twists of each other, of CM curves of prime square conductor; moreover in each case the density of primes of $K$ whose squares arise in this way is in each case $1/\#{{\mathcal{O}}}_K^*$. In Section \[section CM\], we explain this phenomenon, and describe precisely all elliptic curves of prime power conductor and complex multiplication.

Another notable new phenomenon that occurs over imaginary quadratic fields, is the existence of elliptic curves of prime conductor whose residual Galois representations modulo $p$ are irreducible but not absolutely irreducible. When $p$ is odd, this phenomenon does not occur for elliptic curves over $\Q$: the reason is the existence of complex conjugation in the absolute Galois group, whose image under the mod-$p$ representation is similar to $\left(\begin{smallmatrix}1 & 0\\ 0 & -1 \end{smallmatrix}
\right)$. Over an imaginary quadratic field, or any number field which is totally complex (i.e. has no real embedding), the absolute Galois group has no such complex conjugation elements and this argument does not apply.

At the prime $2$ the situation is more exciting! Over $\Q$, if $E$ is a semistable elliptic curve whose discriminant is a square, then its residual image is reducible. There are two different ways to prove such an 