Example: A Rank 3 curve

The curve $E$ with label 5077a has rank $3$. This is the curve with smallest conductor and rank $3$.

sage: E = EllipticCurve('5077a'); E
Elliptic Curve defined by y^2 + y = x^3 - 7*x + 6 
over Rational Field
sage: E.rank()
3

We compute $L(E,s)$ using Dokchitser's algorithm. Note that the order of vanishing appears to be $3$.

sage: E.root_number()
-1
sage: Lser = E.Lseries_dokchitser()
sage: Lser.derivative(1,1)
-5.63436295355925e-22
sage: Lser.derivative(1,2)
2.08600476044634e-21
sage: L = RDF(abs(Lser.derivative(1,3))); L
10.3910994007

That the order of vanishing is really $3$ follows from the Gross-Zagier theorem, which asserts that $L'(E,1)$ is a nonzero multiple of the Néron-Tate canonical height of a certain point on $E$ called a Heegner point. One can explicitly construct this point on $E$ and find that it is torsion, hence has height $0$, so $L'(E,1)=0$. That $L''(E,1)=0$ then follows from the functional equation (see Section 1.3). Finally we compute the other BSD invariants:

sage: Om = RDF(E.omega()); Om
4.15168798309
sage: factor(discriminant(E))
5077
sage: c5077 = 1
sage: Reg = RDF(E.regulator()); Reg
0.417143558758
sage: T = E.torsion_order(); T
1

Putting everything together we see that the conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is $1$.

sage: Sha_conj = (L/6) * T^2 / (Om * Reg * c5077)
sage: Sha_conj
1.0

Note that just as was the case with the curve 389a above, we do not know that the above conjectural order of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E/\mathbb{Q})$ is a rational number, since there are no know theoretical results that relate any of the three real numbers $L^{(3)}(E,1)$, $\Reg (E/\mathbb{Q})$, and $\Omega_{E/\mathbb{Q}}$.