︠6751700c-1592-4ffa-9ff4-3cbcd8fdc9a2s︠ E = EllipticCurve([0,- 1, 0, 0, -1]) E.anlist(20) ︡977a6c8f-15f6-412f-8f70-4b84097cb7ad︡{"stdout":"[0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, -2, 0, 2, 0, -6, 0, -1, 0]\n"}︡{"done":true}︡ ︠de67cba6-5204-49c0-9bf0-398ec73b9303s︠ p = 13; a = 1; b = 1; F = GF(p); R. = 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. = QQ[] K. = NumberField([x^2-31, x^3-x^2-1]) Z = K.zeta_function() Z(2) ︡a95f4fb1-5617-4e3f-baef-03d3146a8e01︡{"stdout":"x^3 + 12*x + 12\n"}︡{"stdout":"19\n"}︡{"stdout":"[0, 1, 0, 3, 0, -2, 0, 4, 0, 6, 0, -2, 0, -5, 0, -6, 0, 4, 0, 2, 0]\n"}︡{"stdout":"1.35165039003399"}︡{"stdout":"\n"}︡{"done":true}︡ ︠929a4cae-c973-47ff-af78-9272b10508bds︠ R. = QQ[] I = R.ideal(x^2-31); h= x^3-x^2-1 h(x+1/3) K. = NumberField([x^2-31, x^3-x^2-1]) K.galois_group() #K. = R.quotient(I); #S. = K[]; #(z^3-z^2-1).factor() K. = NumberField(x^2-31) S. = K[]; (z^3-z^2-1).factor() #K. = 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() ︡5cdadd51-e219-4694-a0ca-edd8e20ba081︡{"stdout":"x^3 - 1/3*x - 29/27\n"}︡{"stdout":"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\n"}︡{"stdout":"z^3 - z^2 - 1\n"}︡{"done":true}︡ ︠61f8b067-58da-4496-bac4-277c6d994172︠ ︡a30d400d-8d49-4204-922d-661005f60f6e︡ ︠87c243c5-a721-483f-be7a-0e62002e007ds︠ R. = QQ[] K. = 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() ︡86c55dd3-875e-451a-966f-a78833949e48︡{"stdout":"[1/3*a^2 + 1/3*a + 1/3, a, a^2]\n"}︡{"stdout":"(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))\n"}︡{"stdout":"(Fractional ideal (3, 1/3*a^2 + 1/3*a + 1/3)) * (Fractional ideal (3, 1/3*a^2 + 1/3*a - 2/3))\n"}︡{"stdout":"9\n"}︡{"done":true}︡ ︠12d64f06-244d-482b-a6d9-afa8d05a62f7s︠ R. = QQ[] K. = NumberField(x^4+x^3+x^2+1) OK = K.ring_of_integers() OK.class_group() ︡1930cb75-7b52-4299-a027-47ef96c4edff︡{"stdout":"Class group of order 1 of Number Field in a with defining polynomial x^4 + x^3 + x^2 + 1\n"}︡{"done":true}︡ ︠ab5d3323-456c-4d45-9ecd-f0aa7539a8a4s︠ R. = QQ[] K. = 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. = 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. = F[] Q = R(P) fac = Q.factor() fac fac1 = fac[0][0] fac2 = fac[1][0] [fac1,fac2] OKR. = 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) ︡72c04e93-6a5a-4888-9815-ed35caeb16ab︡{"stdout":"Class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 - 10\n"}︡{"stdout":"Fractional ideal (3)\n"}︡{"stdout":"-1\n"}︡{"stdout":"(Fractional ideal (3, a + 1))^2\n"}︡{"stderr":"Error in lines 10-10\nTraceback (most recent call last):\n File \"/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py\", line 995, in execute\n exec compile(block+'\\n', '', 'single') in namespace, locals\n File \"\", line 1, in \n File \"/projects/sage/sage-7.5/local/lib/python/site.py\", line 351, in __call__\n raise SystemExit(code)\nSystemExit: 0\n"}︡{"done":true}︡