Explicit Isogenies of Prime Degree Over Quadratic Fields

Abstract

Abstract 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, Özman–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.