2011-08-01 - "obvious" Cremona-Lingham

297 days ago by WilliamStein

K.<a4,a6> = QQ[] E = EllipticCurve([a4,a6]) E.discriminant() 
        
-64*a4^3 - 432*a6^2
-64*a4^3 - 432*a6^2
E = EllipticCurve('49a') 
       
factor(E.discriminant()) 
        
-1 * 7^3
-1 * 7^3
E.conductor() 
        
49
49
F = E.short_weierstrass_model(); F 
        
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
factor(F.discriminant()) 
        
-1 * 2^12 * 3^12 * 7^3
-1 * 2^12 * 3^12 * 7^3

-\Delta = -64x^3 - 432y^2

so

\Delta = 64x^3 + 432y^2

so

\Delta = (4x)^3 + 3\cdot (12y)^2

so

3^3 \Delta = (12x)^3 + 3^4\cdot (12y)^2

so

3^3 \Delta = -(-12x)^3 + (108y)^2

so

Y^2 = X^3 + 3^3\Delta

where

Y = 108y, \qquad X = -12x
factor(432) 
        
2^4 * 3^3
2^4 * 3^3
144*3 
        
432
432
def f(delta): return EllipticCurve([0, 3^3*delta]) 
       
E = f(-6^12*7^3); E 
        
Elliptic Curve defined by y^2 = x^3 - 20159181213696 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 20159181213696 over Rational Field
f(-6^12*7^2).rank() 
        
0
0
E.rank() 
        
1
1
P = E.gens()[0]; P 
        
(578340 : 439796952 : 1)
(578340 : 439796952 : 1)
2*P 
        
(52244145/361 : 376363402857/6859 : 1)
(52244145/361 : 376363402857/6859 : 1)
-P 
        
(578340 : -439796952 : 1)
(578340 : -439796952 : 1)
X, Y, _ = -P; X, Y 
        
(578340, -439796952)
(578340, -439796952)
a4 = X/(-12); a6 = Y/108; a4, a6 
        
(-48195, -4072194)
(-48195, -4072194)
G = EllipticCurve([a4,a6]) G.conductor().factor() 
        
7^2
7^2
G 
        
Elliptic Curve defined by y^2 = x^3 - 48195*x - 4072194 over Rational
Field
Elliptic Curve defined by y^2 = x^3 - 48195*x - 4072194 over Rational Field
F 
        
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
Elliptic Curve defined by y^2 = x^3 - 2835*x - 71442 over Rational Field
H = G.minimal_quadratic_twist()[0]; H 
        
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 37*x - 78 over
Rational Field
Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 37*x - 78 over Rational Field
M = f(-6^12*7^4) M.gens() 
        
[(190512 : 82301184 : 1)]
[(190512 : 82301184 : 1)]
X, Y, _ = -M.gens()[0]; X, Y 
        
(190512, -82301184)
(190512, -82301184)
a4 = X/(-12); a6 = Y/108; a4, a6 
        
(-15876, -762048)
(-15876, -762048)
G = EllipticCurve([a4,a6]) G.conductor().factor() 
        
2^6 * 3^4 * 7^2
2^6 * 3^4 * 7^2
G.minimal_quadratic_twist()[0].conductor().factor() 
        
2^5 * 3^4 * 7^2
2^5 * 3^4 * 7^2