  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.
