Explicit isogenies of prime degree over quadratic fields

Let K be a quadratic field which is not an imaginary quadratic field of class number one. We describe an algorithm to compute a superset of the set of primes p for which there exists an elliptic curve over K admitting a K-rational p-isogeny. Combining this algorithm with recent work on the determination of quadratic points on low-genus modular curves, we determine - conditional upon the Generalised Riemann Hypothesis - the above set of isogeny primes for several quadratic fields, providing the first such examples after Mazur's 1978 determination for K = Q. We will give a live demo of the Sage and PARI/GP implementations of the algorithm.