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 {C} with CM odd (resp. even) if its endomorphism ring End (E) is an odd (resp. even) order in the imaginary quadratic field End (E) {Q}. Suppose that j (E) {R} and let us consider the set J ({R}, E) of all j (E^) where E^ is any elliptic curve that enjoys the following properties: . E^ is isogenous to E; j (E^) {R} ; E^ has the same parity as E. We prove that the closure of J ({R}, E) in {R} 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-Thélène and Alena Pirutka about the distribution of j-invariants of certain elliptic curves of CM type.