Tag Archives: MathOverflow

Elliptic curves with isomorphic cyclic 13-subgroups?

I liked this MathOverflow question, which asks:  are there two non-isogenous elliptic curves over Q, each one of which has a rational cyclic 13-isogeny, and such that the kernels of the two isogenies are isomorphic as Galois modules?

This is precisely to look for rational points on the modular surface S parametrizing pairs (E,E’,C,C’,φ), where E and E’ are elliptic curves, C and C’ are cyclic 13-subgroups, and φ is an isomorphism between C and C’.

S is a quotient of X_1(13) x X_1(13) by the diagonal in the natural (Z/13Z)^* x (Z/13Z)^* action.

Is S general type, rational, what?

 

 

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