how to get rank of elliptic curve of QuadraticField

asked 2014-10-28 11:21:49 +0100

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I try two examples,even elliptic curve of Q cannot run:

http://cjsh716.blog.163.com/blog/stat...

E = EllipticCurve([0,0,0,0,2332])
show(E)
K1.<a> = QuadraticField(-3)
E1 = EllipticCurve(K1,[0,0,0,0,2332]);show(E1);
E.rank()
E1.rank()

E = EllipticCurve([0,0,0,-15,22])
show(E)
K1.<a> = QuadraticField(3)
E1 = EllipticCurve(K1,[0,0,0,-15,22]);show(E1)
E.torsion_points()
E1.torsion_points()
E.rank()
E1.rank()

TypeError: Unable to start pari because the command 'gp --emacs --quiet --stacksize 10000000' failed.

answered 2014-11-27 13:41:13 +0100

John Cremona
701 ●35 ●22

Computing ranks of elliptic curves over number fields is not easy, but should not give the error message posted above. Can you try a smaller example, such as

sage: E1 = EllipticCurve(K1,[0,0,0,0,2])
sage: E1.rank()
2

Secondly, if E is an elliptic curve over Q then its rank over K=Q(sqrt(d)) is E.rank()+E.quadratic_twist(d).rank(), for example here

sage: E = EllipticCurve([0,0,0,0,2])
sage: E.rank() + E.quadratic_twist(-3).rank()
2

or for your examples

sage: E = EllipticCurve([0,0,0,0,2332])
sage: E.rank() + E.quadratic_twist(-3).rank()
2
sage: E = EllipticCurve([0,0,0,-15,22])
sage: E.rank() + E.quadratic_twist(-3).rank()
0

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