ECNF: add Szpiro ratio and Faltings height

[JohnCremona]

We show the Szpiro ratio for elliptic curves over Q, and should also do so for curves over number fields. The definition (Hindry, top of page 8) is simply log(Norm(minimal disc))/log(norm(conductor)). Since both the conductor norm and minimal discriminant norm are stored, this could easily be done on the fly (pending begin precomputed and added to the database in a new column).

Similarly for the Faltings height. Hindry (op.cit.) gives the definition in equation (3.13). This is (I think) too complicated to be computed on the fly, as it involves the period lattice at each infinite place. Another source is Silverman's article in Cornell & Silverman (Prop.1.1 on p.254). It is not clear to me what the formula for the stable Faltings height is.

References:

• Hindry, M., Why is it difficult to compute the Mordell-Weil group ? https://webusers.imj-prg.fr/~marc.hindry/MW-size.pdf
• Faltings G., Calculus on arithmetic surfaces. Annals of Math. 119 (1984), 387–424
• Cornell G., Silverman J. (ed), Arithmetic geometry. Storrs conference, 1984. SpringerVerlag, New York, 1986.

[roed314]

Note that we currently use the modified Szpiro ratio over Q, which is log(max(|c_4|^3, |c_6|^2))/log(N).

[roed314]

As a comment, it would also be nice to have the abc-ratio for elliptic curves over number fields, but we haven't found a definition in the literature.