Hyperelliptic modular curves $X_0(n)$ and isogenies of elliptic curves over quadratic fields

Peter Bruin, Filip Najman

Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperelliptic of genus $\ge 2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over $\mathbb{Q}$. We determine all p…