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 the field C of complex numbers with CM odd (resp. even) if its endomorphism ring $End(E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties. 1) $E'$ is isogenous to E; 2) $j(E')$ is a real number; 3) $E'$ has the same parity as E. We prove that the closure of $J(R,E)$ in the set R of real numbers is the closed semi-infinite interval (-∞,1728] (resp. the whole R) if E is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Th\'el\`ene and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.