Explicit Isogenies of Prime Degree Over Quadratic Fields

Let $K$ be a quadratic field which is not an imaginary of class number one. We describe algorithm to compute the primes $p$ for there exists elliptic curve over admitting $K$-rational $p$-isogeny. This builds on work David, Larson-Vaintrob, and Momose. Combining this with Bruin-Najman, \"{O}zman-Siksek, most recently Box, we determine above set three fields $\mathbb{Q}(\sqrt{-10})$, $\mathbb{Q}(\sqrt{5})$, $\mathbb{Q}(\sqrt{7})$, providing first such examples after Mazur's 1978 determination $K = \mathbb{Q}$. The termination relies Generalised Riemann Hypothesis.