A note on quadratic fields in which a fixed prime number splits completely

Throughout this note,pdenotes a fixed prime number andfdenotes a fixed natural number prime top. It is easy to see and more or less known that for any natural numbern, there exists an elliptic curve over pwhosej-invariant is of degreenoverFpand whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem:for any natural number n, decide whether or not there exists an elliptic curve over pwhose j-invariant is of degree n over Fpand whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.