We call an order O in a quadratic field K odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve E over \({\mathbb {C}}\) with CM odd (resp. even) if its endomorphism ring \(\textrm{End}(E)\) is an odd (resp. even) order in the imaginary quadratic field \(\textrm{End}(E)\otimes {\mathbb {Q}}\) . Suppose that \(j(E)\in {\mathbb {R}}\) and let us consider the set \(\mathscr {J}({\mathbb {R}},E)\) of all \(j(E^{\prime })\) where \(E^{\prime }\) is any elliptic curve that enjoys the following properties:
. \(\bullet\) \(E^{\prime }\) is isogenous to E;
\(\bullet\) \(j(E^{\prime })\in {\mathbb {R}}\) ;
\(\bullet\) \(E^{\prime }\) has the same parity as E.
We prove that the closure of \(\mathscr {J}({\mathbb {R}},E)\) in \({\mathbb {R}}\) is the closed semi-infinite interval \((-\infty ,1728]\) (resp. the whole \({\mathbb {R}}\) ) if E is odd (resp. even).
This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.
