Hyperelliptic modular curves X0(n) and isogenies of elliptic curves over quadratic fields

We study elliptic curves over quadratic fields with isogenies of certain degrees. Let n be a positive integer such that the modular curve X-0 (n) is hyperelliptic of genus and such that its Jacobian has rank 0 over (Q) over bar. We determine all points of X-0 (n) defined over quadratic fields, and we give a moduli interpretation of these points. We show that, with a finite number of exceptions up to (Q) over bar -isomorphism, every elliptic curve over a quadratic field K admitting an n-isogeny is disogenous, for some d n, to the twist of its Galois conjugate by a quadratic extension L of K. We determine d and L explicitly, and we list all exceptions. As a consequence, again with a finite number of exceptions up to (Q) over bar -isomorphism, all elliptic curves with n-isogenies over quadratic fields are in fact (Q) over bar -curves.