An elliptic curve over a characteristic zero field is said to have complex multiplication when its endomorphism ring is larger than Z ("E has extra endomorphisms"). Generic elliptic curves don't have complex multiplication. Often one tries to understand elliptic curves via their Tate modules. When the Tate module of E has extra endomorphisms we say E has formal complex multiplication. Over a number field, E has formal complex multiplication if and only if it has complex multiplication. Over a local field this need not be the case. How often does this happen?