Example: A curve of rank $2$

Let $E$ be the elliptic curve 389a of rank $2$, which is the curve of rank $2$ with smallest conductor.

sage: E = EllipticCurve('389a'); E
Elliptic Curve defined by y^2 + y = x^3 + x^2 - 2*x
over Rational Field
sage: E.rank()
2

Because the curve has rank $2$, we use Dokchitser's $L$-function package to approximate $L^{(2)}(E,1)$ to high precision:

sage: Lser = E.Lseries_dokchitser()
sage: L = RDF(abs(Lser.derivative(1,2))); L
1.51863300058

We compute the regulator, Tamagawa numbers, and torsion as usual:

sage: Om = RDF(E.omega()); Om
4.98042512171
sage: factor(discriminant(E))
389
sage: c389 = 1
sage: Reg = RDF(E.regulator()); Reg
0.152460177943
sage: T = E.torsion_order(); T
1

Finally we solve for the conjectural order of $\char93 {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$.

sage: Sha_conj = (L/2) * T^2 / (Om * Reg * c389)
sage: Sha_conj
1.0

We pause to emphasize that just getting something that looks like an integer by computing

$$\frac{L^{(r)}(E,1)}{r!} \cdot \char93 E(\mathbb{Q})_{\tor }^2 / (\Omega_E \cdot \Reg (E) \cdot \prod c_P)$$ (2.4.1)

is already excellent evidence for Conjecture 2.17. There is also a subtle and deep open problem here:

Open Problem 2.18   Let $E$ be the elliptic curve 389a above. Prove that the quantity (2.4.1) is a rational number.

For curves $E$ of analytic rank $0$ it is easy to prove using modular symbols that the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is a rational number. For curves with analytic rank $1$, this rationality follows from the very deep Gross-Zagier theorem. For curves of analytic rank $\geq 2$ there is not a single example in which the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is known to be a rational number.