matsumi

320 days ago by WilliamStein

E = EllipticCurve([0,-2]); E 
        
Elliptic Curve defined by y^2 = x^3 - 2 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 2 over Rational Field
E.conductor() 
        
1728
1728
divisors(E.conductor()) 
        
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96,
108, 144, 192, 216, 288, 432, 576, 864, 1728]
[1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 96, 108, 144, 192, 216, 288, 432, 576, 864, 1728]
E.discriminant().factor() 
        
-1 * 2^6 * 3^3
-1 * 2^6 * 3^3
E.conductor().factor() 
        
2^6 * 3^3
2^6 * 3^3
E.torsion_order() 
        
1
1
F = EllipticCurve('27a'); F 
        
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
Elliptic Curve defined by y^2 + y = x^3 - 7 over Rational Field
F.torsion_order() 
        
3
3
F.short_weierstrass_model() 
        
Elliptic Curve defined by y^2 = x^3 - 432 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 432 over Rational Field
factor(NumberField(x^3-432,'a').disc()) 
        
-1 * 2^2 * 3^3
-1 * 2^2 * 3^3
factor(NumberField(x^3-2,'a').disc()) 
        
-1 * 2^2 * 3^3
-1 * 2^2 * 3^3
F.q_eigenform(100) 
        
q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 +
11*q^37 + 8*q^43 - 6*q^49 - 10*q^52 - q^61 - 8*q^64 + 5*q^67 - 7*q^73 +
14*q^76 + 17*q^79 - 5*q^91 - 19*q^97 + O(q^100)
q - 2*q^4 - q^7 + 5*q^13 + 4*q^16 - 7*q^19 - 5*q^25 + 2*q^28 - 4*q^31 + 11*q^37 + 8*q^43 - 6*q^49 - 10*q^52 - q^61 - 8*q^64 + 5*q^67 - 7*q^73 + 14*q^76 + 17*q^79 - 5*q^91 - 19*q^97 + O(q^100)
E.q_eigenform(100) 
        
q + q^7 - 5*q^13 - 7*q^19 - 5*q^25 + 4*q^31 - 11*q^37 + 8*q^43 - 6*q^49
+ q^61 + 5*q^67 - 7*q^73 - 17*q^79 - 5*q^91 - 19*q^97 + O(q^100)
q + q^7 - 5*q^13 - 7*q^19 - 5*q^25 + 4*q^31 - 11*q^37 + 8*q^43 - 6*q^49 + q^61 + 5*q^67 - 7*q^73 - 17*q^79 - 5*q^91 - 19*q^97 + O(q^100)
E.q_eigenform(100) - F.q_eigenform(100) 
        
2*q^4 + 2*q^7 - 10*q^13 - 4*q^16 - 2*q^28 + 8*q^31 - 22*q^37 + 10*q^52 +
2*q^61 + 8*q^64 - 14*q^76 - 34*q^79 + O(q^100)
2*q^4 + 2*q^7 - 10*q^13 - 4*q^16 - 2*q^28 + 8*q^31 - 22*q^37 + 10*q^52 + 2*q^61 + 8*q^64 - 14*q^76 - 34*q^79 + O(q^100)