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 users manual 200 tutorial 50

Pari-gp 2006/7/12 3-1 tutorial GAP 50

Pari-gp 2006/7/12 4 Pari-gp http://pari.math.u-bordeaux.fr/ GNU Public License Linux, FreeBSD Unix clone, Apple MacOS X, Microsoft Windows

Pari-gp 2006/7/12 5 Pari-gp Q: A:

Pari-gp 2006/7/12 6 Pari-gp Q: A: gp 4MB 500,000

Pari-gp 2006/7/12 7 Pari-gp Q: C gp A: gp for() C gp2c gp C

Pari-gp 2006/7/12 8 Pari-gp Q: A: Abel Smith Normal Form, Abel Hermite Normal Form,

Pari-gp 2006/7/12 9 Pari-gp Q: A: H. Cohen GTM 138, GTM 193

Pari-gp 2006/7/12 10 Pari-gp Q: A: GRH GRH Heuristic 2

Pari-gp 2006/7/12 11 Microsoft Windows 1. http://pari.math.u-bordeaux.fr/downloads Microsoft Windows Pari-2-3-0.exe. 2.

Pari-gp 2006/7/12 12 gp p =

Pari-gp 2006/7/12 12-1 Pari Perl (Math::Pari), Python, Lisp (CLISP) Risa/Asir

Pari-gp 2006/7/12 13 gp exact GMP p 10 28. 16

Pari-gp 2006/7/12 13-1 \p \ps. gp? \p 56 56

Pari-gp 2006/7/12 14 gp [1,2,3] : [1,2,3]~ : [1,2,3;4,5,6;7,8,9] = 1 2 3 4 5 6 7 8 9 *: [1,2,3]*[1,2,3]~ = 14

Pari-gp 2006/7/12 14-1 matdet(), matrank(), mateigen(), charpoly().

Pari-gp 2006/7/12 15 gp? a = 1;? a = [1, 2, 3];? a[1] 1

Pari-gp 2006/7/12 16 sin(): p gp> sin(i) time = 0 ms. %12 = 0.E-28 + 1.175201193643801456882381851*I (e 1 e)/2i = 1.175201193643801456882381851 I.

Pari-gp 2006/7/12 17 factorint(): MPQS, ECM, ρ, etc... factor(): 1

Pari-gp 2006/7/12 17-1 fermat(n)=2^(2^n)+1. sizedigit(n) n 10 debugging level 3 \g 3

Pari-gp 2006/7/12 18 quad qfb quaddisc(x) Q( x) qfbclassno(x) Q( x) Shanks quadclassunit(x) Q( x)

Pari-gp 2006/7/12 19 qfbclassno(x)? qfbclassno (-3299) %10 = 27 qfbclassno (-3299,1) %11 = 27 Euler

Pari-gp 2006/7/12 19-1 Qfb(a, b, c) ax 2 + bxy + cy 2.

Pari-gp 2006/7/12 20 quadclassunit() quadclassunit(x,{flag}, {tech=[]}) Q( x) Buchmann-McCurley sub-exponential D < 10 25, D > 10 10 qfbclassno()? quadclassunit(-10^25-3) %20 = [491852207132, [245926103566, 2], [Qfb(7, 1, 357142857142857142857143), Qfb(13, 13, 192307692307692307692311)], 1]

Pari-gp 2006/7/12 21 quadclassunit(x) flag=1 D > 0 tech=[c 1, c 2 ] c 1, c 2 > 0 c 2 = c 0.1 c 2.0 c = 6 GRH GRH bnf...

Pari-gp 2006/7/12 21-1? quadclassunit(-3299,,[6,6]) %21 = [27, [9, 3], [Qfb(3, 1, 275), Qfb(23, -17, 39)], 1] GRH

Pari-gp 2006/7/12 22 nf bnf big number field/buchman s number field x 2 + 3299 = 0 2 Q( 3299)? T = x^2 + 3299;? bnf = bnfinit(t);? bnf.clgp %69 [27, [9, 3], [[3, 2; 0, 1], [23, 8; 0, 1]]] bnf.no= =27 bnf.cyc= = Z/9Z Z/3Z, bnf.gen=

Pari-gp 2006/7/12 23 bnfinit() bnfcertify() : bnfinit() 1? bnf = bnfinit(t);? bnfcertify(bnf) %10 = 1

Pari-gp 2006/7/12 24 2 Hilbert quadhilbert(d): 2 Q( D) Hilbert Q( D)?quadhilbert(-3299) time = 33 ms. %11 = x^27-125*x^26 + 411202*x^25-5273842*x^24 + 5927663*x^23 + 514861*x^22 + 756180179*x^21-533202511*x^20 + 3726905423*x^19-4436877539*x^18 + 1430746075*x^17-1211962575*x^16-2629273159*x^15 + 4851196931*x^14-690886193*x^13 + 208639789*x^12-104541139*x^11-1178105817*x^10-839425855*x^9 + 691789163*x^8 + 554710685*x^7-15194969*x^6-9973495*x^5 + 11421563*x^4-2710350*x^3 + 347998*x^2 + 259*x + 1

Pari-gp 2006/7/12 25 monic Z T nfinit(t), bnfinit(t). T θ Q Q(θ) x 4 + 24x 2 + 585x + 1791 = 0 K? T = x^4 + 24*x^2 + 585*x + 1791;? bnf = bnfinit(t);? bnf.sign %68 [0,4]? bnf.clgp %69 [4, [4], [[7, 4, 5, 6; 0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1]]] bnf.no= = 4, bnf.cyc= 4 bnf.gen=

Pari-gp 2006/7/12 26 bnf bnf.zk Z HNF idealprimedec() 2? bnfisprincipal(bnf,idealpow(bnf,bnf.gen[1],4)) %190 = [[0]~, [3, -6, 2, -1]~] bnf bnf.gen[1] 4=bnf.no

Pari-gp 2006/7/12 27 polcompositum(pol1, pol2, flag=0: pol1 pol2? polcompositum(x^2+23, polsubcyclo(9,3)); %82 = [x^6 + 63*x^4 + 2*x^3 + 1596*x^2-144*x + 1554]? qfb 23 3=bnfinit(%82[1]); Q( 23) 9 3 Q( 23) Z 3 1st layer.

Pari-gp 2006/7/12 28 Q Galois? G17 = galoisinit(polcyclo(17));? galoisisabelian(g17) %203 = [16]? galoispermtopol(g17,g17.gen[1]) %204 = x^7? galoisfixedfield(g17,g17.gen[1]^4,1) %207 = x^4 + x^3-6*x^2 - x + 1? galoisexport(galoisinit(%)) %208 = "Group((1, 3, 2, 4))"

Pari-gp 2006/7/12 28-1 polcyclo(17) 17 Galois 16 σ 7 σ 4 x^4 + x^3-6*x^2 - x + 1 Q Galois GAP Group((1, 3, 2, 4))

Pari-gp 2006/7/12 29

Pari-gp 2006/7/12 30 if() gp if(,, ) for() for(,, )? for(d=1,100,if(isfundamental(d), h=qfbclassno(d); if(mod(h,3)==mod(0,3),print(d, ", ", h))))

Pari-gp 2006/7/12 31 (,...) = { }

Pari-gp 2006/7/12 32 file \r file read(file ) write(file,...)... file

Pari-gp 2006/7/12 33 ray class group/ray class field bnr rnf gp2c: gp C gp C Unix

Pari-gp 2006/7/12 34