What is the largest prime number $p$ for which one knows examples of nonisogenous elliptic curves $E\_1$ and $E\_2$ over $\mathbb{Q}$ with isomorphic mod $p$ Galois representations: $E\_1[p] \cong E\_2[p]$? I do not require the isomorphism to preserve the Weil pairing.



Elliptic curves with the same mod $p$ representation
The largest known is for $p = 17$. This can be found by searching Cremona's tables. In particular, in [this paper](https://www.dpmms.cam.ac.uk/~taf1000/papers/highercongr.pdf), Tom Fisher mentions the examples of
curves 3675b1 and 47775b1, which are


$E\_{1} : y^{2} + xy + y = x^{3} + x^{2} - 393x - 9654$


and


$E\_{2} : y^{2} + xy + y = x^{3} + x^{2} + 398095697x + 4077685984826$.


The conjecture that if $p > 17$, there are no such curves is known as the Frey-Mazur conjecture. (Frey and Mazur originally asked whether this can occur for $p > 5$, because in that situation the curve $X\_{E}(p)$, the moduli space of elliptic curves with the same mod $p$ Galois representation as $E$, has genus $> 1$.)


