If I understand FC's remark under the post "[Very strong multiplicity one for Hecke eigenforms](https://mathoverflow.net/questions/3352/very-strong-multiplicity-one-for-hecke-eigenforms)," in the course of Faltings's proof of the Tate conjecture, Faltings proves the following statement: let E/Q and F/Q be elliptic curves and write Q(E[p]) (resp. Q(F[p])) for the number field obtained by adjoining the x and y coordinates of the p-torsion of E to Q. Then if Q(E[p]) = Q(F[p]) for infinitely many primes p, E/Q and F/Q are isogenous.


Learning of this result prompted me to wonder: suppose P is a *finite* set of primes. Then do there exist E/Q and F/Q such that Q(E[p]) = Q(F[p]) for each p in P with E/Q and F/Q *not isogenous*?


If not in general, what is known about the particular P for which the above question does (or doesn't) have an affirmative answer?



An inverse problem: Number fields attached to elliptic curves over Q
Most people would say no. Indeed, there's a conjecture, most commonly ascribed to Frey, that for p a SINGLE large enough prime, E[p] isomorphic to F[p] implies that E and F are isogenous. I believe p=37 is thought to be large enough, but don't hold me to it.


