sage: E = EllipticCurve('37a'); E Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field sage: E.rank() 1
We next compute the value . The corresponding function in SAGE takes a bound on the number of terms of the -series to use, and returns an approximate to along with a bound on the error (coming from the tail end of the series).
sage: L, error = E.Lseries_deriv_at1(200); L, error (0.305999773834879, 2.10219814818300e-90) sage: L = RDF(L); L 0.305999773835
We compute and the Tamagawa number, regulator, and torsion as above.
sage: Om = RDF(E.omega()); Om 5.98691729246 sage: factor(discriminant(E)) 37 sage: c37 = 1 sage: Reg = RDF(E.regulator()); Reg 0.05111140824 sage: T = E.torsion_order(); T 1
Finally, we solve and find that the conjectural order of is .
sage: Sha_conj = L * T^2 / (Om * Reg * c37); Sha_conj 1.0