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Let $n$ be an integer such that the modular curve $X_0(n)$ is hyperellipticof genus $\ge2$ and such that the Jacobian of $X_0(n)$ has rank $0$ over$\mathbb Q$. We determine all points of $X_0(n)$ defined over quadratic fields,and we give a moduli interpretation of these points. As a consequence, we showthat up to $\overline{\mathbb Q}$-isomorphism, all but finitely many ellipticcurves with $n$-isogenies over quadratic fields are in fact $\mathbb Q$-curves,and we list all exceptions. We also show that, again with finitely manyexceptions up to $\overline{\mathbb Q}$-isomorphism, every $\mathbb Q$-curve$E$ over a quadratic field $K$ admitting an $n$-isogeny is $d$-isogenous, forsome $d\mid n$, to the twist of its Galois conjugate by some quadraticextension $L$ of $K$; we determine $d$ and $L$ explicitly.

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