  For each prime number $\ell$ and for each imaginary quadratic order of class
number one or two, we determine all the possible $\ell$-adic Galois
representations that occur for any elliptic curve with complex multiplication
by such an order over its minimal field of definition, and then we determine
all the isomorphism classes of elliptic curves that have a prescribed
$\ell$-adic Galois representation.
