  We provide an explicit and algorithmic version of a theorem of Momose
classifying isogenies of prime degree of elliptic curves over number fields,
which we implement in Sage and PARI/GP. Combining this algorithm with recent
work of Box-Gajovi\'c-Goodman we determine the first instances of isogenies of
prime degree for cubic number fields, as well as for several quadratic fields
not previously known. While the correctness of the general algorithm relies on
the Generalised Riemann Hypothesis, the algorithm is unconditional for the
restricted class of semistable elliptic curves.
