Constructing genus-3 hyperelliptic Jacobians with CM

Given a sextic CM field K , we give an explicit method for finding all genus- $3$ hyperelliptic curves defined over 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 [J. Ramanujan Math. Soc. 16 (2001) no. 4, 339–372], 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 Fₚ with a given zeta function by finding roots of the Rosenhain minimal polynomials modulo p .