cx * x above primes in FA (complete factorization if !FA)*/
651 : static GEN
652 293983 : idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
653 : {
654 293983 : const long N = lg(x)-1;
655 : long i, j, k, l, v;
656 293983 : GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
657 :
658 294010 : l = lg(vp);
659 294010 : i = cx? expi(cx)+1: 1;
660 294014 : vP = cgetg((l+i-2)*N+1, t_COL);
661 294007 : vE = cgetg((l+i-2)*N+1, t_COL);
662 734740 : for (i = k = 1; i < l; i++)
663 : {
664 440729 : GEN L, p = gel(vp,i);
665 440729 : long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
666 440727 : if (vc)
667 : {
668 48617 : L = idealprimedec(nf,p);
669 48617 : if (is_pm1(cx)) cx = NULL;
670 : }
671 : else
672 392110 : L = idealprimedec_limit_f(nf,p,Nval);
673 1018691 : for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
674 : {
675 577961 : GEN P = gel(L,j);
676 577961 : pari_sp av = avma;
677 577961 : v = idealHNF_val(x, P, Nval, Zval);
678 577934 : set_avma(av);
679 577939 : Nval -= v*pr_get_f(P);
680 577944 : v += vc * pr_get_e(P); if (!v) continue;
681 483519 : gel(vP,k) = P;
682 483519 : gel(vE,k) = utoipos(v); k++;
683 : }
684 491659 : if (vc) for (; j<lg(L); j++)
685 : {
686 50937 : GEN P = gel(L,j);
687 50937 : gel(vP,k) = P;
688 50937 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
689 : }
690 : }
691 294011 : if (cx && !FA)
692 : { /* complete factorization */
693 73365 : GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
694 73363 : long lc = lg(cP);
695 159419 : for (i=1; i<lc; i++)
696 : {
697 86054 : GEN p = gel(cP,i), L = idealprimedec(nf,p);
698 86056 : long vc = itos(gel(cE,i));
699 188317 : for (j=1; j<lg(L); j++)
700 : {
701 102261 : GEN P = gel(L,j);
702 102261 : gel(vP,k) = P;
703 102261 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
704 : }
705 : }
706 : }
707 294011 : setlg(vP, k);
708 294009 : setlg(vE, k); return mkmat2(vP, vE);
709 : }
710 : /* true nf, x integral ideal */
711 : static GEN
712 240109 : idealHNF_factor(GEN nf, GEN x, ulong lim)
713 : {
714 240109 : GEN cx, F = NULL;
715 240109 : if (lim)
716 : {
717 : GEN P, E;
718 : long i;
719 : /* strict useless because of prime table */
720 77 : F = absZ_factor_limit(gcoeff(x,1,1), lim);
721 77 : P = gel(F,1);
722 77 : E = gel(F,2);
723 : /* filter out entries > lim */
724 126 : for (i = lg(P)-1; i; i--)
725 126 : if (cmpiu(gel(P,i), lim) <= 0) break;
726 77 : setlg(P, i+1);
727 77 : setlg(E, i+1);
728 : }
729 240109 : x = Q_primitive_part(x, &cx);
730 240083 : return idealHNF_factor_i(nf, x, cx, F);
731 : }
732 : /* c * vector(#L,i,L[i].e), assume results fit in ulong */
733 : static GEN
734 28514 : prV_e_muls(GEN L, long c)
735 : {
736 28514 : long j, l = lg(L);
737 28514 : GEN z = cgetg(l, t_COL);
738 58721 : for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
739 28498 : return z;
740 : }
741 : /* true nf, y in Q */
742 : static GEN
743 28508 : Q_nffactor(GEN nf, GEN y, ulong lim)
744 : {
745 : GEN f, P, E;
746 : long l, i;
747 28508 : if (typ(y) == t_INT)
748 : {
749 28485 : if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
750 28471 : if (is_pm1(y)) return trivial_fact();
751 : }
752 18001 : y = Q_abs_shallow(y);
753 18004 : if (!lim) f = Q_factor(y);
754 : else
755 : {
756 188 : f = Q_factor_limit(y, lim);
757 189 : P = gel(f,1);
758 189 : E = gel(f,2);
759 273 : for (i = lg(P)-1; i > 0; i--)
760 238 : if (abscmpiu(gel(P,i), lim) < 0) break;
761 189 : setlg(P,i+1); setlg(E,i+1);
762 : }
763 18001 : P = gel(f,1); l = lg(P); if (l == 1) return f;
764 17966 : E = gel(f,2);
765 46500 : for (i = 1; i < l; i++)
766 : {
767 28540 : gel(P,i) = idealprimedec(nf, gel(P,i));
768 28513 : gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
769 : }
770 17960 : P = shallowconcat1(P); gel(f,1) = P; settyp(P, t_COL);
771 17972 : E = shallowconcat1(E); gel(f,2) = E; return f;
772 : }
773 :
774 : GEN
775 25515 : idealfactor_partial(GEN nf, GEN x, GEN L)
776 : {
777 25515 : pari_sp av = avma;
778 : long i, j, l;
779 : GEN P, E;
780 25515 : if (!L) return idealfactor(nf, x);
781 24661 : if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
782 24633 : l = lg(L); if (l == 1) return trivial_fact();
783 23863 : P = cgetg(l, t_VEC);
784 89964 : for (i = 1; i < l; i++)
785 : {
786 66101 : GEN p = gel(L,i);
787 66101 : gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
788 : }
789 23863 : P = shallowconcat1(P); settyp(P, t_COL);
790 23863 : P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
791 23863 : E = cgetg_copy(P, &l);
792 114695 : for (i = j = 1; i < l; i++)
793 : {
794 90832 : long v = idealval(nf, x, gel(P,i));
795 90832 : if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
796 : }
797 23863 : setlg(P,j);
798 23863 : setlg(E,j); return gc_GEN(av, mkmat2(P, E));
799 : }
800 : GEN
801 268741 : idealfactor_limit(GEN nf, GEN x, ulong lim)
802 : {
803 268741 : pari_sp av = avma;
804 : GEN fa, y;
805 268741 : long tx = idealtyp(&x, NULL);
806 :
807 268717 : if (tx == id_PRIME)
808 : {
809 119 : if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
810 112 : retmkmat2(mkcolcopy(x), mkcol(gen_1));
811 : }
812 268598 : nf = checknf(nf);
813 268596 : if (tx == id_PRINCIPAL)
814 : {
815 29864 : y = nf_to_scalar_or_basis(nf, x);
816 29864 : if (typ(y) != t_COL) return gc_GEN(av, Q_nffactor(nf, y, lim));
817 : }
818 240083 : y = idealnumden(nf, x);
819 240095 : fa = idealHNF_factor(nf, gel(y,1), lim);
820 240083 : if (!isint1(gel(y,2)))
821 14 : fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
822 240082 : fa = gc_GEN(av, fa);
823 240098 : return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
824 : }
825 : GEN
826 268296 : idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
827 : GEN
828 189 : gpidealfactor(GEN nf, GEN x, GEN lim)
829 : {
830 189 : ulong L = 0;
831 189 : if (lim)
832 : {
833 70 : if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
834 70 : L = itou(lim);
835 : }
836 189 : return idealfactor_limit(nf, x, L);
837 : }
838 :
839 : static GEN
840 7749 : ramified_root(GEN nf, GEN R, GEN A, long n)
841 : {
842 7749 : GEN v, P = gel(idealfactor(nf, R), 1);
843 7749 : long i, l = lg(P);
844 7749 : v = cgetg(l, t_VECSMALL);
845 8414 : for (i = 1; i < l; i++)
846 : {
847 672 : long w = idealval(nf, A, gel(P,i));
848 672 : if (w % n) return NULL;
849 665 : v[i] = w / n;
850 : }
851 7742 : return idealfactorback(nf, P, v, 0);
852 : }
853 : static int
854 7 : ramified_root_simple(GEN nf, long n, GEN P, GEN v)
855 : {
856 7 : long i, l = lg(v);
857 21 : for (i = 1; i < l; i++)
858 : {
859 14 : long w = v[i] % n;
860 14 : if (w)
861 : {
862 7 : GEN vpr = idealprimedec(nf, gel(P,i));
863 7 : long lpr = lg(vpr), j;
864 14 : for (j = 1; j < lpr; j++)
865 : {
866 7 : long e = pr_get_e(gel(vpr,j));
867 7 : if ((e * w) % n) return 0;
868 : }
869 : }
870 : }
871 7 : return 1;
872 : }
873 : /* true nf, n > 1, A a non-zero integral ideal; check whether A is the n-th
874 : * power of an ideal and set *pB to its n-th root if so */
875 : static long
876 7756 : idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
877 : {
878 : GEN C, root;
879 : long i, l;
880 :
881 7756 : if (typ(A) == t_MAT && ZM_isscalar(A, NULL)) A = gcoeff(A,1,1);
882 7756 : if (typ(A) == t_INT) /* > 0 */
883 : {
884 5551 : GEN P = nf_get_ramified_primes(nf), v, q;
885 5551 : l = lg(P); v = cgetg(l, t_VECSMALL);
886 24962 : for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
887 5551 : C = gen_1;
888 5551 : if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
889 5551 : if (!pB) return ramified_root_simple(nf, n, P, v);
890 5544 : q = factorback2(P, v);
891 5544 : root = ramified_root(nf, q, q, n);
892 5544 : if (!root) return 0;
893 5544 : if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
894 5544 : *pB = root; return 1;
895 : }
896 : /* compute valuations at ramified primes */
897 2205 : root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
898 2205 : if (!root) return 0;
899 : /* remove ramified primes */
900 2198 : if (isint1(root))
901 1834 : root = matid(nf_get_degree(nf));
902 : else
903 364 : A = idealdivexact(nf, A, idealpows(nf,root,n));
904 2198 : A = Q_primitive_part(A, &C);
905 2198 : if (C)
906 : {
907 7 : if (!Z_ispowerall(C,n,&C)) return 0;
908 0 : if (pB) root = ZM_Z_mul(root, C);
909 : }
910 :
911 : /* compute final n-th root, at most degree(nf)-1 iterations */
912 2191 : for (i = 0;; i++)
913 2079 : {
914 4270 : GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
915 4270 : if (is_pm1(a)) break;
916 2107 : if (!Z_ispowerall(a,n,&b)) return 0;
917 2079 : J = idealadd(nf, b, A);
918 2079 : A = idealdivexact(nf, idealpows(nf,J,n), A);
919 : /* div and not divexact here */
920 2079 : if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
921 : }
922 2163 : if (pB) *pB = root;
923 2163 : return 1;
924 : }
925 :
926 : /* A is assumed to be the n-th power of an ideal in nf
927 : returns its n-th root. */
928 : long
929 3906 : idealispower(GEN nf, GEN A, long n, GEN *pB)
930 : {
931 3906 : pari_sp av = avma;
932 : GEN v, N, D;
933 3906 : nf = checknf(nf);
934 3906 : if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
935 3906 : if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
936 3899 : v = idealnumden(nf,A);
937 3899 : if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
938 3899 : if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
939 3857 : if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
940 3857 : if (pB) *pB = gc_upto(av, idealdiv(nf,N,D)); else set_avma(av);
941 3857 : return 1;
942 : }
943 :
944 : /* x t_INT or integral nonzero ideal in HNF */
945 : static GEN
946 118369 : idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
947 : {
948 : GEN cx, y, U, N, F, Q;
949 118369 : if (typ(x) == t_INT)
950 : {
951 63644 : if (!signe(x) || is_pm1(x)) return gen_1;
952 3549 : F = Z_factor_limit(x, B);
953 3549 : gel(F,2) = gdiventgs(gel(F,2), k);
954 3549 : return ginv(factorback(F));
955 : }
956 54725 : N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
957 53909 : F = absZ_factor_limit_strict(N, B, &U);
958 53909 : if (U)
959 : {
960 147 : GEN M = powii(gel(U,1), gel(U,2));
961 147 : y = hnfmodid(x, M); /* coprime part to B! */
962 147 : if (!idealispower(nf, y, k, &U)) U = NULL;
963 147 : x = hnfmodid(x, diviiexact(N, M));
964 : }
965 : /* x = B-smooth part of initial x */
966 53909 : x = Q_primitive_part(x, &cx);
967 53908 : F = idealHNF_factor_i(nf, x, cx, F);
968 53909 : gel(F,2) = gdiventgs(gel(F,2), k);
969 53909 : Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
970 53909 : if (U) Q = idealmul(nf,Q,U);
971 53909 : if (typ(Q) == t_INT) return Q;
972 13153 : y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
973 13153 : return gdiv(y, gcoeff(Q,1,1));
974 : }
975 : GEN
976 59192 : idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
977 : {
978 59192 : pari_sp av = avma;
979 : GEN a, b;
980 59192 : nf = checknf(nf);
981 59192 : if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
982 59192 : x = idealnumden(nf, x);
983 59192 : a = gel(x,1);
984 59192 : if (isintzero(a)) { set_avma(av); return gen_1; }
985 59185 : a = idealredmodpower_i(nf, gel(x,1), n, B);
986 59185 : b = idealredmodpower_i(nf, gel(x,2), n, B);
987 59185 : if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
988 59185 : return gc_GEN(av, a);
989 : }
990 :
991 : /* P prime ideal in idealprimedec format. Return valuation(A) at P */
992 : long
993 9806073 : idealval(GEN nf, GEN A, GEN P)
994 : {
995 9806073 : pari_sp av = avma;
996 : GEN p, cA;
997 9806073 : long vcA, v, Zval, tx = idealtyp(&A, NULL);
998 :
999 9806062 : if (tx == id_PRINCIPAL) return nfval(nf,A,P);
1000 9693442 : checkprid(P);
1001 9693714 : if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
1002 : /* id_MAT */
1003 9693686 : nf = checknf(nf);
1004 9693857 : A = Q_primitive_part(A, &cA);
1005 9693983 : p = pr_get_p(P);
1006 9694015 : vcA = cA? Q_pval(cA,p): 0;
1007 9694017 : if (pr_is_inert(P)) return gc_long(av,vcA);
1008 9295938 : Zval = Z_pval(gcoeff(A,1,1), p);
1009 9295947 : if (!Zval) v = 0;
1010 : else
1011 : {
1012 3806973 : long Nval = idealHNF_norm_pval(A, p, Zval);
1013 3807072 : v = idealHNF_val(A, P, Nval, Zval);
1014 : }
1015 9295877 : return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
1016 : }
1017 : GEN
1018 7119 : gpidealval(GEN nf, GEN ix, GEN P)
1019 : {
1020 7119 : long v = idealval(nf,ix,P);
1021 7105 : return v == LONG_MAX? mkoo(): stoi(v);
1022 : }
1023 :
1024 : /* gcd and generalized Bezout */
1025 :
1026 : GEN
1027 83578 : idealadd(GEN nf, GEN x, GEN y)
1028 : {
1029 83578 : pari_sp av = avma;
1030 : long tx, ty;
1031 : GEN z, a, dx, dy, dz;
1032 :
1033 83578 : tx = idealtyp(&x, NULL);
1034 83578 : ty = idealtyp(&y, NULL); nf = checknf(nf);
1035 83578 : if (tx != id_MAT) x = idealhnf_shallow(nf,x);
1036 83578 : if (ty != id_MAT) y = idealhnf_shallow(nf,y);
1037 83578 : if (lg(x) == 1) return gc_GEN(av,y);
1038 82703 : if (lg(y) == 1) return gc_GEN(av,x); /* check for 0 ideal */
1039 82108 : dx = Q_denom(x);
1040 82108 : dy = Q_denom(y); dz = lcmii(dx,dy);
1041 82108 : if (is_pm1(dz)) dz = NULL; else {
1042 5975 : x = Q_muli_to_int(x, dz);
1043 5975 : y = Q_muli_to_int(y, dz);
1044 : }
1045 82108 : a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
1046 82108 : if (is_pm1(a))
1047 : {
1048 22889 : long N = lg(x)-1;
1049 22889 : if (!dz) { set_avma(av); return matid(N); }
1050 1085 : return gc_upto(av, scalarmat(ginv(dz), N));
1051 : }
1052 59219 : z = ZM_hnfmodid(shallowconcat(x,y), a);
1053 59219 : if (dz) z = RgM_Rg_div(z,dz);
1054 59219 : return gc_upto(av,z);
1055 : }
1056 :
1057 : static GEN
1058 28 : trivial_merge(GEN x)
1059 28 : { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
1060 : /* true nf */
1061 : static GEN
1062 803208 : _idealaddtoone(GEN nf, GEN x, GEN y, long red)
1063 : {
1064 : GEN a;
1065 803208 : long tx = idealtyp(&x, NULL);
1066 803185 : long ty = idealtyp(&y, NULL);
1067 : long ea;
1068 803187 : if (tx != id_MAT) x = idealhnf_shallow(nf, x);
1069 803223 : if (ty != id_MAT) y = idealhnf_shallow(nf, y);
1070 803223 : if (lg(x) == 1)
1071 14 : a = trivial_merge(y);
1072 803209 : else if (lg(y) == 1)
1073 14 : a = trivial_merge(x);
1074 : else
1075 803195 : a = hnfmerge_get_1(x, y);
1076 803189 : if (!a) pari_err_COPRIME("idealaddtoone",x,y);
1077 803175 : if (red && (ea = gexpo(a)) > 10)
1078 : {
1079 4557 : GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
1080 4557 : b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
1081 4557 : if (gexpo(b) < ea) a = b;
1082 : }
1083 803175 : return a;
1084 : }
1085 : /* true nf */
1086 : GEN
1087 17633 : idealaddtoone_i(GEN nf, GEN x, GEN y)
1088 17633 : { return _idealaddtoone(nf, x, y, 1); }
1089 : /* true nf */
1090 : GEN
1091 785580 : idealaddtoone_raw(GEN nf, GEN x, GEN y)
1092 785580 : { return _idealaddtoone(nf, x, y, 0); }
1093 :
1094 : GEN
1095 98 : idealaddtoone(GEN nf, GEN x, GEN y)
1096 : {
1097 98 : GEN z = cgetg(3,t_VEC), a;
1098 98 : pari_sp av = avma;
1099 98 : nf = checknf(nf);
1100 98 : a = gc_upto(av, idealaddtoone_i(nf,x,y));
1101 84 : gel(z,1) = a;
1102 84 : gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
1103 84 : return z;
1104 : }
1105 :
1106 : /* assume elements of list are integral ideals */
1107 : GEN
1108 35 : idealaddmultoone(GEN nf, GEN list)
1109 : {
1110 35 : pari_sp av = avma;
1111 35 : long N, i, l, nz, tx = typ(list);
1112 : GEN H, U, perm, L;
1113 :
1114 35 : nf = checknf(nf); N = nf_get_degree(nf);
1115 35 : if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
1116 35 : l = lg(list);
1117 35 : L = cgetg(l, t_VEC);
1118 35 : if (l == 1)
1119 0 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1120 35 : nz = 0; /* number of nonzero ideals in L */
1121 98 : for (i=1; i<l; i++)
1122 : {
1123 70 : GEN I = gel(list,i);
1124 70 : if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
1125 70 : if (lg(I) != 1)
1126 : {
1127 42 : nz++; RgM_check_ZM(I,"idealaddmultoone");
1128 35 : if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
1129 : }
1130 63 : gel(L,i) = I;
1131 : }
1132 28 : H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
1133 28 : if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
1134 7 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1135 49 : for (i=1; i<=N; i++)
1136 49 : if (perm[i] == 1) break;
1137 21 : U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
1138 21 : nz = 0;
1139 63 : for (i=1; i<l; i++)
1140 : {
1141 42 : GEN c = gel(L,i);
1142 42 : if (lg(c) == 1)
1143 14 : c = gen_0;
1144 : else {
1145 28 : c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
1146 28 : nz++;
1147 : }
1148 42 : gel(L,i) = c;
1149 : }
1150 21 : return gc_GEN(av, L);
1151 : }
1152 :
1153 : /* multiplication */
1154 :
1155 : /* x integral ideal (without archimedean component) in HNF form
1156 : * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
1157 : * alpha a ZV or a ZM (multiplication table). Multiply them */
1158 : static GEN
1159 975146 : idealHNF_mul_two(GEN nf, GEN x, GEN y)
1160 : {
1161 975146 : GEN m, a = gel(y,1), alpha = gel(y,2);
1162 : long i, N;
1163 :
1164 975146 : if (typ(alpha) != t_MAT)
1165 : {
1166 625563 : alpha = zk_scalar_or_multable(nf, alpha);
1167 625562 : if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
1168 14279 : return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
1169 : }
1170 960866 : N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
1171 3807755 : for (i=1; i<=N; i++) gel(m,i) = ZM_ZC_mul(alpha,gel(x,i));
1172 3806726 : for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
1173 960162 : return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
1174 : }
1175 :
1176 : /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
1177 : * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
1178 : GEN
1179 474545 : idealHNF_mul(GEN nf, GEN x, GEN y)
1180 : {
1181 : GEN z;
1182 474545 : if (typ(y) == t_VEC)
1183 301224 : z = idealHNF_mul_two(nf,x,y);
1184 : else
1185 : { /* reduce one ideal to two-elt form. The smallest */
1186 173321 : GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
1187 173321 : if (cmpii(xZ, yZ) < 0)
1188 : {
1189 35945 : if (is_pm1(xZ)) return gcopy(y);
1190 20343 : z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
1191 : }
1192 : else
1193 : {
1194 137376 : if (is_pm1(yZ)) return gcopy(x);
1195 36305 : z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
1196 : }
1197 : }
1198 357887 : return z;
1199 : }
1200 :
1201 : /* operations on elements in factored form */
1202 :
1203 : GEN
1204 224909 : famat_mul_shallow(GEN f, GEN g)
1205 : {
1206 224909 : if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
1207 224909 : if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
1208 224909 : if (lgcols(f) == 1) return g;
1209 167620 : if (lgcols(g) == 1) return f;
1210 165677 : return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
1211 165676 : shallowconcat(gel(f,2), gel(g,2)));
1212 : }
1213 : GEN
1214 88347 : famat_mulpow_shallow(GEN f, GEN g, GEN e)
1215 : {
1216 88347 : if (!signe(e)) return f;
1217 54585 : return famat_mul_shallow(f, famat_pow_shallow(g, e));
1218 : }
1219 :
1220 : GEN
1221 147668 : famat_mulpows_shallow(GEN f, GEN g, long e)
1222 : {
1223 147668 : if (e==0) return f;
1224 118740 : return famat_mul_shallow(f, famat_pows_shallow(g, e));
1225 : }
1226 :
1227 : GEN
1228 10311 : famat_div_shallow(GEN f, GEN g)
1229 10311 : { return famat_mul_shallow(f, famat_inv_shallow(g)); }
1230 :
1231 : GEN
1232 376293 : Z_to_famat(GEN x)
1233 : {
1234 : long k;
1235 376293 : if (equali1(x)) return trivial_fact();
1236 192492 : k = Z_isanypower(x, &x) ;
1237 192492 : return to_famat_shallow(x, k? utoi(k): gen_1);
1238 : }
1239 : GEN
1240 197089 : Q_to_famat(GEN x)
1241 : {
1242 197089 : if (typ(x) == t_INT) return Z_to_famat(x);
1243 179204 : return famat_div(Z_to_famat(gel(x,1)), Z_to_famat(gel(x,2)));
1244 : }
1245 : GEN
1246 0 : to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
1247 : GEN
1248 2769796 : to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
1249 :
1250 : /* concat the single elt x; not gconcat since x may be a t_COL */
1251 : static GEN
1252 155937 : append(GEN v, GEN x)
1253 : {
1254 155937 : long i, l = lg(v);
1255 155937 : GEN w = cgetg(l+1, typ(v));
1256 661454 : for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
1257 155937 : gel(w,i) = gcopy(x); return w;
1258 : }
1259 : /* add x^1 to famat f */
1260 : static GEN
1261 162630 : famat_add(GEN f, GEN x)
1262 : {
1263 162630 : GEN h = cgetg(3,t_MAT);
1264 162630 : if (lgcols(f) == 1)
1265 : {
1266 13200 : gel(h,1) = mkcolcopy(x);
1267 13200 : gel(h,2) = mkcol(gen_1);
1268 : }
1269 : else
1270 : {
1271 149430 : gel(h,1) = append(gel(f,1), x);
1272 149430 : gel(h,2) = gconcat(gel(f,2), gen_1);
1273 : }
1274 162630 : return h;
1275 : }
1276 : /* add x^-1 to famat f */
1277 : static GEN
1278 20914 : famat_sub(GEN f, GEN x)
1279 : {
1280 20914 : GEN h = cgetg(3,t_MAT);
1281 20914 : if (lgcols(f) == 1)
1282 : {
1283 14407 : gel(h,1) = mkcolcopy(x);
1284 14407 : gel(h,2) = mkcol(gen_m1);
1285 : }
1286 : else
1287 : {
1288 6507 : gel(h,1) = append(gel(f,1), x);
1289 6507 : gel(h,2) = gconcat(gel(f,2), gen_m1);
1290 : }
1291 20914 : return h;
1292 : }
1293 :
1294 : GEN
1295 451458 : famat_mul(GEN f, GEN g)
1296 : {
1297 : GEN h;
1298 451458 : if (typ(g) != t_MAT) {
1299 30519 : if (typ(f) == t_MAT) return famat_add(f, g);
1300 0 : h = cgetg(3, t_MAT);
1301 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1302 0 : gel(h,2) = mkcol2(gen_1, gen_1);
1303 0 : return h;
1304 : }
1305 420939 : if (typ(f) != t_MAT) return famat_add(g, f);
1306 288828 : if (lgcols(f) == 1) return gcopy(g);
1307 265495 : if (lgcols(g) == 1) return gcopy(f);
1308 259501 : h = cgetg(3,t_MAT);
1309 259501 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1310 259501 : gel(h,2) = gconcat(gel(f,2), gel(g,2));
1311 259501 : return h;
1312 : }
1313 :
1314 : GEN
1315 200125 : famat_div(GEN f, GEN g)
1316 : {
1317 : GEN h;
1318 200125 : if (typ(g) != t_MAT) {
1319 20872 : if (typ(f) == t_MAT) return famat_sub(f, g);
1320 0 : h = cgetg(3, t_MAT);
1321 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1322 0 : gel(h,2) = mkcol2(gen_1, gen_m1);
1323 0 : return h;
1324 : }
1325 179253 : if (typ(f) != t_MAT) return famat_sub(g, f);
1326 179211 : if (lgcols(f) == 1) return famat_inv(g);
1327 260 : if (lgcols(g) == 1) return gcopy(f);
1328 260 : h = cgetg(3,t_MAT);
1329 260 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1330 260 : gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
1331 260 : return h;
1332 : }
1333 :
1334 : GEN
1335 22984 : famat_sqr(GEN f)
1336 : {
1337 : GEN h;
1338 22984 : if (typ(f) != t_MAT) return to_famat(f,gen_2);
1339 22984 : if (lgcols(f) == 1) return gcopy(f);
1340 13139 : h = cgetg(3,t_MAT);
1341 13139 : gel(h,1) = gcopy(gel(f,1));
1342 13139 : gel(h,2) = gmul2n(gel(f,2),1);
1343 13139 : return h;
1344 : }
1345 :
1346 : GEN
1347 26947 : famat_inv_shallow(GEN f)
1348 : {
1349 26947 : if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
1350 10451 : if (lgcols(f) == 1) return f;
1351 10451 : return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
1352 : }
1353 : GEN
1354 199358 : famat_inv(GEN f)
1355 : {
1356 199358 : if (typ(f) != t_MAT) return to_famat(f,gen_m1);
1357 199358 : if (lgcols(f) == 1) return gcopy(f);
1358 180814 : retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
1359 : }
1360 : GEN
1361 60642 : famat_pow(GEN f, GEN n)
1362 : {
1363 60642 : if (typ(f) != t_MAT) return to_famat(f,n);
1364 60642 : if (lgcols(f) == 1) return gcopy(f);
1365 60642 : retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
1366 : }
1367 : GEN
1368 62032 : famat_pow_shallow(GEN f, GEN n)
1369 : {
1370 62032 : if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
1371 35696 : if (typ(f) != t_MAT) return to_famat_shallow(f,n);
1372 8016 : if (lgcols(f) == 1) return f;
1373 6103 : return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
1374 : }
1375 :
1376 : GEN
1377 151899 : famat_pows_shallow(GEN f, long n)
1378 : {
1379 151899 : if (n==1) return f;
1380 35007 : if (n==-1) return famat_inv_shallow(f);
1381 35000 : if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
1382 26742 : if (lgcols(f) == 1) return f;
1383 26742 : return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
1384 : }
1385 :
1386 : GEN
1387 0 : famat_Z_gcd(GEN M, GEN n)
1388 : {
1389 0 : pari_sp av=avma;
1390 0 : long i, j, l=lgcols(M);
1391 0 : GEN F=cgetg(3,t_MAT);
1392 0 : gel(F,1)=cgetg(l,t_COL);
1393 0 : gel(F,2)=cgetg(l,t_COL);
1394 0 : for (i=1, j=1; i<l; i++)
1395 : {
1396 0 : GEN p = gcoeff(M,i,1);
1397 0 : GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
1398 0 : if (signe(e))
1399 : {
1400 0 : gcoeff(F,j,1)=p;
1401 0 : gcoeff(F,j,2)=e;
1402 0 : j++;
1403 : }
1404 : }
1405 0 : setlg(gel(F,1),j); setlg(gel(F,2),j);
1406 0 : return gc_GEN(av,F);
1407 : }
1408 :
1409 : /* x assumed to be a t_MATs (factorization matrix), or compatible with
1410 : * the element_* functions. */
1411 : static GEN
1412 33946 : ext_sqr(GEN nf, GEN x)
1413 33946 : { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
1414 : static GEN
1415 61062 : ext_mul(GEN nf, GEN x, GEN y)
1416 61062 : { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
1417 : static GEN
1418 20407 : ext_inv(GEN nf, GEN x)
1419 20407 : { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
1420 : static GEN
1421 0 : ext_pow(GEN nf, GEN x, GEN n)
1422 0 : { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
1423 :
1424 : GEN
1425 0 : famat_to_nf(GEN nf, GEN f)
1426 : {
1427 : GEN t, x, e;
1428 : long i;
1429 0 : if (lgcols(f) == 1) return gen_1;
1430 0 : x = gel(f,1);
1431 0 : e = gel(f,2);
1432 0 : t = nfpow(nf, gel(x,1), gel(e,1));
1433 0 : for (i=lg(x)-1; i>1; i--)
1434 0 : t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
1435 0 : return t;
1436 : }
1437 :
1438 : GEN
1439 0 : famat_idealfactor(GEN nf, GEN x)
1440 : {
1441 : long i, l;
1442 0 : GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
1443 0 : for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
1444 0 : h = famat_reduce(famatV_factorback(h,e));
1445 0 : return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
1446 : }
1447 :
1448 : GEN
1449 308204 : famat_reduce(GEN fa)
1450 : {
1451 : GEN E, G, L, g, e;
1452 : long i, k, l;
1453 :
1454 308204 : if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
1455 297261 : g = gel(fa,1); l = lg(g);
1456 297261 : e = gel(fa,2);
1457 297261 : L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
1458 297262 : G = cgetg(l, t_COL);
1459 297262 : E = cgetg(l, t_COL);
1460 : /* merge */
1461 2366633 : for (k=i=1; i<l; i++,k++)
1462 : {
1463 2069370 : gel(G,k) = gel(g,L[i]);
1464 2069370 : gel(E,k) = gel(e,L[i]);
1465 2069370 : if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
1466 : {
1467 839993 : gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
1468 839994 : k--;
1469 : }
1470 : }
1471 : /* kill 0 exponents */
1472 297263 : l = k;
1473 1526641 : for (k=i=1; i<l; i++)
1474 1229378 : if (!gequal0(gel(E,i)))
1475 : {
1476 1204612 : gel(G,k) = gel(G,i);
1477 1204612 : gel(E,k) = gel(E,i); k++;
1478 : }
1479 297263 : setlg(G, k);
1480 297263 : setlg(E, k); return mkmat2(G,E);
1481 : }
1482 : GEN
1483 77 : matreduce(GEN f)
1484 77 : { pari_sp av = avma;
1485 77 : switch(typ(f))
1486 : {
1487 35 : case t_VEC: case t_COL:
1488 : {
1489 35 : GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
1490 35 : return gc_GEN(av, mkmat2(f, zc_to_ZC(e)));
1491 : }
1492 35 : case t_MAT:
1493 35 : if (lg(f) == 3) break;
1494 : default:
1495 14 : pari_err_TYPE("matreduce", f);
1496 : }
1497 28 : if (typ(gel(f,1)) == t_VECSMALL)
1498 0 : f = famatsmall_reduce(f);
1499 : else
1500 : {
1501 28 : if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
1502 21 : f = famat_reduce(f);
1503 : }
1504 21 : return gc_GEN(av, f);
1505 : }
1506 :
1507 : GEN
1508 185892 : famatsmall_reduce(GEN fa)
1509 : {
1510 : GEN E, G, L, g, e;
1511 : long i, k, l;
1512 185892 : if (lgcols(fa) == 1) return fa;
1513 185892 : g = gel(fa,1); l = lg(g);
1514 185892 : e = gel(fa,2);
1515 185892 : L = vecsmall_indexsort(g);
1516 185893 : G = cgetg(l, t_VECSMALL);
1517 185893 : E = cgetg(l, t_VECSMALL);
1518 : /* merge */
1519 504731 : for (k=i=1; i<l; i++,k++)
1520 : {
1521 318838 : G[k] = g[L[i]];
1522 318838 : E[k] = e[L[i]];
1523 318838 : if (k > 1 && G[k] == G[k-1])
1524 : {
1525 7974 : E[k-1] += E[k];
1526 7974 : k--;
1527 : }
1528 : }
1529 : /* kill 0 exponents */
1530 185893 : l = k;
1531 496757 : for (k=i=1; i<l; i++)
1532 310864 : if (E[i])
1533 : {
1534 307219 : G[k] = G[i];
1535 307219 : E[k] = E[i]; k++;
1536 : }
1537 185893 : setlg(G, k);
1538 185893 : setlg(E, k); return mkmat2(G,E);
1539 : }
1540 :
1541 : GEN
1542 67007 : famat_remove_trivial(GEN fa)
1543 : {
1544 67007 : GEN P, E, p = gel(fa,1), e = gel(fa,2);
1545 67007 : long j, k, l = lg(p);
1546 67007 : P = cgetg(l, t_COL);
1547 67007 : E = cgetg(l, t_COL);
1548 1826705 : for (j = k = 1; j < l; j++)
1549 1759698 : if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
1550 67007 : setlg(P, k); setlg(E, k); return mkmat2(P,E);
1551 : }
1552 :
1553 : GEN
1554 13814 : famatV_factorback(GEN v, GEN e)
1555 : {
1556 13814 : long i, l = lg(e);
1557 : GEN V;
1558 13814 : if (l == 1) return trivial_fact();
1559 13429 : V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
1560 56788 : for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
1561 13429 : return V;
1562 : }
1563 :
1564 : GEN
1565 56273 : famatV_zv_factorback(GEN v, GEN e)
1566 : {
1567 56273 : long i, l = lg(e);
1568 : GEN V;
1569 56273 : if (l == 1) return trivial_fact();
1570 53816 : V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
1571 177813 : for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
1572 53815 : return V;
1573 : }
1574 :
1575 : GEN
1576 568151 : ZM_famat_limit(GEN fa, GEN limit)
1577 : {
1578 : pari_sp av;
1579 : GEN E, G, g, e, r;
1580 : long i, k, l, n, lG;
1581 :
1582 568151 : if (lgcols(fa) == 1) return fa;
1583 568144 : g = gel(fa,1); l = lg(g);
1584 568144 : e = gel(fa,2);
1585 1137435 : for(n=0, i=1; i<l; i++)
1586 569292 : if (cmpii(gel(g,i),limit)<=0) n++;
1587 568143 : lG = n<l-1 ? n+2 : n+1;
1588 568143 : G = cgetg(lG, t_COL);
1589 568143 : E = cgetg(lG, t_COL);
1590 568143 : av = avma;
1591 1137435 : for (i=1, k=1, r = gen_1; i<l; i++)
1592 : {
1593 569291 : if (cmpii(gel(g,i),limit)<=0)
1594 : {
1595 569159 : gel(G,k) = gel(g,i);
1596 569159 : gel(E,k) = gel(e,i);
1597 569159 : k++;
1598 133 : } else r = mulii(r, powii(gel(g,i), gel(e,i)));
1599 : }
1600 568144 : if (k<i)
1601 : {
1602 133 : gel(G, k) = gc_INT(av, r);
1603 133 : gel(E, k) = gen_1;
1604 : }
1605 568144 : return mkmat2(G,E);
1606 : }
1607 :
1608 : /* assume pr has degree 1 and coprime to Q_denom(x) */
1609 : static GEN
1610 122255 : to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1611 : {
1612 122255 : GEN d, r, p = modpr_get_p(modpr);
1613 122255 : x = nf_to_scalar_or_basis(nf,x);
1614 122255 : if (typ(x) != t_COL) return Rg_to_Fp(x,p);
1615 121016 : x = Q_remove_denom(x, &d);
1616 121016 : r = zk_to_Fq(x, modpr);
1617 121015 : if (d) r = Fp_div(r, d, p);
1618 121015 : return r;
1619 : }
1620 :
1621 : /* pr coprime to all denominators occurring in x */
1622 : static GEN
1623 665 : famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1624 : {
1625 665 : GEN p = modpr_get_p(modpr);
1626 665 : GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
1627 665 : long i, l = lg(g);
1628 3808 : for (i = 1; i < l; i++)
1629 : {
1630 3143 : GEN n = modii(gel(e,i), q);
1631 3143 : if (signe(n))
1632 : {
1633 3122 : GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
1634 3122 : h = Fp_pow(h, n, p);
1635 3122 : t = t? Fp_mul(t, h, p): h;
1636 : }
1637 : }
1638 665 : return t? modii(t, p): gen_1;
1639 : }
1640 :
1641 : /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
1642 : GEN
1643 119798 : nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1644 : {
1645 665 : return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
1646 120463 : : to_Fp_coprime(nf, x, modpr);
1647 : }
1648 :
1649 : static long
1650 4331903 : zk_pvalrem(GEN x, GEN p, GEN *py)
1651 4331903 : { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
1652 : /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
1653 : * such that x = p^v (newx / dx); dx = NULL if 1 */
1654 : static GEN
1655 4377720 : nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
1656 : {
1657 : long vcx;
1658 : GEN dx;
1659 4377720 : x = nf_to_scalar_or_basis(nf, x);
1660 4377721 : x = Q_remove_denom(x, &dx);
1661 4377731 : if (dx)
1662 : {
1663 61229 : vcx = - Z_pvalrem(dx, p, &dx);
1664 61229 : if (!vcx) vcx = zk_pvalrem(x, p, &x);
1665 61229 : if (isint1(dx)) dx = NULL;
1666 : }
1667 : else
1668 : {
1669 4316502 : vcx = zk_pvalrem(x, p, &x);
1670 4316492 : dx = NULL;
1671 : }
1672 4377721 : *pv = vcx;
1673 4377721 : *pdx = dx; return x;
1674 : }
1675 : /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
1676 : * if p inert (instead of 1) */
1677 : static GEN
1678 95075 : p_makecoprime(GEN pr)
1679 : {
1680 95075 : GEN B = pr_get_tau(pr), b;
1681 : long i, e;
1682 :
1683 95075 : if (typ(B) == t_INT) return NULL;
1684 72318 : b = gel(B,1); /* B = multiplication table by b */
1685 72318 : e = pr_get_e(pr);
1686 72318 : if (e == 1) return b;
1687 : /* one could also divide (exactly) by p in each iteration */
1688 45383 : for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
1689 22178 : return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
1690 : }
1691 :
1692 : /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
1693 : * Method: modify each g[i] so that it becomes coprime to pr,
1694 : * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
1695 : * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
1696 : * Optimizations:
1697 : * 1) remove all powers of p from contents, and consider extra generator p^vp;
1698 : * modified as p * (b/p)^e = b^e / p^(e-1)
1699 : * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
1700 : *
1701 : * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
1702 : * case the e[i] are large */
1703 : GEN
1704 2220947 : famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
1705 : {
1706 2220947 : GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
1707 2220949 : long i, l = lg(g);
1708 :
1709 2220949 : G = cgetg(l+1, t_VEC);
1710 2220958 : E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
1711 6598677 : for (i=1; i < l; i++)
1712 : {
1713 : long vcx;
1714 4377721 : GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
1715 4377712 : if (vcx) /* = v_p(content(g[i])) */
1716 : {
1717 142267 : GEN a = mulsi(vcx, gel(e,i));
1718 142283 : vp = vp? addii(vp, a): a;
1719 : }
1720 : /* x integral, content coprime to p; dx coprime to p */
1721 4377727 : if (typ(x) == t_INT)
1722 : { /* x coprime to p, hence to pr */
1723 1136100 : x = modii(x, prkZ);
1724 1136099 : if (dx) x = Fp_div(x, dx, prkZ);
1725 : }
1726 : else
1727 : {
1728 3241627 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1729 3241594 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1730 3241628 : if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
1731 : }
1732 4377711 : gel(G,i) = x;
1733 4377711 : gel(E,i) = gel(e,i);
1734 : }
1735 :
1736 2220956 : t = vp? p_makecoprime(pr): NULL;
1737 2220958 : if (!t)
1738 : { /* no need for extra generator */
1739 2148717 : setlg(G,l);
1740 2148716 : setlg(E,l);
1741 : }
1742 : else
1743 : {
1744 72241 : gel(G,i) = FpC_red(t, prkZ);
1745 72241 : gel(E,i) = vp;
1746 : }
1747 2220957 : return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
1748 : }
1749 :
1750 : /* simplified version of famat_makecoprime for X = SUnits[1] */
1751 : GEN
1752 98 : sunits_makecoprime(GEN X, GEN pr, GEN prk)
1753 : {
1754 98 : GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
1755 98 : long i, l = lg(X);
1756 :
1757 98 : G = cgetg(l, t_VEC);
1758 9205 : for (i = 1; i < l; i++)
1759 : {
1760 9107 : GEN x = gel(X,i);
1761 9107 : if (typ(x) == t_INT) /* a prime */
1762 1491 : x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
1763 : else
1764 : {
1765 7616 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1766 7616 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1767 : }
1768 9107 : gel(G,i) = x;
1769 : }
1770 98 : return G;
1771 : }
1772 :
1773 : /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
1774 : GEN
1775 18606 : famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
1776 : {
1777 18606 : GEN t, cyc = bid_get_cyc(bid);
1778 18606 : if (lg(cyc) == 1)
1779 0 : t = gen_1;
1780 : else
1781 18606 : t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
1782 : cyc_get_expo(cyc));
1783 18606 : return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
1784 : }
1785 :
1786 : GEN
1787 16015377 : vecmul(GEN x, GEN y)
1788 : {
1789 16015377 : if (!is_vec_t(typ(x))) return gmul(x,y);
1790 3529110 : pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
1791 : }
1792 :
1793 : GEN
1794 185983 : vecsqr(GEN x)
1795 : {
1796 185983 : if (!is_vec_t(typ(x))) return gsqr(x);
1797 46606 : pari_APPLY_same(vecsqr(gel(x,i)))
1798 : }
1799 :
1800 : GEN
1801 826 : vecinv(GEN x)
1802 : {
1803 826 : if (!is_vec_t(typ(x))) return ginv(x);
1804 56 : pari_APPLY_same(vecinv(gel(x,i)))
1805 : }
1806 :
1807 : GEN
1808 0 : vecpow(GEN x, GEN n)
1809 : {
1810 0 : if (!is_vec_t(typ(x))) return powgi(x,n);
1811 0 : pari_APPLY_same(vecpow(gel(x,i), n))
1812 : }
1813 :
1814 : GEN
1815 903 : vecdiv(GEN x, GEN y)
1816 : {
1817 903 : if (!is_vec_t(typ(x))) return gdiv(x,y);
1818 903 : pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
1819 : }
1820 :
1821 : /* A ideal as a square t_MAT */
1822 : static GEN
1823 266228 : idealmulelt(GEN nf, GEN x, GEN A)
1824 : {
1825 : long i, lx;
1826 : GEN dx, dA, D;
1827 266228 : if (lg(A) == 1) return cgetg(1, t_MAT);
1828 266228 : x = nf_to_scalar_or_basis(nf,x);
1829 266228 : if (typ(x) != t_COL)
1830 71769 : return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
1831 194459 : x = Q_remove_denom(x, &dx);
1832 194459 : A = Q_remove_denom(A, &dA);
1833 194459 : x = zk_multable(nf, x);
1834 194459 : D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
1835 194459 : x = zkC_multable_mul(A, x);
1836 194459 : settyp(x, t_MAT); lx = lg(x);
1837 : /* x may contain scalars (at most 1 since the ideal is nonzero)*/
1838 751432 : for (i=1; i<lx; i++)
1839 566226 : if (typ(gel(x,i)) == t_INT)
1840 : {
1841 9253 : if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
1842 9253 : gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
1843 9253 : break;
1844 : }
1845 194459 : x = ZM_hnfmodid(x, D);
1846 194459 : dx = mul_denom(dx,dA);
1847 194459 : return dx? gdiv(x,dx): x;
1848 : }
1849 :
1850 : /* nf a true nf, tx <= ty */
1851 : static GEN
1852 512055 : idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
1853 : {
1854 : GEN z, cx, cy;
1855 512055 : switch(tx)
1856 : {
1857 304262 : case id_PRINCIPAL:
1858 304262 : switch(ty)
1859 : {
1860 37586 : case id_PRINCIPAL:
1861 37586 : return idealhnf_principal(nf, nfmul(nf,x,y));
1862 448 : case id_PRIME:
1863 : {
1864 448 : GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
1865 448 : if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
1866 :
1867 217 : x = nf_to_scalar_or_basis(nf, x);
1868 217 : switch(typ(x))
1869 : {
1870 203 : case t_INT:
1871 203 : if (!signe(x)) return cgetg(1,t_MAT);
1872 203 : return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
1873 7 : case t_FRAC:
1874 7 : return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
1875 : }
1876 : /* t_COL */
1877 7 : x = Q_primitive_part(x, &cx);
1878 7 : x = zk_multable(nf, x);
1879 7 : z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
1880 7 : z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
1881 7 : return cx? ZM_Q_mul(z, cx): z;
1882 : }
1883 266228 : default: /* id_MAT */
1884 266228 : return idealmulelt(nf, x,y);
1885 : }
1886 42850 : case id_PRIME:
1887 42850 : if (ty==id_PRIME)
1888 4347 : { y = pr_hnf(nf,y); cy = NULL; }
1889 : else
1890 38503 : y = Q_primitive_part(y, &cy);
1891 42850 : y = idealHNF_mul_two(nf,y,x);
1892 42851 : return cy? ZM_Q_mul(y,cy): y;
1893 :
1894 164943 : default: /* id_MAT */
1895 : {
1896 164943 : long N = nf_get_degree(nf);
1897 164943 : if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
1898 164929 : x = Q_primitive_part(x, &cx);
1899 164929 : y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
1900 164929 : y = idealHNF_mul(nf,x,y);
1901 164929 : return cx? ZM_Q_mul(y,cx): y;
1902 : }
1903 : }
1904 : }
1905 :
1906 : /* output the ideal product x.y */
1907 : GEN
1908 512055 : idealmul(GEN nf, GEN x, GEN y)
1909 : {
1910 : pari_sp av;
1911 : GEN res, ax, ay, z;
1912 512055 : long tx = idealtyp(&x,&ax);
1913 512055 : long ty = idealtyp(&y,&ay), f;
1914 512055 : if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
1915 512055 : f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1916 512055 : av = avma;
1917 512055 : z = gc_upto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
1918 512042 : if (!f) return z;
1919 28039 : if (ax && ay)
1920 26541 : ax = ext_mul(nf, ax, ay);
1921 : else
1922 1498 : ax = gcopy(ax? ax: ay);
1923 28039 : gel(res,1) = z; gel(res,2) = ax; return res;
1924 : }
1925 :
1926 : /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
1927 : * nf = true nf */
1928 : static GEN
1929 317550 : idealsqrprime(GEN nf, GEN pr, GEN *pc)
1930 : {
1931 317550 : GEN p = pr_get_p(pr), q, gen;
1932 317550 : long e = pr_get_e(pr), f = pr_get_f(pr);
1933 :
1934 317553 : q = (e == 1)? sqri(p): p;
1935 317549 : if (e <= 2 && e * f == nf_get_degree(nf))
1936 : { /* pr^e = (p) */
1937 45692 : *pc = q;
1938 45692 : return mkvec2(gen_1,gen_0);
1939 : }
1940 271856 : gen = nfsqr(nf, pr_get_gen(pr));
1941 271854 : gen = FpC_red(gen, q);
1942 271842 : *pc = NULL;
1943 271842 : return mkvec2(q, gen);
1944 : }
1945 : /* cf idealpow_aux */
1946 : static GEN
1947 39245 : idealsqr_aux(GEN nf, GEN x, long tx)
1948 : {
1949 39245 : GEN T = nf_get_pol(nf), m, cx, a, alpha;
1950 39245 : long N = degpol(T);
1951 39245 : switch(tx)
1952 : {
1953 385 : case id_PRINCIPAL:
1954 385 : return idealhnf_principal(nf, nfsqr(nf,x));
1955 10785 : case id_PRIME:
1956 10785 : if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
1957 10617 : x = idealsqrprime(nf, x, &cx);
1958 10617 : x = idealhnf_two(nf,x);
1959 10617 : return cx? ZM_Z_mul(x, cx): x;
1960 28075 : default:
1961 28075 : x = Q_primitive_part(x, &cx);
1962 28075 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
1963 28075 : alpha = nfsqr(nf,alpha);
1964 28075 : m = zk_scalar_or_multable(nf, alpha);
1965 28075 : if (typ(m) == t_INT) {
1966 1642 : x = gcdii(sqri(a), m);
1967 1642 : if (cx) x = gmul(x, gsqr(cx));
1968 1642 : x = scalarmat(x, N);
1969 : }
1970 : else
1971 : { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
1972 26433 : x = ZM_hnfmodid(m, sqri(a));
1973 26433 : if (cx) cx = gsqr(cx);
1974 26433 : if (cx) x = ZM_Q_mul(x, cx);
1975 : }
1976 28075 : return x;
1977 : }
1978 : }
1979 : GEN
1980 39245 : idealsqr(GEN nf, GEN x)
1981 : {
1982 : pari_sp av;
1983 : GEN res, ax, z;
1984 39245 : long tx = idealtyp(&x,&ax);
1985 39245 : res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1986 39245 : av = avma;
1987 39245 : z = gc_upto(av, idealsqr_aux(checknf(nf), x, tx));
1988 39245 : if (!ax) return z;
1989 33946 : gel(res,1) = z;
1990 33946 : gel(res,2) = ext_sqr(nf, ax); return res;
1991 : }
1992 :
1993 : /* norm of an ideal */
1994 : GEN
1995 106058 : idealnorm(GEN nf, GEN x)
1996 : {
1997 : pari_sp av;
1998 : long tx;
1999 :
2000 106058 : switch(idealtyp(&x, NULL))
2001 : {
2002 4935 : case id_PRIME: return pr_norm(x);
2003 11179 : case id_MAT: return RgM_det_triangular(x);
2004 : }
2005 : /* id_PRINCIPAL */
2006 89944 : nf = checknf(nf); av = avma;
2007 89944 : x = nfnorm(nf, x);
2008 89944 : tx = typ(x);
2009 89944 : if (tx == t_INT) return gc_INT(av, absi(x));
2010 420 : if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
2011 420 : return gc_upto(av, Q_abs(x));
2012 : }
2013 :
2014 : /* x \cap Z */
2015 : GEN
2016 3031 : idealdown(GEN nf, GEN x)
2017 : {
2018 3031 : pari_sp av = avma;
2019 : GEN y, c;
2020 3031 : switch(idealtyp(&x, NULL))
2021 : {
2022 7 : case id_PRIME: return icopy(pr_get_p(x));
2023 2121 : case id_MAT: return gcopy(gcoeff(x,1,1));
2024 : }
2025 : /* id_PRINCIPAL */
2026 903 : nf = checknf(nf); av = avma;
2027 903 : x = nf_to_scalar_or_basis(nf, x);
2028 903 : if (is_rational_t(typ(x))) return Q_abs(x);
2029 14 : x = Q_primitive_part(x, &c);
2030 14 : y = zkmultable_capZ(zk_multable(nf, x));
2031 14 : return gc_GEN(av, mul_content(c, y));
2032 : }
2033 :
2034 : /* true nf */
2035 : static GEN
2036 42 : idealismaximal_int(GEN nf, GEN p)
2037 : {
2038 : GEN L;
2039 42 : if (!BPSW_psp(p)) return NULL;
2040 77 : if (!dvdii(nf_get_index(nf), p) &&
2041 49 : !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
2042 28 : L = idealprimedec(nf, p);
2043 28 : return (lg(L) == 2 && pr_get_e(gel(L,1)) == 1)? gel(L,1): NULL;
2044 : }
2045 : /* true nf */
2046 : static GEN
2047 21 : idealismaximal_mat(GEN nf, GEN x)
2048 : {
2049 : GEN p, c, L;
2050 : long i, l, f;
2051 21 : x = Q_primitive_part(x, &c);
2052 21 : p = gcoeff(x,1,1);
2053 21 : if (c)
2054 : {
2055 7 : if (typ(c) == t_FRAC || !equali1(p)) return NULL;
2056 7 : return idealismaximal_int(nf, c);
2057 : }
2058 14 : if (!BPSW_psp(p)) return NULL;
2059 14 : l = lg(x); f = 1;
2060 35 : for (i = 2; i < l; i++)
2061 : {
2062 21 : c = gcoeff(x,i,i);
2063 21 : if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
2064 : }
2065 14 : L = idealprimedec_limit_f(nf, p, f);
2066 28 : for (i = lg(L)-1; i; i--)
2067 : {
2068 28 : GEN pr = gel(L,i);
2069 28 : if (pr_get_f(pr) != f) break;
2070 28 : if (idealval(nf, x, pr) == 1) return pr;
2071 : }
2072 0 : return NULL;
2073 : }
2074 : /* true nf */
2075 : static GEN
2076 77 : idealismaximal_i(GEN nf, GEN x)
2077 : {
2078 : GEN L, p, pr, c;
2079 : long i, l;
2080 77 : switch(idealtyp(&x, NULL))
2081 : {
2082 7 : case id_PRIME: return x;
2083 21 : case id_MAT: return idealismaximal_mat(nf, x);
2084 : }
2085 : /* id_PRINCIPAL */
2086 49 : x = nf_to_scalar_or_basis(nf, x);
2087 49 : switch(typ(x))
2088 : {
2089 35 : case t_INT: return idealismaximal_int(nf, absi_shallow(x));
2090 0 : case t_FRAC: return NULL;
2091 : }
2092 14 : x = Q_primitive_part(x, &c);
2093 14 : if (c) return NULL;
2094 14 : p = zkmultable_capZ(zk_multable(nf, x));
2095 14 : if (!BPSW_psp(p)) return NULL;
2096 7 : L = idealprimedec(nf, p); l = lg(L); pr = NULL;
2097 21 : for (i = 1; i < l; i++)
2098 : {
2099 14 : long v = ZC_nfval(x, gel(L,i));
2100 14 : if (v > 1 || (v && pr)) return NULL;
2101 14 : pr = gel(L,i);
2102 : }
2103 7 : return pr;
2104 : }
2105 : GEN
2106 77 : idealismaximal(GEN nf, GEN x)
2107 : {
2108 77 : pari_sp av = avma;
2109 77 : x = idealismaximal_i(checknf(nf), x);
2110 77 : if (!x) { set_avma(av); return gen_0; }
2111 49 : return gc_GEN(av, x);
2112 : }
2113 :
2114 : /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
2115 : *
2116 : * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
2117 : * nf[5][7] = same in 2-elt form.
2118 : * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
2119 : GEN
2120 212260 : idealHNF_inv_Z(GEN nf, GEN I)
2121 : {
2122 212260 : GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
2123 212260 : if (isint1(IZ)) return matid(lg(I)-1);
2124 197797 : J = idealHNF_mul(nf,I, gmael(nf,5,7));
2125 : /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
2126 : * missing content cancels while solving the linear equation */
2127 197799 : dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
2128 197799 : return ZM_hnfmodid(dual, IZ);
2129 : }
2130 : /* I HNF with rational coefficients (denominator d). */
2131 : GEN
2132 69867 : idealHNF_inv(GEN nf, GEN I)
2133 : {
2134 69867 : GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
2135 69867 : J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
2136 69867 : return equali1(IQ)? J: RgM_Rg_div(J, IQ);
2137 : }
2138 :
2139 : /* return p * P^(-1) [integral] */
2140 : GEN
2141 38687 : pr_inv_p(GEN pr)
2142 : {
2143 38687 : if (pr_is_inert(pr)) return matid(pr_get_f(pr));
2144 38015 : return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
2145 : }
2146 : GEN
2147 17864 : pr_inv(GEN pr)
2148 : {
2149 17864 : GEN p = pr_get_p(pr);
2150 17864 : if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
2151 17591 : return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
2152 : }
2153 :
2154 : GEN
2155 115839 : idealinv(GEN nf, GEN x)
2156 : {
2157 : GEN res, ax;
2158 : pari_sp av;
2159 115839 : long tx = idealtyp(&x,&ax), N;
2160 :
2161 115839 : res = ax? cgetg(3,t_VEC): NULL;
2162 115839 : nf = checknf(nf); av = avma;
2163 115839 : N = nf_get_degree(nf);
2164 115839 : switch (tx)
2165 : {
2166 62967 : case id_MAT:
2167 62967 : if (lg(x)-1 != N) pari_err_DIM("idealinv");
2168 62967 : x = idealHNF_inv(nf,x); break;
2169 35840 : case id_PRINCIPAL:
2170 35840 : x = nf_to_scalar_or_basis(nf, x);
2171 35840 : if (typ(x) != t_COL)
2172 35791 : x = idealhnf_principal(nf,ginv(x));
2173 : else
2174 : { /* nfinv + idealhnf where we already know (x) \cap Z */
2175 : GEN c, d;
2176 49 : x = Q_remove_denom(x, &c);
2177 49 : x = zk_inv(nf, x);
2178 49 : x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
2179 49 : if (!d) /* x and x^(-1) integral => x a unit */
2180 14 : x = c? scalarmat(c, N): matid(N);
2181 : else
2182 : {
2183 35 : c = c? gdiv(c,d): ginv(d);
2184 35 : x = zk_multable(nf, x);
2185 35 : x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
2186 : }
2187 : }
2188 35840 : break;
2189 17032 : case id_PRIME:
2190 17032 : x = pr_inv(x); break;
2191 : }
2192 115839 : x = gc_upto(av,x); if (!ax) return x;
2193 20407 : gel(res,1) = x;
2194 20407 : gel(res,2) = ext_inv(nf, ax); return res;
2195 : }
2196 :
2197 : /* write x = A/B, A,B coprime integral ideals */
2198 : GEN
2199 389740 : idealnumden(GEN nf, GEN x)
2200 : {
2201 389740 : pari_sp av = avma;
2202 : GEN x0, c, d, A, B, J;
2203 389740 : long tx = idealtyp(&x, NULL);
2204 389740 : nf = checknf(nf);
2205 389744 : switch (tx)
2206 : {
2207 7 : case id_PRIME:
2208 7 : retmkvec2(idealhnf(nf, x), gen_1);
2209 148183 : case id_PRINCIPAL:
2210 : {
2211 : GEN xZ, mx;
2212 148183 : x = nf_to_scalar_or_basis(nf, x);
2213 148183 : switch(typ(x))
2214 : {
2215 88060 : case t_INT: return gc_GEN(av, mkvec2(absi_shallow(x),gen_1));
2216 2639 : case t_FRAC:return gc_GEN(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
2217 : }
2218 : /* t_COL */
2219 57484 : x = Q_remove_denom(x, &d);
2220 57484 : if (!d) return gc_GEN(av, mkvec2(idealhnf_shallow(nf, x), gen_1));
2221 105 : mx = zk_multable(nf, x);
2222 105 : xZ = zkmultable_capZ(mx);
2223 105 : x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
2224 105 : x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
2225 105 : break;
2226 : }
2227 241554 : default: /* id_MAT */
2228 : {
2229 241554 : long n = lg(x)-1;
2230 241554 : if (n == 0) return mkvec2(gen_0, gen_1);
2231 241554 : if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
2232 241554 : x0 = x = Q_remove_denom(x, &d);
2233 241551 : if (!d) return gc_GEN(av, mkvec2(x, gen_1));
2234 21 : break;
2235 : }
2236 : }
2237 126 : J = hnfmodid(x, d); /* = d/B */
2238 126 : c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
2239 126 : B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
2240 126 : if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
2241 126 : A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
2242 126 : A = ZM_Z_divexact(A, d); /* = A ! */
2243 126 : return gc_GEN(av, mkvec2(A, B));
2244 : }
2245 :
2246 : /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
2247 : * nf = true nf */
2248 : static GEN
2249 1309169 : idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
2250 : {
2251 1309169 : GEN p = pr_get_p(pr), q, gen;
2252 :
2253 1309154 : *pc = NULL;
2254 1309154 : if (is_pm1(n)) /* n = 1 special cased for efficiency */
2255 : {
2256 636176 : q = p;
2257 636176 : if (typ(pr_get_tau(pr)) == t_INT) /* inert */
2258 : {
2259 0 : *pc = (signe(n) >= 0)? p: ginv(p);
2260 0 : return mkvec2(gen_1,gen_0);
2261 : }
2262 636158 : if (signe(n) >= 0) gen = pr_get_gen(pr);
2263 : else
2264 : {
2265 170800 : gen = pr_get_tau(pr); /* possibly t_MAT */
2266 170817 : *pc = ginv(p);
2267 : }
2268 : }
2269 673052 : else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
2270 : else
2271 : {
2272 366119 : long e = pr_get_e(pr), f = pr_get_f(pr);
2273 366125 : GEN r, m = truedvmdis(n, e, &r);
2274 366098 : if (e * f == nf_get_degree(nf))
2275 : { /* pr^e = (p) */
2276 76661 : if (signe(m)) *pc = powii(p,m);
2277 76662 : if (!signe(r)) return mkvec2(gen_1,gen_0);
2278 36563 : q = p;
2279 36563 : gen = nfpow(nf, pr_get_gen(pr), r);
2280 : }
2281 : else
2282 : {
2283 289449 : m = absi_shallow(m);
2284 289452 : if (signe(r)) m = addiu(m,1);
2285 289452 : q = powii(p,m); /* m = ceil(|n|/e) */
2286 289459 : if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
2287 : else
2288 : {
2289 43151 : gen = pr_get_tau(pr);
2290 43151 : if (typ(gen) == t_MAT) gen = gel(gen,1);
2291 43151 : n = negi(n);
2292 43151 : gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
2293 43148 : *pc = ginv(q);
2294 : }
2295 : }
2296 326026 : gen = FpC_red(gen, q);
2297 : }
2298 962172 : return mkvec2(q, gen);
2299 : }
2300 :
2301 : /* True nf. x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
2302 : GEN
2303 828044 : idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
2304 : {
2305 : GEN c, cx, y;
2306 828044 : long N = nf_get_degree(nf);
2307 :
2308 828031 : if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
2309 :
2310 : /* inert, special cased for efficiency */
2311 828024 : if (pr_is_inert(pr))
2312 : {
2313 76993 : GEN q = powii(pr_get_p(pr), n);
2314 74941 : return typ(x) == t_MAT? RgM_Rg_mul(x,q)
2315 151931 : : scalarmat_shallow(gmul(Q_abs(x),q), N);
2316 : }
2317 :
2318 751037 : y = idealpowprime(nf, pr, n, &c);
2319 751012 : if (typ(x) == t_MAT)
2320 748669 : { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
2321 : else
2322 2343 : { cx = x; x = NULL; }
2323 750812 : cx = mul_content(c,cx);
2324 750837 : if (x)
2325 574409 : x = idealHNF_mul_two(nf,x,y);
2326 : else
2327 176428 : x = idealhnf_two(nf,y);
2328 751221 : if (cx) x = ZM_Q_mul(x,cx);
2329 750910 : return x;
2330 : }
2331 : GEN
2332 13007 : idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
2333 : {
2334 13007 : return idealmulpowprime(nf,x,pr, negi(n));
2335 : }
2336 :
2337 : /* nf = true nf */
2338 : static GEN
2339 942139 : idealpow_aux(GEN nf, GEN x, long tx, GEN n)
2340 : {
2341 942139 : GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
2342 942139 : long N = degpol(T), s = signe(n);
2343 942140 : if (!s) return matid(N);
2344 927137 : switch(tx)
2345 : {
2346 75528 : case id_PRINCIPAL:
2347 75528 : return idealhnf_principal(nf, nfpow(nf,x,n));
2348 656634 : case id_PRIME:
2349 656634 : if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
2350 558113 : x = idealpowprime(nf, x, n, &cx);
2351 558101 : x = idealhnf_two(nf,x);
2352 558120 : return cx? ZM_Q_mul(x, cx): x;
2353 194975 : default:
2354 194975 : if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
2355 69173 : n1 = (s < 0)? negi(n): n;
2356 :
2357 69173 : x = Q_primitive_part(x, &cx);
2358 69173 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
2359 69173 : alpha = nfpow(nf,alpha,n1);
2360 69173 : m = zk_scalar_or_multable(nf, alpha);
2361 69173 : if (typ(m) == t_INT) {
2362 553 : x = gcdii(powii(a,n1), m);
2363 553 : if (s<0) x = ginv(x);
2364 553 : if (cx) x = gmul(x, powgi(cx,n));
2365 553 : x = scalarmat(x, N);
2366 : }
2367 : else
2368 : { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
2369 68620 : x = ZM_hnfmodid(m, powii(a,n1));
2370 68620 : if (cx) cx = powgi(cx,n);
2371 68620 : if (s<0) {
2372 7 : GEN xZ = gcoeff(x,1,1);
2373 7 : cx = cx ? gdiv(cx, xZ): ginv(xZ);
2374 7 : x = idealHNF_inv_Z(nf,x);
2375 : }
2376 68620 : if (cx) x = ZM_Q_mul(x, cx);
2377 : }
2378 69173 : return x;
2379 : }
2380 : }
2381 :
2382 : /* raise the ideal x to the power n (in Z) */
2383 : GEN
2384 942140 : idealpow(GEN nf, GEN x, GEN n)
2385 : {
2386 : pari_sp av;
2387 : long tx;
2388 : GEN res, ax;
2389 :
2390 942140 : if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
2391 942140 : tx = idealtyp(&x,&ax);
2392 942141 : res = ax? cgetg(3,t_VEC): NULL;
2393 942141 : av = avma;
2394 942141 : x = gc_upto(av, idealpow_aux(checknf(nf), x, tx, n));
2395 942140 : if (!ax) return x;
2396 0 : gel(res,1) = x;
2397 0 : gel(res,2) = ext_pow(nf, ax, n);
2398 0 : return res;
2399 : }
2400 :
2401 : /* Return ideal^e in number field nf. e is a C integer. */
2402 : GEN
2403 313385 : idealpows(GEN nf, GEN ideal, long e)
2404 : {
2405 313385 : long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
2406 313385 : affsi(e,court); return idealpow(nf,ideal,court);
2407 : }
2408 :
2409 : static GEN
2410 28606 : _idealmulred(GEN nf, GEN x, GEN y)
2411 28606 : { return idealred(nf,idealmul(nf,x,y)); }
2412 : static GEN
2413 35759 : _idealsqrred(GEN nf, GEN x)
2414 35759 : { return idealred(nf,idealsqr(nf,x)); }
2415 : static GEN
2416 11385 : _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
2417 : static GEN
2418 35759 : _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
2419 :
2420 : /* compute x^n (x ideal, n integer), reducing along the way */
2421 : GEN
2422 80385 : idealpowred(GEN nf, GEN x, GEN n)
2423 : {
2424 80385 : pari_sp av = avma, av2;
2425 : long s;
2426 : GEN y;
2427 :
2428 80385 : if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
2429 80385 : s = signe(n); if (s == 0) return idealpow(nf,x,n);
2430 80385 : y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
2431 80385 : av2 = avma;
2432 80385 : if (s < 0) y = idealinv(nf,y);
2433 80385 : if (s < 0 || is_pm1(n)) y = idealred(nf,y);
2434 80386 : return avma == av2? gc_GEN(av,y): gc_upto(av,y);
2435 : }
2436 :
2437 : GEN
2438 17221 : idealmulred(GEN nf, GEN x, GEN y)
2439 : {
2440 17221 : pari_sp av = avma;
2441 17221 : return gc_upto(av, _idealmulred(nf,x,y));
2442 : }
2443 :
2444 : long
2445 91 : isideal(GEN nf,GEN x)
2446 : {
2447 91 : long N, i, j, lx, tx = typ(x);
2448 : pari_sp av;
2449 : GEN T, xZ;
2450 :
2451 91 : nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
2452 91 : if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
2453 91 : switch(tx)
2454 : {
2455 14 : case t_INT: case t_FRAC: return 1;
2456 7 : case t_POL: return varn(x) == varn(T);
2457 7 : case t_POLMOD: return RgX_equal_var(T, gel(x,1));
2458 14 : case t_VEC: return get_prid(x)? 1 : 0;
2459 42 : case t_MAT: break;
2460 7 : default: return 0;
2461 : }
2462 42 : N = degpol(T);
2463 42 : if (lx-1 != N) return (lx == 1);
2464 28 : if (nbrows(x) != N) return 0;
2465 :
2466 28 : av = avma; x = Q_primpart(x);
2467 28 : if (!ZM_ishnf(x)) return 0;
2468 14 : xZ = gcoeff(x,1,1);
2469 21 : for (j=2; j<=N; j++)
2470 14 : if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
2471 14 : for (i=2; i<=N; i++)
2472 14 : for (j=2; j<=N; j++)
2473 7 : if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
2474 7 : return gc_long(av,1);
2475 : }
2476 :
2477 : GEN
2478 39744 : idealdiv(GEN nf, GEN x, GEN y)
2479 : {
2480 39744 : pari_sp av = avma;
2481 39744 : return gc_upto(av, idealmul(nf, x, idealinv(nf,y)));
2482 : }
2483 :
2484 : /* This routine computes the quotient x/y of two ideals in the number field nf.
2485 : * It assumes that the quotient is an integral ideal. The idea is to find an
2486 : * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1. Then
2487 : *
2488 : * x + (Nx/Nz) x
2489 : * ----------- = ---
2490 : * y + (Ny/Nz) y
2491 : *
2492 : * Proof: we can assume x and y are integral. Let p be any prime ideal
2493 : *
2494 : * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
2495 : * product of the integers N(x/y) and N(y/z)). Both the numerator and the
2496 : * denominator on the left will be coprime to p. So will x/y, since x/y is
2497 : * assumed integral and its norm N(x/y) is coprime to p.
2498 : *
2499 : * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
2500 : * Hence v_p (x + Nx/Nz) = v_p(x). Likewise for the denominators. QED.
2501 : *
2502 : * Peter Montgomery. July, 1994. */
2503 : static void
2504 7 : err_divexact(GEN x, GEN y)
2505 7 : { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
2506 0 : gen_1,mkvec2(x,y)); }
2507 : GEN
2508 5263 : idealdivexact(GEN nf, GEN x0, GEN y0)
2509 : {
2510 5263 : pari_sp av = avma;
2511 : GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
2512 :
2513 5263 : nf = checknf(nf);
2514 5263 : x = idealhnf_shallow(nf, x0);
2515 5263 : y = idealhnf_shallow(nf, y0);
2516 5263 : if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
2517 5256 : if (lg(x) == 1) retgc_const(av, cgetg(1, t_MAT)); /* numerator is zero */
2518 5256 : y = Q_primitive_part(y, &cy);
2519 5256 : if (cy) x = RgM_Rg_div(x,cy);
2520 5256 : xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
2521 5249 : yZ = gcoeff(y,1,1); if (isint1(yZ)) return gc_GEN(av, x);
2522 2870 : Nx = idealnorm(nf,x);
2523 2870 : Ny = idealnorm(nf,y);
2524 2870 : if (typ(Nx) != t_INT) err_divexact(x,y);
2525 2870 : q = dvmdii(Nx,Ny, &r);
2526 2870 : if (signe(r)) err_divexact(x,y);
2527 2870 : if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
2528 : /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
2529 616 : for (Nz = Ny;;) /* q = Nx/Nz */
2530 533 : {
2531 1149 : GEN p1 = gcdii(Nz, q);
2532 1149 : if (is_pm1(p1)) break;
2533 533 : Nz = diviiexact(Nz,p1);
2534 533 : q = mulii(q,p1);
2535 : }
2536 616 : xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
2537 616 : if (!equalii(xZ,q))
2538 : { /* Replace x/y by x+(Nx/Nz) / y+(Ny/Nz) */
2539 468 : x = ZM_hnfmodid(x, q);
2540 : /* y reduced to unit ideal ? */
2541 468 : if (Nz == Ny) return gc_upto(av, x);
2542 :
2543 146 : yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
2544 146 : y = ZM_hnfmodid(y, q);
2545 : }
2546 294 : yZ = gcoeff(y,1,1);
2547 294 : y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
2548 294 : return gc_upto(av, ZM_Z_divexact(y, yZ));
2549 : }
2550 :
2551 : GEN
2552 21 : idealintersect(GEN nf, GEN x, GEN y)
2553 : {
2554 21 : pari_sp av = avma;
2555 : GEN z, dx, dy;
2556 :
2557 21 : nf = checknf(nf);
2558 21 : x = idealhnf_shallow(nf,x);
2559 21 : y = idealhnf_shallow(nf,y);
2560 21 : if (lg(x) == 1 || lg(y) == 1) retgc_const(av, cgetg(1, t_MAT));
2561 14 : x = Q_remove_denom(x, &dx);
2562 14 : y = Q_remove_denom(y, &dy);
2563 14 : if (dx) y = ZM_Z_mul(y, dx);
2564 14 : if (dy) x = ZM_Z_mul(x, dy);
2565 14 : dx = mul_denom(dx,dy);
2566 14 : z = ZM_hnfintersectmod(x,y, lcmii(gcoeff(x,1,1), gcoeff(y,1,1)));
2567 14 : if (dx) z = RgM_Rg_div(z,dx);
2568 14 : return gc_upto(av,z);
2569 : }
2570 :
2571 : /*******************************************************************/
2572 : /* */
2573 : /* T2-IDEAL REDUCTION */
2574 : /* */
2575 : /*******************************************************************/
2576 :
2577 : static GEN
2578 21 : chk_vdir(GEN nf, GEN vdir)
2579 : {
2580 21 : long i, l = lg(vdir);
2581 : GEN v;
2582 21 : if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
2583 14 : switch(typ(vdir))
2584 : {
2585 0 : case t_VECSMALL: return vdir;
2586 14 : case t_VEC: break;
2587 0 : default: pari_err_TYPE("idealred",vdir);
2588 : }
2589 14 : v = cgetg(l, t_VECSMALL);
2590 56 : for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
2591 14 : return v;
2592 : }
2593 :
2594 : static void
2595 12709 : twistG(GEN G, long r1, long i, long v)
2596 : {
2597 12709 : long j, lG = lg(G);
2598 12709 : if (i <= r1) {
2599 37275 : for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
2600 : } else {
2601 648 : long k = (i<<1) - r1;
2602 4640 : for (j=1; j<lG; j++)
2603 : {
2604 3992 : gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
2605 3992 : gcoeff(G,k ,j) = gmul2n(gcoeff(G,k ,j), v);
2606 : }
2607 : }
2608 12709 : }
2609 :
2610 : GEN
2611 139185 : nf_get_Gtwist(GEN nf, GEN vdir)
2612 : {
2613 : long i, l, v, r1;
2614 : GEN G;
2615 :
2616 139185 : if (!vdir) return nf_get_roundG(nf);
2617 21 : if (typ(vdir) == t_MAT)
2618 : {
2619 0 : long N = nf_get_degree(nf);
2620 0 : if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
2621 0 : return vdir;
2622 : }
2623 21 : vdir = chk_vdir(nf, vdir);
2624 14 : G = RgM_shallowcopy(nf_get_G(nf));
2625 14 : r1 = nf_get_r1(nf);
2626 14 : l = lg(vdir);
2627 56 : for (i=1; i<l; i++)
2628 : {
2629 42 : v = vdir[i]; if (!v) continue;
2630 42 : twistG(G, r1, i, v);
2631 : }
2632 14 : return RM_round_maxrank(G);
2633 : }
2634 : GEN
2635 12667 : nf_get_Gtwist1(GEN nf, long i)
2636 : {
2637 12667 : GEN G = RgM_shallowcopy( nf_get_G(nf) );
2638 12667 : long r1 = nf_get_r1(nf);
2639 12667 : twistG(G, r1, i, 10);
2640 12667 : return RM_round_maxrank(G);
2641 : }
2642 :
2643 : GEN
2644 98529 : RM_round_maxrank(GEN G0)
2645 : {
2646 98529 : long e, r = lg(G0)-1;
2647 98529 : pari_sp av = avma;
2648 98529 : for (e = 4; ; e <<= 1, set_avma(av))
2649 0 : {
2650 98529 : GEN G = gmul2n(G0, e), H = ground(G);
2651 98528 : if (ZM_rank(H) == r) return H; /* maximal rank ? */
2652 : }
2653 : }
2654 :
2655 : GEN
2656 139178 : idealred0(GEN nf, GEN I, GEN vdir)
2657 : {
2658 139178 : pari_sp av = avma;
2659 139178 : GEN G, aI, IZ, J, y, my, dyi, yi, c1 = NULL;
2660 : long N;
2661 :
2662 139178 : nf = checknf(nf);
2663 139178 : N = nf_get_degree(nf);
2664 : /* put first for sanity checks, unused when I obviously principal */
2665 139178 : G = nf_get_Gtwist(nf, vdir);
2666 139171 : switch (idealtyp(&I,&aI))
2667 : {
2668 37341 : case id_PRIME:
2669 37341 : if (pr_is_inert(I)) {
2670 585 : if (!aI) { set_avma(av); return matid(N); }
2671 585 : c1 = gel(I,1); I = matid(N);
2672 585 : goto END;
2673 : }
2674 36756 : IZ = pr_get_p(I);
2675 36756 : J = pr_inv_p(I);
2676 36757 : I = idealhnf_two(nf,I);
2677 36757 : break;
2678 101802 : case id_MAT:
2679 101802 : if (lg(I)-1 != N) pari_err_DIM("idealred");
2680 101795 : I = Q_primitive_part(I, &c1);
2681 101795 : IZ = gcoeff(I,1,1);
2682 101795 : if (is_pm1(IZ))
2683 : {
2684 9083 : if (!aI) { set_avma(av); return matid(N); }
2685 8999 : goto END;
2686 : }
2687 92712 : J = idealHNF_inv_Z(nf, I);
2688 92712 : break;
2689 21 : default: /* id_PRINCIPAL, silly case */
2690 21 : if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
2691 21 : if (!aI) return I;
2692 14 : goto END;
2693 : }
2694 : /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
2695 129469 : y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
2696 129469 : if (ZV_isscalar(y))
2697 : { /* already reduced */
2698 71085 : if (!aI) return gc_GEN(av, I);
2699 67907 : goto END;
2700 : }
2701 :
2702 58384 : my = zk_multable(nf, y);
2703 58383 : I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
2704 58382 : c1 = mul_content(c1, IZ);
2705 58382 : if (equali1(c1)) c1 = NULL; /* can be simplified with IZ */
2706 58382 : yi = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
2707 58384 : dyi = Q_denom(yi); /* generates (y) \cap Z */
2708 58384 : I = hnfmodid(I, dyi);
2709 58384 : if (!aI) return gc_upto(av, I);
2710 56389 : if (typ(aI) == t_MAT)
2711 : {
2712 39309 : GEN nyi = Q_muli_to_int(yi, dyi);
2713 39309 : if (gexpo(nyi) >= gexpo(y))
2714 20830 : aI = famat_div(aI, y); /* yi "larger" than y, keep the latter */
2715 : else
2716 : { /* use yi */
2717 18479 : aI = famat_mul(aI, nyi);
2718 18479 : c1 = div_content(c1, dyi);
2719 : }
2720 39309 : if (c1) { aI = famat_mul(aI, Q_to_famat(c1)); c1 = NULL; }
2721 : }
2722 : else
2723 17080 : c1 = c1? RgC_Rg_mul(yi, c1): yi;
2724 133894 : END:
2725 133894 : if (c1) aI = ext_mul(nf, aI,c1);
2726 133892 : return gc_GEN(av, mkvec2(I, aI));
2727 : }
2728 :
2729 : /* I integral ZM (not HNF), G ZM, rounded Cholesky form of a weighted
2730 : * T2 matrix. Reduce I wrt G */
2731 : GEN
2732 1346022 : idealpseudored(GEN I, GEN G)
2733 1346022 : { return ZM_mul(I, ZM_lll(ZM_mul(G, I), 0.99, LLL_IM)); }
2734 :
2735 : /* Same I, G; m in I with T2(m) small */
2736 : GEN
2737 142664 : idealpseudomin(GEN I, GEN G)
2738 : {
2739 142664 : GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
2740 142662 : return ZM_ZC_mul(I, gel(u,1));
2741 : }
2742 : /* Same I,G; irrational m in I with T2(m) small */
2743 : GEN
2744 0 : idealpseudomin_nonscalar(GEN I, GEN G)
2745 : {
2746 0 : GEN u = ZM_lll(ZM_mul(G, I), 0.99, LLL_IM);
2747 0 : GEN m = ZM_ZC_mul(I, gel(u,1));
2748 0 : if (ZV_isscalar(m) && lg(u) > 2) m = ZM_ZC_mul(I, gel(u,2));
2749 0 : return m;
2750 : }
2751 : /* Same I,G; t_VEC of irrational m in I with T2(m) small */
2752 : GEN
2753 1254115 : idealpseudominvec(GEN I, GEN G)
2754 : {
2755 1254115 : long i, j, k, n = lg(I)-1;
2756 1254115 : GEN x, L, b = idealpseudored(I, G);
2757 1254114 : L = cgetg(1 + (n*(n+1))/2, t_VEC);
2758 4418490 : for (i = k = 1; i <= n; i++)
2759 : {
2760 3164376 : x = gel(b,i);
2761 3164376 : if (!ZV_isscalar(x)) gel(L,k++) = x;
2762 : }
2763 3164375 : for (i = 2; i <= n; i++)
2764 : {
2765 1910261 : long J = minss(i, 4);
2766 4733589 : for (j = 1; j < J; j++)
2767 : {
2768 2823328 : x = ZC_add(gel(b,i),gel(b,j));
2769 2823328 : if (!ZV_isscalar(x)) gel(L,k++) = x;
2770 : }
2771 : }
2772 1254114 : setlg(L,k); return L;
2773 : }
2774 :
2775 : GEN
2776 13188 : idealred_elt(GEN nf, GEN I)
2777 : {
2778 13188 : pari_sp av = avma;
2779 13188 : GEN u = idealpseudomin(I, nf_get_roundG(nf));
2780 13188 : return gc_upto(av, u);
2781 : }
2782 :
2783 : GEN
2784 7 : idealmin(GEN nf, GEN x, GEN vdir)
2785 : {
2786 7 : pari_sp av = avma;
2787 : GEN y, dx;
2788 7 : nf = checknf(nf);
2789 7 : switch( idealtyp(&x, NULL) )
2790 : {
2791 0 : case id_PRINCIPAL: return gcopy(x);
2792 0 : case id_PRIME: x = pr_hnf(nf,x); break;
2793 7 : case id_MAT: if (lg(x) == 1) return gen_0;
2794 : }
2795 7 : x = Q_remove_denom(x, &dx);
2796 7 : y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
2797 7 : if (dx) y = RgC_Rg_div(y, dx);
2798 7 : return gc_upto(av, y);
2799 : }
2800 :
2801 : /*******************************************************************/
2802 : /* */
2803 : /* APPROXIMATION THEOREM */
2804 : /* */
2805 : /*******************************************************************/
2806 : /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
2807 : * and ppo(a,b) = Z_ppo(a,b) */
2808 : /* return gcd(a,b),ppi(a,b),ppo(a,b) */
2809 : GEN
2810 986167 : Z_ppio(GEN a, GEN b)
2811 : {
2812 986167 : GEN x, y, d = gcdii(a,b);
2813 986167 : if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
2814 757533 : x = d; y = diviiexact(a,d);
2815 : for(;;)
2816 131117 : {
2817 888650 : GEN g = gcdii(x,y);
2818 888650 : if (is_pm1(g)) return mkvec3(d, x, y);
2819 131117 : x = mulii(x,g); y = diviiexact(y,g);
2820 : }
2821 : }
2822 : /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
2823 : * and pple all others */
2824 : /* return gcd(a,b),ppg(a,b),pple(a,b) */
2825 : GEN
2826 0 : Z_ppgle(GEN a, GEN b)
2827 : {
2828 0 : GEN x, y, g, d = gcdii(a,b);
2829 0 : if (equalii(a, d)) return mkvec3(a, gen_1, a);
2830 0 : x = diviiexact(a,d); y = d;
2831 : for(;;)
2832 : {
2833 0 : g = gcdii(x,y);
2834 0 : if (is_pm1(g)) return mkvec3(d, x, y);
2835 0 : x = mulii(x,g); y = diviiexact(y,g);
2836 : }
2837 : }
2838 : static void
2839 0 : Z_dcba_rec(GEN L, GEN a, GEN b)
2840 : {
2841 : GEN x, r, v, g, h, c, c0;
2842 : long n;
2843 0 : if (is_pm1(b)) {
2844 0 : if (!is_pm1(a)) vectrunc_append(L, a);
2845 0 : return;
2846 : }
2847 0 : v = Z_ppio(a,b);
2848 0 : a = gel(v,2);
2849 0 : r = gel(v,3);
2850 0 : if (!is_pm1(r)) vectrunc_append(L, r);
2851 0 : v = Z_ppgle(a,b);
2852 0 : g = gel(v,1);
2853 0 : h = gel(v,2);
2854 0 : x = c0 = gel(v,3);
2855 0 : for (n = 1; !is_pm1(h); n++)
2856 : {
2857 : GEN d, y;
2858 : long i;
2859 0 : v = Z_ppgle(h,sqri(g));
2860 0 : g = gel(v,1);
2861 0 : h = gel(v,2);
2862 0 : c = gel(v,3); if (is_pm1(c)) continue;
2863 0 : d = gcdii(c,b);
2864 0 : x = mulii(x,d);
2865 0 : y = d; for (i=1; i < n; i++) y = sqri(y);
2866 0 : Z_dcba_rec(L, diviiexact(c,y), d);
2867 : }
2868 0 : Z_dcba_rec(L,diviiexact(b,x), c0);
2869 : }
2870 : static GEN
2871 6828640 : Z_cba_rec(GEN L, GEN a, GEN b)
2872 : {
2873 : GEN g;
2874 : /* a few naive steps before switching to dcba */
2875 6828640 : if (lg(L) > 10) { Z_dcba_rec(L, a, b); return veclast(L); }
2876 6828640 : if (is_pm1(a)) return b;
2877 4062408 : g = gcdii(a,b);
2878 4062408 : if (is_pm1(g)) { vectrunc_append(L, a); return b; }
2879 3035634 : a = diviiexact(a,g);
2880 3035634 : b = diviiexact(b,g);
2881 3035634 : return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
2882 : }
2883 : GEN
2884 757372 : Z_cba(GEN a, GEN b)
2885 : {
2886 757372 : GEN L = vectrunc_init(expi(a) + expi(b) + 2);
2887 757372 : GEN t = Z_cba_rec(L, a, b);
2888 757372 : if (!is_pm1(t)) vectrunc_append(L, t);
2889 757372 : return L;
2890 : }
2891 : /* P = coprime base, extend it by b; TODO: quadratic for now */
2892 : GEN
2893 49 : ZV_cba_extend(GEN P, GEN b)
2894 : {
2895 49 : long i, l = lg(P);
2896 49 : GEN w = cgetg(l+1, t_VEC);
2897 175 : for (i = 1; i < l; i++)
2898 : {
2899 126 : GEN v = Z_cba(gel(P,i), b);
2900 126 : long nv = lg(v)-1;
2901 126 : gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
2902 126 : b = gel(v,nv);
2903 : }
2904 49 : gel(w,l) = b; return shallowconcat1(w);
2905 : }
2906 : GEN
2907 28 : ZV_cba(GEN v)
2908 : {
2909 28 : long i, l = lg(v);
2910 : GEN P;
2911 28 : if (l <= 2) return v;
2912 14 : P = Z_cba(gel(v,1), gel(v,2));
2913 42 : for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
2914 14 : return P;
2915 : }
2916 :
2917 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2918 : GEN
2919 19280335 : Z_ppo(GEN x, GEN f)
2920 : {
2921 19280335 : (void)Z_pvalrem(x, f, &x);
2922 : for (;;)
2923 : {
2924 58205170 : f = gcdii(x, f); if (is_pm1(f)) break;
2925 38925241 : x = diviiexact(x, f);
2926 : }
2927 19279967 : return x;
2928 : }
2929 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2930 : ulong
2931 70518377 : u_ppo(ulong x, ulong f)
2932 : {
2933 : for (;;)
2934 : {
2935 70518377 : f = ugcd(x, f); if (f == 1) break;
2936 16212790 : x /= f;
2937 : }
2938 54305543 : return x;
2939 : }
2940 :
2941 : /* result known to be representable as an ulong */
2942 : static ulong
2943 1645036 : lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
2944 :
2945 : /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
2946 : * set *pd = gcd(x,N) */
2947 : ulong
2948 5915420 : Fl_invgen(ulong x, ulong N, ulong *pd)
2949 : {
2950 : ulong d, d0, e, v, v1;
2951 : long s;
2952 5915420 : *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
2953 5916274 : if (s > 0) v = N - v;
2954 5916274 : if (d == 1) return v;
2955 : /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
2956 2758066 : e = N / d;
2957 2758066 : d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
2958 2758190 : if (d0 == 1) return v;
2959 1644994 : e = lcmuu(e, d / d0);
2960 1645036 : return u_chinese_coprime(v, 1, e, d0, e*d0);
2961 : }
2962 :
2963 : /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
2964 : static GEN
2965 126 : nf_coprime_part(GEN nf, GEN x, GEN listpr)
2966 : {
2967 126 : long v, j, lp = lg(listpr), N = nf_get_degree(nf);
2968 : GEN x1, x2, ex;
2969 :
2970 : #if 0 /*1) via many gcds. Expensive ! */
2971 : GEN f = idealprodprime(nf, listpr);
2972 : f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
2973 : x = scalarmat(x, N);
2974 : for (;;)
2975 : {
2976 : if (gequal1(gcoeff(f,1,1))) break;
2977 : x = idealdivexact(nf, x, f);
2978 : f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
2979 : }
2980 : x2 = x;
2981 : #else /*2) from prime decomposition */
2982 126 : x1 = NULL;
2983 350 : for (j=1; j<lp; j++)
2984 : {
2985 224 : GEN pr = gel(listpr,j);
2986 224 : v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
2987 :
2988 126 : ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
2989 126 : x1 = x1? idealmulpowprime(nf, x1, pr, ex)
2990 126 : : idealpow(nf, pr, ex);
2991 : }
2992 126 : x = scalarmat(x, N);
2993 126 : x2 = x1? idealdivexact(nf, x, x1): x;
2994 : #endif
2995 126 : return x2;
2996 : }
2997 :
2998 : /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f */
2999 : GEN
3000 10920 : make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
3001 : {
3002 : GEN fZ, t, L, D2, d1, d2, d;
3003 :
3004 10920 : L = Q_remove_denom(L0, &d);
3005 10920 : if (!d) return L0;
3006 :
3007 : /* L0 = L / d, L integral */
3008 518 : fZ = gcoeff(f,1,1);
3009 518 : if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
3010 : /* Kill denom part coprime to fZ */
3011 126 : d2 = Z_ppo(d, fZ);
3012 126 : t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
3013 126 : if (equalii(d, d2)) return L;
3014 :
3015 126 : d1 = diviiexact(d, d2);
3016 : /* L0 = (L / d1) mod f. d1 not coprime to f
3017 : * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
3018 126 : D2 = nf_coprime_part(nf, d1, listpr);
3019 126 : t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
3020 126 : L = nfmuli(nf,t,L);
3021 :
3022 : /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
3023 126 : return Q_div_to_int(L, d1); /* exact division */
3024 : }
3025 :
3026 : /* assume L is a list of prime ideals. Return the product */
3027 : GEN
3028 666 : idealprodprime(GEN nf, GEN L)
3029 : {
3030 666 : long l = lg(L), i;
3031 : GEN z;
3032 666 : if (l == 1) return matid(nf_get_degree(nf));
3033 666 : z = pr_hnf(nf, gel(L,1));
3034 694 : for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
3035 666 : return z;
3036 : }
3037 :
3038 : /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
3039 : GEN
3040 1064 : idealprod(GEN nf, GEN I)
3041 : {
3042 1064 : long i, l = lg(I);
3043 : GEN z;
3044 2450 : for (i = 1; i < l; i++)
3045 2443 : if (!equali1(gel(I,i))) break;
3046 1064 : if (i == l) return gen_1;
3047 1057 : z = gel(I,i);
3048 1855 : for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
3049 1057 : return z;
3050 : }
3051 :
3052 : /* v_pr(idealprod(nf,I)) */
3053 : long
3054 1917 : idealprodval(GEN nf, GEN I, GEN pr)
3055 : {
3056 1917 : long i, l = lg(I), v = 0;
3057 10371 : for (i = 1; i < l; i++)
3058 8454 : if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
3059 1917 : return v;
3060 : }
3061 :
3062 : /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
3063 : GEN
3064 75625 : factorbackprime(GEN nf, GEN L, GEN e)
3065 : {
3066 75625 : long l = lg(L), i;
3067 : GEN z;
3068 :
3069 75625 : if (l == 1) return matid(nf_get_degree(nf));
3070 61002 : z = idealpow(nf, gel(L,1), gel(e,1));
3071 113931 : for (i=2; i<l; i++)
3072 52929 : if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
3073 61002 : return z;
3074 : }
3075 :
3076 : /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
3077 : * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
3078 : GEN
3079 58441 : pr_uniformizer(GEN pr, GEN F)
3080 : {
3081 58441 : GEN p = pr_get_p(pr), t = pr_get_gen(pr);
3082 58441 : if (!equalii(F, p))
3083 : {
3084 36884 : long e = pr_get_e(pr);
3085 36884 : GEN u, v, q = (e == 1)? sqri(p): p;
3086 36884 : u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
3087 36884 : v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
3088 36884 : if (pr_is_inert(pr))
3089 28 : t = addii(mulii(p, v), u);
3090 : else
3091 : {
3092 36856 : t = ZC_Z_mul(t, v);
3093 36856 : gel(t,1) = addii(gel(t,1), u); /* return u + vt */
3094 : }
3095 : }
3096 58441 : return t;
3097 : }
3098 : /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
3099 : GEN
3100 81194 : prV_lcm_capZ(GEN L)
3101 : {
3102 81194 : long i, r = lg(L);
3103 : GEN F;
3104 81194 : if (r == 1) return gen_1;
3105 68460 : F = pr_get_p(gel(L,1));
3106 121852 : for (i = 2; i < r; i++)
3107 : {
3108 53392 : GEN pr = gel(L,i), p = pr_get_p(pr);
3109 53392 : if (!dvdii(F, p)) F = mulii(F,p);
3110 : }
3111 68460 : return F;
3112 : }
3113 : /* v vector of prid. Return underlying list of rational primes */
3114 : GEN
3115 66234 : prV_primes(GEN v)
3116 : {
3117 66234 : long i, l = lg(v);
3118 66234 : GEN w = cgetg(l,t_VEC);
3119 219768 : for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
3120 66234 : return ZV_sort_uniq(w);
3121 : }
3122 :
3123 : /* Given a prime ideal factorization with possibly zero or negative
3124 : * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
3125 : * and v_pr(b) >= 0 for all other pr.
3126 : * For optimal performance, all [anti-]uniformizers should be precomputed,
3127 : * but no support for this yet. If nored, do not reduce result. */
3128 : static GEN
3129 54089 : idealapprfact_i(GEN nf, GEN x, int nored)
3130 : {
3131 54089 : GEN d = NULL, z, L, e, e2, F;
3132 : long i, r;
3133 54089 : int hasden = 0;
3134 :
3135 54089 : nf = checknf(nf);
3136 54089 : L = gel(x,1);
3137 54089 : e = gel(x,2);
3138 54089 : F = prV_lcm_capZ(L);
3139 54089 : z = NULL; r = lg(e);
3140 136923 : for (i = 1; i < r; i++)
3141 : {
3142 82834 : long s = signe(gel(e,i));
3143 : GEN pi, q;
3144 82834 : if (!s) continue;
3145 54073 : if (s < 0) hasden = 1;
3146 54073 : pi = pr_uniformizer(gel(L,i), F);
3147 54073 : q = nfpow(nf, pi, gel(e,i));
3148 54073 : z = z? nfmul(nf, z, q): q;
3149 : }
3150 54089 : if (!z) return gen_1;
3151 26994 : if (hasden) /* denominator */
3152 : {
3153 10103 : z = Q_remove_denom(z, &d);
3154 10103 : d = diviiexact(d, Z_ppo(d, F));
3155 : }
3156 26994 : if (nored || typ(z) != t_COL) return d? gdiv(z, d): z;
3157 10103 : e2 = cgetg(r, t_VEC);
3158 28673 : for (i = 1; i < r; i++) gel(e2,i) = addiu(gel(e,i), 1);
3159 10103 : x = factorbackprime(nf, L, e2);
3160 10103 : if (d) x = RgM_Rg_mul(x, d);
3161 10103 : z = ZC_reducemodlll(z, x);
3162 10103 : return d? RgC_Rg_div(z,d): z;
3163 : }
3164 :
3165 : GEN
3166 0 : idealapprfact(GEN nf, GEN x) {
3167 0 : pari_sp av = avma;
3168 0 : return gc_upto(av, idealapprfact_i(nf, x, 0));
3169 : }
3170 : GEN
3171 14 : idealappr(GEN nf, GEN x) {
3172 14 : pari_sp av = avma;
3173 14 : if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
3174 14 : return gc_upto(av, idealapprfact_i(nf, x, 0));
3175 : }
3176 :
3177 : /* OBSOLETE */
3178 : GEN
3179 14 : idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
3180 :
3181 : static GEN
3182 21 : mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
3183 : {
3184 21 : GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
3185 21 : long i, r = lg(E);
3186 84 : for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
3187 21 : return idealapprfact_i(nf,F,1);
3188 : }
3189 :
3190 : static void
3191 14 : not_in_ideal(GEN a) {
3192 14 : pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
3193 0 : }
3194 : /* x integral in HNF, a an 'nf' */
3195 : static int
3196 28 : in_ideal(GEN x, GEN a)
3197 : {
3198 28 : switch(typ(a))
3199 : {
3200 14 : case t_INT: return dvdii(a, gcoeff(x,1,1));
3201 7 : case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
3202 7 : default: return 0;
3203 : }
3204 : }
3205 :
3206 : /* Given an integral ideal x and a in x, gives a b such that
3207 : * x = aZ_K + bZ_K using the approximation theorem */
3208 : GEN
3209 42 : idealtwoelt2(GEN nf, GEN x, GEN a)
3210 : {
3211 42 : pari_sp av = avma;
3212 : GEN cx, b;
3213 :
3214 42 : nf = checknf(nf);
3215 42 : a = nf_to_scalar_or_basis(nf, a);
3216 42 : x = idealhnf_shallow(nf,x);
3217 42 : if (lg(x) == 1)
3218 : {
3219 14 : if (!isintzero(a)) not_in_ideal(a);
3220 7 : set_avma(av); return gen_0;
3221 : }
3222 28 : x = Q_primitive_part(x, &cx);
3223 28 : if (cx) a = gdiv(a, cx);
3224 28 : if (!in_ideal(x, a)) not_in_ideal(a);
3225 21 : b = mat_ideal_two_elt2(nf, x, a);
3226 21 : if (typ(b) == t_COL)
3227 : {
3228 14 : GEN mod = idealhnf_principal(nf,a);
3229 14 : b = ZC_hnfrem(b,mod);
3230 14 : if (ZV_isscalar(b)) b = gel(b,1);
3231 : }
3232 : else
3233 : {
3234 7 : GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
3235 7 : b = centermodii(b, aZ, shifti(aZ,-1));
3236 : }
3237 21 : b = cx? gmul(b,cx): gcopy(b);
3238 21 : return gc_upto(av, b);
3239 : }
3240 :
3241 : /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
3242 : * beta * x is an integral ideal coprime to y */
3243 : GEN
3244 37191 : idealcoprimefact(GEN nf, GEN x, GEN fy)
3245 : {
3246 37191 : GEN L = gel(fy,1), e;
3247 37191 : long i, r = lg(L);
3248 :
3249 37191 : e = cgetg(r, t_COL);
3250 76055 : for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
3251 37191 : return idealapprfact_i(nf, mkmat2(L,e), 0);
3252 : }
3253 : GEN
3254 84 : idealcoprime(GEN nf, GEN x, GEN y)
3255 : {
3256 84 : pari_sp av = avma;
3257 84 : return gc_upto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
3258 : }
3259 :
3260 : GEN
3261 7 : nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3262 : {
3263 7 : pari_sp av = avma;
3264 7 : GEN z, p, pr = modpr, T;
3265 :
3266 7 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3267 0 : x = nf_to_Fq(nf,x,modpr);
3268 0 : y = nf_to_Fq(nf,y,modpr);
3269 0 : z = Fq_mul(x,y,T,p);
3270 0 : return gc_upto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3271 : }
3272 :
3273 : GEN
3274 0 : nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3275 : {
3276 0 : pari_sp av = avma;
3277 0 : nf = checknf(nf);
3278 0 : return gc_upto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
3279 : }
3280 :
3281 : GEN
3282 0 : nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
3283 : {
3284 0 : pari_sp av=avma;
3285 0 : GEN z, T, p, pr = modpr;
3286 :
3287 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3288 0 : z = nf_to_Fq(nf,x,modpr);
3289 0 : z = Fq_pow(z,k,T,p);
3290 0 : return gc_upto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3291 : }
3292 :
3293 : GEN
3294 0 : nfkermodpr(GEN nf, GEN x, GEN modpr)
3295 : {
3296 0 : pari_sp av = avma;
3297 0 : GEN T, p, pr = modpr;
3298 :
3299 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3300 0 : if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
3301 0 : x = nfM_to_FqM(x, nf, modpr);
3302 0 : return gc_GEN(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
3303 : }
3304 :
3305 : GEN
3306 0 : nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
3307 : {
3308 0 : const char *f = "nfsolvemodpr";
3309 0 : pari_sp av = avma;
3310 : GEN T, p, modpr;
3311 :
3312 0 : nf = checknf(nf);
3313 0 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3314 0 : if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
3315 0 : a = nfM_to_FqM(a, nf, modpr);
3316 0 : switch(typ(b))
3317 : {
3318 0 : case t_MAT:
3319 0 : b = nfM_to_FqM(b, nf, modpr);
3320 0 : b = FqM_gauss(a,b,T,p);
3321 0 : if (!b) pari_err_INV(f,a);
3322 0 : a = FqM_to_nfM(b, modpr);
3323 0 : break;
3324 0 : case t_COL:
3325 0 : b = nfV_to_FqV(b, nf, modpr);
3326 0 : b = FqM_FqC_gauss(a,b,T,p);
3327 0 : if (!b) pari_err_INV(f,a);
3328 0 : a = FqV_to_nfV(b, modpr);
3329 0 : break;
3330 0 : default: pari_err_TYPE(f,b);
3331 : }
3332 0 : return gc_GEN(av, a);
3333 : }
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