M = ModularForms(Gamma0(3),2) M.set_precision(14) M.basis()
[ 1 + 12*q + 36*q^2 + 12*q^3 + 84*q^4 + 72*q^5 + 36*q^6 + 96*q^7 + 180*q^8 + 12*q^9 + 216*q^10 + 144*q^11 + 84*q^12 + 168*q^13 + O(q^14) ]
EllipticCurve([0,0,1,0,-61]) phi = EllipticCurve([0,0,1,0,-61]).modular_parametrization() print(phi) phi.power_series(prec = 100) print(parent(phi))
Elliptic Curve defined by y^2 + y = x^3 - 61 over Rational Field Modular parameterization from the upper half plane to Elliptic Curve defined by y^2 + y = x^3 - 61 over Rational Field (q^-2 + q + 8*q^4 - 19*q^7 + 43*q^10 - 70*q^13 + 160*q^16 - 324*q^19 + 739*q^22 - 1473*q^25 + 3000*q^28 - 5855*q^31 + 11635*q^34 - 22901*q^37 + 45184*q^40 - 88149*q^43 + 171524*q^46 - 331954*q^49 + 641584*q^52 - 1236088*q^55 + 2376836*q^58 - 4557893*q^61 + 8723040*q^64 - 16660735*q^67 + 31769685*q^70 - 60481943*q^73 + 114976336*q^76 - 218264380*q^79 + 413813619*q^82 - 783621456*q^85 + 1482270144*q^88 - 2800885598*q^91 + 5287367236*q^94 - 9972031158*q^97 + O(q^98), -q^-3 - 2 + 18*q^3 - 23*q^6 + 81*q^9 - 171*q^12 + 511*q^15 - 1215*q^18 + 2925*q^21 - 6303*q^24 + 13851*q^27 - 29727*q^30 + 64308*q^33 - 135999*q^36 + 285669*q^39 - 591601*q^42 + 1219374*q^45 - 2494755*q^48 + 5080932*q^51 - 10285461*q^54 + 20724210*q^57 - 41558899*q^60 + 83021112*q^63 - 165233601*q^66 + 327779385*q^69 - 648177471*q^72 + 1278087300*q^75 - 2513373628*q^78 + 4930361865*q^81 - 9649221597*q^84 + 18843716794*q^87 - 36724618674*q^90 + 71436441195*q^93 - 138708240983*q^96 + O(q^97)) <type 'instance'>
Q.<x> = QQ[] P = x^2-5 K.<w> = P.splitting_field(); K OK.basis() OK = K.ring_of_integers() I = OK.ideal(2) I I.factor() L.<a,b> = NumberField([x^2-5,x^2-13]) O = L.ring_of_integers() O.basis() J = O.ideal(2) J.factor() J = O.ideal(3); J.factor()
Number Field in w with defining polynomial x^2 - 5 [1/2*w + 1/2, w] Fractional ideal (2) Fractional ideal (2) [3/2*a - b + 1/2, 3*a - 2*b, (-1/4*b + 11/4)*a - 7/4*b + 9/4, 11/2*a - 7/2*b] (Fractional ideal (1/2*a + 1/2*b + 1)) * (Fractional ideal ((-1/4*b + 3/4)*a - 1/4*b + 7/4)) (Fractional ideal (1/2*b - 1/2)) * (Fractional ideal (1/2*b + 1/2))
a = -7; b = -12 -4*a^3-27*b^2 for a in [-40..-1] : for b in [1..40] : D=-4*a^3-27*b^2 d = sqrt(D) if d.is_integer() : [a,b,D,d]
-2516 [-39, 19, 227529, 477] [-39, 26, 219024, 468] [-37, 37, 165649, 407] [-31, 30, 94864, 308] [-27, 27, 59049, 243] [-21, 7, 35721, 189] [-21, 17, 29241, 171] [-21, 20, 26244, 162] [-21, 28, 15876, 126] [-21, 35, 3969, 63] [-21, 37, 81, 9] [-19, 19, 17689, 133] [-19, 30, 3136, 56] [-13, 12, 4900, 70] [-13, 13, 4225, 65] [-12, 8, 5184, 72] [-12, 16, 0, 0] [-9, 9, 729, 27] [-7, 6, 400, 20] [-7, 7, 49, 7] [-3, 1, 81, 9] [-3, 2, 0, 0]
Q.<x> = QQ[] #P = x^4+x^3+x^2+x+1 P = x^3-39*x-19 P.factor() K.<w> = P.splitting_field(); K type(K) OK = K.ring_of_integers() base = OK.basis(); base M.<u,v,t> = ZZ[] OK.class_group() G = K.galois_group(); t = [[0,0,0],[0,0,0],[0,0,0]]; for m in [0..len(base)-1] : for n in [0..len(G)-1] : t[m][n] = (G[1]^n)(base[m]) t I = OK.ideal(2,w) type(I) type(K.pari_nf()) R.<a,b,c> = K[] T = [a,b,c] no = 1; for n in [0..len(G)-1] : tmp = 0; for m in [0..len(base)-1] : tmp = tmp+t[m][n]*T[m] no = no*tmp no no.factor()
x^3 - 39*x - 19 Number Field in w with defining polynomial x^3 - 39*x - 19 <class 'sage.rings.number_field.number_field.NumberField_absolute_with_category'> [2/53*w^2 + 23/53*w + 1/53, w, w^2] Class group of order 1 of Number Field in w with defining polynomial x^3 - 39*x - 19 [[2/53*w^2 + 23/53*w + 1/53, -7/53*w^2 - 1/53*w + 235/53, 5/53*w^2 - 22/53*w - 77/53], [w, -13/53*w^2 - 17/53*w + 338/53, 13/53*w^2 - 36/53*w - 338/53], [w^2, -36/53*w^2 + 169/53*w + 2314/53, -17/53*w^2 - 169/53*w + 1820/53]] <class 'sage.rings.number_field.number_field_ideal.NumberFieldFractionalIdeal'> <type 'sage.libs.pari.gen.gen'> a^3 + 9*a^2*b + 24*a*b^2 + 19*b^3 - 21*a^2*c - 57*a*b*c - 252*a*c^2 - 741*b*c^2 + 361*c^3 (a + (3/53*w^2 + 8/53*w + 81/53)*b + (-32/53*w^2 - 103/53*w + 461/53)*c) * (a + (-4/53*w^2 + 7/53*w + 263/53)*b + (47/53*w^2 - 69/53*w - 1593/53)*c) * (a + (1/53*w^2 - 15/53*w + 133/53)*b + (-15/53*w^2 + 172/53*w + 19/53)*c)