Everett W. Howe: Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian, Proc. Amer. Math. Soc. 129 (2001) 1647–1657, MR 2002a:14028, Zbl 0974.14021.
(An official and an unofficial electronic version are available.)
We show how for every integer n one can explicitly construct n distinct plane quartics and one hyperelliptic curve over C all of whose Jacobians are isomorphic to one another as abelian varieties without polarization. When we say that the curves can be constructed “explicitly”, we mean that the coefficients of the defining equations of the curves are simple rational expressions in algebraic numbers in R whose minimal polynomials over Q can be given exactly and whose decimal approximations can be given to as many places as is necessary to distinguish them from their conjugates. We also prove a simply-stated theorem that allows one to decide whether or not two plane quartics over C, each with a pair of commuting involutions, are isomorphic to one another.