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The main result of this paper is to extend from   Q   to each of the nine imaginary quadratic fields of class number 1 a result of [Serre,  Duke Math. J . 54 (1987) 179–230] and [Mestre–Oesterlé,  J. reine. angew. Math . 400 (1989) 173–184], namely that if   E   is an elliptic curve of prime conductor, then either   E   or a 2‐, 3‐ or 5‐isogenous curve has prime discriminant. For four of the nine fields, the theorem holds with no change, while for the remaining five fields the discriminant of a curve with prime conductor is (up to isogeny) either prime or the square of a prime. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and second that a certain level‐lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non‐trivial torsion over each of the nine fields: in the case of 2‐torsion, we find that such curves either have CM or with a small finite number of exceptions arise from a family analogous to the Setzer–Neumann family over   Q  .

On elliptic curves of prime power conductor over imaginary quadratic fields with class number 1