Example: A Rank 0 curve with nontrivial Sha

Consider the curve $E$ with label 681b. This curve has rank $0$, and we compute the conjectural order of $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ as in the previous section:

sage: E = EllipticCurve('681b'); E
Elliptic Curve defined by y^2 + x*y  = x^3 + x^2 - 1154*x - 15345 
over Rational Field
sage: E.rank()
0
sage: L = RDF(E.Lseries(1)); L
1.84481520613
sage: Om = RDF(E.omega()); Om
0.81991786939

There are two primes of bad reduction this time.

sage: factor(681)
3 * 227
sage: factor(discriminant(E))
3^10 * 227^2
sage: c3 = E.tamagawa_number(3); c227 = E.tamagawa_number(227)
sage: c3, c227
(2, 2)
sage: Reg = RDF(E.regulator()); Reg
1.0
sage: T = E.torsion_order(); T
4

In this case it turns out that $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is conjecturally $9$.

sage: Sha_conj = L * T^2 / (Om * Reg * c3*c227); Sha_conj
9.0