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 , , , , and . All these special values look like they are , except for which is about , hence clearly nonzero. One can prove that (e.g., using denominator bounds coming from modular symbols), hence since the root number is , we have either or , and of course suspect (but cannot prove yet) that .
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 .
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 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 , , and 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.
William 2007-05-25