  We show how for every integer n one can explicitly construct n distinct plane
quartics and one hyperelliptic curve over the complex numbers 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 the complex numbers, each with a pair of commuting involutions,
are isomorphic to one another.
