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 , we use Dokchitser's -function package to approximate 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 .
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
William 2007-05-25