sage: E = EllipticCurve('11a'); E Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field sage: E.rank() 0
Next we compute the number to double precision (as an element of the real double field RDF):
sage: L = RDF(E.Lseries(1)); L 0.253841860856
We next compute the real period:
sage: Om = RDF(E.omega()); Om 1.26920930428
To compute we factor the discriminant of . It turns at that only divides the discriminant, and since the reduction at is split multiplicative the Tamagawa number is .
sage: factor(discriminant(E)) -1 * 11^5 sage: c11 = E.tamagawa_number(11); c11 5
Next we compute the regulator, which is since rank .
sage: Reg = RDF(E.regulator()); Reg 1.0
The torsion subgroup has order .
sage: T = E.torsion_order(); T 5
Putting everything together in (2.3.1) and solving for the conjectural order of , we see that Conjecture 2.17 for is equivalent to the assertion that has order .
sage: Sha_conj = L * T^2 / (Om * Reg * c11); Sha_conj 1.0
William 2007-05-25