Hey all,
I just ran into a question of Mazur from his 1978 paper **'Rational isogenies of Prime Degree'** and I wonder what is the status of his question:


"Are there examples of elliptic curves $E/\mathbf{Q}$, $E'/\mathbf{Q}$ which are not isogenous over $\mathbf{Q}$ such that $E[N] \cong E'[N]$ (as $\text{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ modules) for some $N \geq 7$?"


Also the case $N=6$ is interesting, as it is the only genus $1$ case. For $N\leq 5$ the genus of $X(N)$ is zero, hence for all twists it will have infinitely many rational points.


Thanks for your answers.



Elliptic curves over QQ with isomorphic n-torsion
This question comes down to a question about a certain Hilbert modular surface X\_N, which parametrizes pairs of elliptic curves with isomorphic N-torsion -- you can think of it as


X(N) x X(N) / PSL\_2(Z/NZ)


where the group acts diagonally on the two factors.


This surface, like all Hilbert modular surfaces, has some modular curves lying around, which in this case parametrize pairs of elliptic curves that are isogenous. So you're asking: do these surfaces have rational points other than the ones lying on the special curves you know about? Well... questions about rational points on surfaces are hard. But for N = 7, the surface is rational so the points should be dense there. (Since only finitely many of the modular curves have non-cuspidal points over Q, this is enough!) When N = 11, the surface is elliptic. But for large N the surface becomes general type. Of course this still leaves us knowing basically nothing for certain about the presence or absence of rational points. Anyway, what I know about this all comes from [David Carlton's thesis](http://arxiv.org/abs/math/9903011). As far as I know there's no substantial progress on the general question, which is one of my favorites! 


