this talk will present a new method for computing the igusa class polynomials of a primitive quartic $cm$ field. for a primitive quartic $cm$ field, $k$, we compute the igusa class polynomials modulo $p$ for certain small primes $p$ and then use the chinese remainder theorem and a conjectural bound on the denominators to construct the class polynomials. we also provide an extension to genus $2$ of the algorithm for determining endomorphism rings of elliptic curves. our algorithm can be used to generate genus $2$ curves over a finite field with a given zeta function. \vskip .1in \noindent joint work with kirsten eisentraeger, university of michigan.