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 using Dokchitser's algorithm. Note that the order of vanishing appears to be .
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 follows from the Gross-Zagier theorem, which asserts that is a nonzero multiple of the Néron-Tate canonical height of a certain point on called a Heegner point. One can explicitly construct this point on and find that it is torsion, hence has height , so . That 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 is .
sage: Sha_conj = (L/6) * T^2 / (Om * Reg * c5077) sage: Sha_conj 1.0Note that just as was the case with the curve 389a above, we do not know that the above conjectural order of is a rational number, since there are no know theoretical results that relate any of the three real numbers , , and .
William 2007-05-25