I will describe joint work with Andrew Sutherland, giving deterministic and probabilistic algorithms to decide whether a given monic irreducible polynomial H in Z[X] is a Hilbert class polynomial, and if so, which one. These algorithms can be used to determine whether a given algebraic integer is the jinvariant of an elliptic curve with complex multiplication (CM), and if so, the associated CM discriminant. Our algorithms admit simple implementations that are asymptotically and practically faster than existing approaches