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