Example: A Curve of Rank 0

Consider the elliptic curve $E$ with Cremona label 11a, which is one the 3 curves of smallest conductor. We now compute each of the quantities in Conjecture 2.17. First we define the curve $E$ in SAGE and compute its rank:

sage: E = EllipticCurve('11a'); E
Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 
over Rational Field
sage: E.rank()
0

Next we compute the number $L(E,1)$ to double precision (as an element of the real double field RDF):

sage: L = RDF(E.Lseries(1)); L
0.253841860856

We next compute the real period:

sage: Om = RDF(E.omega()); Om
1.26920930428

To compute $\prod c_p$ we factor the discriminant of $E$. It turns at that only $11$ divides the discriminant, and since the reduction at $11$ is split multiplicative the Tamagawa number is $5=\ord _{11}(\Delta_E)$.

sage: factor(discriminant(E))
-1 * 11^5
sage: c11 = E.tamagawa_number(11); c11
5

Next we compute the regulator, which is $1$ since $E$ rank $0$.

sage: Reg = RDF(E.regulator()); Reg
1.0

The torsion subgroup has order $5$.

sage: T = E.torsion_order(); T
5

Putting everything together in (2.3.1) and solving for the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$, we see that Conjecture 2.17 for $E$ is equivalent to the assertion that ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ has order $1$.

sage: Sha_conj = L * T^2 / (Om * Reg * c11); Sha_conj
1.0