For any imaginary quadratic field $F$, the Hilbert class field $H$ is generated by the $j$-invariant of any elliptic curve with complex multiplication (CM) by $\mathcal O$, the ring of algebraic integers of $F$.

What is the simplest generalization of this well known and useful fact? Since an imaginary quadratic field is the simplest example of a CM field, one is led to ask:

Is there an (albeit conjectural) invariant of abelian surfaces which generates the Hilbert class field of a CM field $K$ (of degree $4$ over $\mathbb Q$)?

Note that $K$ is a totally imaginary quadratic extension of a real quadratic field.