Pari-gp /7/5 1 Pari-gp 3 pq

Size: px

Start display at page:

Download "Pari-gp /7/5 1 Pari-gp 3 pq"

さなえふしはら
4 years ago
Views:

1 Pari-gp /7/5 1 Pari-gp 3 pq

2 Pari-gp /7/ pq 3 2. Pari-gp 3. p p 4. p Abel

3 Pari-gp /7/5 3 pq 3

4 Pari-gp /7/5 4 p q 1 (mod 9) p q 3 (3, 3) Abel 3

5 Pari-gp /7/5 5 p polcyclo(n): n n polsubcyclo(n,d,v=x): finds an equation (in variable v) for the d-th degree subfields of Q(ζ n ). Output is a polynomial or a vector of polynomials is there are several such fields, or none. p 1 (mod 9) polsubcyclo(p,3) Q(ζ p ) 3

6 Pari-gp /7/5 6 polcompositum(p,q,flag=0): vector of all possible compositums of the number fields defined by the polynomials P and Q. > p=19;q=37; polcompositum(polsubcyclo(p,3),polsubcyclo(q,3)) time = 2 ms. %100 = [x^9-56*x^7 + 60*x^ *x^5-834*x^4-4393*x^ *x^ *x ]

7 Pari-gp /7/5 7 nfsubfields(pol,d=0): find all subfields of degree d of number field defined by pol (all subfields if d is null or omitted). Result is a vector of subfields, each being given by [g,h], where g is an absolute equation and h expresses one of the roots of g in terms of the root x of the polynomial defining nf. > nfsubfields(%100[1],3) time = 4 ms. %102 =......

8 Pari-gp /7/5 8 bnfinit(p,flag=0,tech=[]): compute the necessary data for future use in ideal and unit group computations, including fundamental units if they are not too large. flag and tech are both optional. flag can be any of 0: default, 1: insist on having fundamental units, 2: do not compute units, 3: small bnfinit, which can be converted to a big one using bnfmake. See manual for details about tech.

9 Pari-gp /7/5 9 (16:00) gp > %102[1][1] time = 0 ms. %103 = x^3-111*x (16:02) gp > bnf=bnfinit(%102[1][1]); time = 12 ms. (16:02) gp > bnf.clgp time = 0 ms. %105 = [1, [], []] (16:02) gp > bnf=bnfinit(%102[2][1]); time = 1 ms. (16:02) gp > bnf.clgp time = 0 ms. %107 = [1, [], []] (16:02) gp > bnf=bnfinit(%102[3][1]); time = 3 ms.

10 Pari-gp /7/5 10 (16:02) gp > bnf.clgp time = 0 ms. %109 = [12, [6, 2], [[3, 0, 2; 0, 1, 0; 0, 0, 1], [9, 3, 0; 0, 3, 0; 0, 0, 1]]]

11 Pari-gp /7/5 11 p q 1 (mod 9) p q 3 K 3 h(k) 9 h(k)

12 Pari-gp /7/5 12 { forprime(q = 10, 150, if(q%9 == 1, fq = polsubcyclo(q, 3); forprime(p = q, 1200, if(p%9 == 1, u = polcompositum(polsubcyclo(p, 3), fq); sbfds = nfsubfields(u[1], 3); for(i = 1, length(sbfds), bnf = bnfinit(sbfds[i][1],1); if(bnf.no%9 == 0, if(bnfcertify(bnf),,print("failed certify")); print(p, ", ", q, ", ", bnf.no, ", ", bnf.pol))))))) }

13 Pari-gp /7/ n 3, (J. Math. Soc. Japan, 26, 1974) M.-N. Gras 3, (J. Reine Angew. Math., 277, 1975) V. Ennola and R. Turunen 3, (Math. Comp., 45, 1985)

14 Pari-gp /7/5 14 n 3 K. Uchida, Class numbers of cubic cyclic fields, J. Math. Soc. Japan, 26, 1974 (available online). n N (fix)., l n a = qq 1 q 2 N, q, q 1, q 2 (2/q) 1 or (3/q) 1 2 mod q i l i = 1, 2 3 mod q 1 l 3 mod q 2 l p = a2n f(x) = x 3 + px 2 + 2px + p = 0 3 n p

15 Pari-gp /7/5 15 M.-N. Gras 3 M.-N. Gras, Méthodes et algorithmes pour le calcul numérique du nombre de classes et des unités des extensions cubiques cycliques de Q, J. Reine Angew. Math., 277, 1975 < 4, 000 cyclic cubic fields η h : h c(h) := 16 3 R(η)/ log2 m 3 3. q c(h) h

16 Pari-gp /7/5 16 V. Ennola and R. Turunen 3 V. Ennola and R. Turunen, On cyclic cubic fields, Math. Comp., 45 (172), 1985n, On Totally Real Cubic Fields, Math. Comp., 44 (170), Voronoi 16, 000 cyclic cubic fields

17 Pari-gp /7/5 17

Pari-gp 2006/7/12 2 1. Pari-gp 2. Microsoft Windows 3. 4. Pari-gp 5. 2 6. 7. Galois 8.

Pari-gp 2006/7/12 1 Pari-GP 2006 7 12 Pari-gp 2006/7/12 2 1. Pari-gp 2. Microsoft Windows 3. 4. Pari-gp 5. 2 6. 7. Galois 8. Pari-gp 2006/7/12 2-1 +ε. Pari-gp 2006/7/12 3 Pari-gp Pari C gp GP gp K. Belabas