  Given a sextic CM field $K$, we give an explicit method for finding all genus
3 hyperelliptic curves defined over $\mathbb{C}$ whose Jacobians are simple and
have complex multiplication by the maximal order of this field, via an
approximation of their Rosenhain invariants. Building on the work of Weng, we
give an algorithm which works in complete generality, for any CM sextic field
$K$, and computes minimal polynomials of the Rosenhain invariants for any
period matrix of the Jacobian. This algorithm can be used to generate genus 3
hyperelliptic curves over a finite field $\mathbb{F}_p$ with a given zeta
function by finding roots of the Rosenhain minimal polynomials modulo $p$.
