Frey–Mazur conjecture

Let $E/\mathbb{Q}$ and $E'/\mathbb{Q}$ be elliptic curves. Their $p$-torsion modules are the corresponding Galois modules $E[p]$ and $E'[p]$.

Frey–Mazur conjecture. If $E[p]$ and $E'[p]$ are isomorphic as Galois modules for some prime $p>17$, then $E$ and $E'$ are $\mathbb{Q}$-isogenous.

This is the stronger formulation of the Frey–Mazur conjecture, whose original formulation was weaker. The source gives no resolution evidence, so the conjecture remains open.