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how to get the general formula of SHA-TATE GROUP?

asked 2014-07-10 02:23:26 -0500

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how to get SHA general formula by parametrization a1, a2, a3, a4, a6,no a number

E=EllipticCurve([0,1]);E;S=E.sha();S

Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
Tate-Shafarevich group for the Elliptic Curve defined by y^2 = x^3 + 1
over Rational Field

 R.<a1,a2,a3,a4,a6>=QQ[];show(R);E=EllipticCurve([a1,a2,a3,a4,a6]);show(E)

Multivariate Polynomial Ring in a1, a2, a3, a4, a6 over Rational Field
Elliptic Curve defined by y^2 + a1*x*y + a3*y = x^3 + a2*x^2 + a4*x + a6
over Multivariate Polynomial Ring in a1, a2, a3, a4, a6 over Rational
Field

S=E.sha();S

Traceback (click to the left of this block for traceback)
...
AttributeError: 'EllipticCurve_generic_with_category' object has no
attribute 'sha'
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answered 2014-07-10 08:34:15 -0500

kcrisman gravatar image

The Tate-Shafarevich group is apparently only implemented for elliptic curves over the rational field. See the documentation and note that over generic rings or fields it isn't there. Perhaps this is an enhancement request?

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thank kcrisman master very much! I see about it http://wstein.org/projects/ 19 Ariel Shwayder's Junior Project on "Visualizing L(E,s)" ------there some datas images about L(E,s) by PARI. 12 Dimitar Jetchev's senior thesis Visibility of Shafarevich-Tate Group -----there no datas images about sha-tate group.

cjsh gravatar imagecjsh ( 2014-07-10 22:52:07 -0500 )edit

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Asked: 2014-07-10 02:23:26 -0500

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Last updated: Jul 10 '14