Odd and Even Elliptic Curves with Complex Multiplication

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.