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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.

Explicit Isogenies of Prime Degree Over Quadratic Fields