Plane quartics with Jacobians isomorphic to a hyperelliptic Jacobian

We show how for every integer $n$ one can explicitly construct $n$ distinct plane quartics and one hyperelliptic curve over ${\mathbf C}$ 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 ${\mathbf R}$ whose minimal polynomials over ${\mathbf Q}$ 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 ${\mathbf C}$, each with a pair of commuting involutions, are isomorphic to one another.