Building on Mazur’s 1978 work on prime degree isogenies, Kenku determined in 1981 all possible cyclic isogenies of elliptic curves over
Q
\mathbb {Q}
. Although more than 40 years have passed, the determination of cyclic isogenies of elliptic curves over a single other number field has hitherto not been realised.
In this paper we develop a procedure to assist in establishing such a determination for a given quadratic field. Executing this procedure on all quadratic fields
Q
(
d
)
\mathbb {Q}(\sqrt {d})
with
|
d
|
>
10
4
|d| > 10^4
we obtain, conditional on the Generalised Riemann Hypothesis, the determination of cyclic isogenies of elliptic curves over
19
19
quadratic fields, including
Q
(
213
)
\mathbb {Q}(\sqrt {213})
and
Q
(
−
2289
)
\mathbb {Q}(\sqrt {-2289})
. To make this procedure work, we determine all of the finitely many quadratic points on the modular curves
X
0
(
125
)
X_0(125)
and
X
0
(
169
)
X_0(169)
, which may be of independent interest.