Example: A Rank 4 curve

Let $E$ be the curve of rank $4$ with label 234446b. It is likely that this is the curve with smallest conductor and rank $4$ (a big calculation of the author et al. shows that there are no rank $4$ curves with smaller prime conductor).

sage: E = EllipticCurve([1, -1, 0, -79, 289]); E
Elliptic Curve defined by y^2 + x*y  = x^3 - x^2 - 79*x + 289 
over Rational Field
sage: E.rank()
4

We next compute $L(E,1)$, $L'(E,1)$, $L^{(2)}(E,1)$, $L^{(3)}(E,1)$, and $L^{(4)}(E,1)$. All these special values look like they are $0$, except for $L^{(4)}(E,1)$ which is about $214$, hence clearly nonzero. One can prove that $L(E,1)=0$ (e.g., using denominator bounds coming from modular symbols), hence since the root number is $+1$, we have either $r_{E,{\rm an}} = 2$ or $r_{E,{\rm an}}=4$, and of course suspect (but cannot prove yet) that $r_{E,{\rm an}}=4$.

sage: E.root_number()
1
sage: Lser = E.Lseries_dokchitser()
sage: Lser(1)
1.43930352980778e-18
sage: Lser.derivative(1,1)
-4.59277879927938e-24
sage: Lser.derivative(1,2)
-8.85707917856308e-22
sage: Lser.derivative(1,3)
1.01437455701212e-20
sage: L = RDF(abs(Lser.derivative(1,4))); L
214.652337502

As above, we compute the other BSD invariants of $E$.

sage: Om = RDF(E.omega()); Om
2.97267184726
sage: factor(discriminant(E))
2^2 * 117223
sage: c2 = 2
sage: c117223 = 1
sage: Reg = RDF(E.regulator()); Reg
1.50434488828
sage: T = E.torsion_order(); T
1

Finally, 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/24) * T^2 / (Om * Reg * c2 * c117223)
sage: Sha_conj
1.0

Again we emphasize that we do not even know that the conjectural order computed above is a rational number.

It seems almost a miracle that $L^{(4)}(E,1) = 214.65\ldots$, $\Omega_E = 2.97\ldots$, and $\Reg (E) = 1.50\ldots$ have anything to do with each other, but indeed they do:

sage: L/24, 2*Om*Reg
(8.9438473959, 8.9438473959)

That these two numbers are the same to several decimal places is a fact, independent of any conjectures.