CoCalc Shared Files2016-12-15-191547.sagewsOpen in CoCalc with one click!
Author: acx01bc .
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Description: test splitting prime ideals modulo p
E = EllipticCurve([0,- 1, 0, 0, -1]) E.anlist(20)
[0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, -2, 0, 2, 0, -6, 0, -1, 0]
p = 13; a = 1; b = 1; F = GF(p); R.<x> = F[]; (x^3-a*x-b).factor() E = EllipticCurve(F,[0, 0, 0, -a, -b]) E.cardinality() E2 = EllipticCurve([0, 0, 0, -a, -b]) E2.anlist(20) R.<x> = QQ[] K.<a,b> = NumberField([x^2-31, x^3-x^2-1]) Z = K.zeta_function() Z(2)
x^3 + 12*x + 12 19 [0, 1, 0, 3, 0, -2, 0, 4, 0, 6, 0, -2, 0, -5, 0, -6, 0, 4, 0, 2, 0] 1.35165039003399
R.<x> = QQ[] I = R.ideal(x^2-31); h= x^3-x^2-1 h(x+1/3) K.<a,b> = NumberField([x^2-31, x^3-x^2-1]) K.galois_group() #K.<t> = R.quotient(I); #S.<z> = K[]; #(z^3-z^2-1).factor() K.<a> = NumberField(x^2-31) S.<z> = K[]; (z^3-z^2-1).factor() #K.<a> = NumberField(x^2-31) #OK = K.ring_of_integers() #OK.basis() #I = OK.ideal([3,1+a]) #J = OK.ideal([3,1-a]) #I*J #I.norm() #I.factor() #J.factor()# #I.basis() #H = OK.ideal(1+a) #H.factor()
x^3 - 1/3*x - 29/27 Galois group PARI group [12, -1, 3, "D(6) = S(3)[x]2"] of degree 6 of the Number Field in a with defining polynomial x^2 - 31 over its base field z^3 - z^2 - 1
R.<x> = QQ[] K.<a> = NumberField(x^3-19) OK = K.ring_of_integers() OK.basis() I = OK.ideal(3) I.factor() J = OK.ideal(3)+OK.ideal(-1+a) J.factor() J.norm()
[1/3*a^2 + 1/3*a + 1/3, a, a^2] (Fractional ideal (3, 1/3*a^2 + 1/3*a + 1/3))^2 * (Fractional ideal (3, 1/3*a^2 + 1/3*a - 2/3)) (Fractional ideal (3, 1/3*a^2 + 1/3*a + 1/3)) * (Fractional ideal (3, 1/3*a^2 + 1/3*a - 2/3)) 9
R.<x> = QQ[] K.<a> = NumberField(x^4+x^3+x^2+1) OK = K.ring_of_integers() OK.class_group()
Class group of order 1 of Number Field in a with defining polynomial x^4 + x^3 + x^2 + 1
R.<x> = QQ[] K.<a> = NumberField(x^2-10) OK = K.ring_of_integers() OK.class_group() I = OK.ideal(3,1+a) J = OK.ideal(3,1-a) I*J (3-a)*(3+a) (I*I).factor() exit(0) P = x^5+7*x^4-7*x^3+7*x+1 ['is prime : ', P.is_prime()] K.<a> = NumberField(P) ['is galois : ',K.is_galois()] OK = K.ring_of_integers() OK.class_group().gen() I = OK.ideal(3,a^2+a+1); I I.basis() exit(0); p = 3 OK(p).factor() F = GF(p) R.<y> = F[] Q = R(P) fac = Q.factor() fac fac1 = fac[0][0] fac2 = fac[1][0] [fac1,fac2] OKR.<z> = OK[] fac1R = OKR(fac1) fac2R = OKR(fac2) fac2R I1 = OK.ideal([p,fac1R(a)]) I2 = OK.ideal([p,fac2R(a)]) [fac1R(a),fac2R(a)] [I1,I2] I1*I2 I2.intersection(3) fac1R(a)*fac2R(a)/3 fac1.gcd(fac2)
Class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 - 10 Fractional ideal (3) -1 (Fractional ideal (3, a + 1))^2
Error in lines 10-10 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 995, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-7.5/local/lib/python/site.py", line 351, in __call__ raise SystemExit(code) SystemExit: 0