TITLE: Elliptic curves over QQ with isomorphic n-torsion
QUESTION [16 upvotes]: 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.

REPLY [12 votes]: Tom Fisher has a paper where he shows that there are infinitely many examples for $N = 9$ and $N = 11$. See this link.