Congruences modulo $23$ to $y^2=x^3-23$ are trivial

We say that two elliptic curves $E$ and $F$ over $\mathbb{Q}$ are congruent modulo a prime $p$ if their $p$-torsion Galois modules (over the algebraic closure of $\mathbb{Q}$) are isomorphic. Such a congruence is called trivial if there is a rational isogeny between $E$ and $F$ with degree prime to $p$. A version of the Frey-Mazur conjecture states that any congruence modulo any prime $p\ge 19$ is trivial. Given an elliptic curve $E\mathrm{/}\mathbb{Q}$ and a prime $p$, it is well-known that there is a twist of the classical modular curve $X(p)$ whose rational points describe the elliptic curves congruent to $E$ modulo $p$. In this article, we apply Mazur’s strategy to determine the rational points of such a twisted modular curve under certain assumptions. This involves, among others, the determination of the previously unknown Tate module of its Jacobian and new instances of the Birch and Swinnerton–Dyer conjecture (for abelian varieties not of ${\mathrm{G}\mathrm{L}}_2$-type). In particular, we determine an explicit bound on the conductor of any elliptic curve congruent modulo $p$ to $y^2=x^3-p$ when $p$ is prime and congruent to $5$ modulo $9$, and deduce that any congruence modulo $23$ to $y^2=x^3-23$ is trivial.