Families of Pairs of Genus 2 Curves with Isomorphic Unpolarized Jacobians

We construct three families of pairs of genus 2 curves over a field K, whose Jacobians are isomorphic as unpolarized abelian varieties. Each family is parameterized by an open subset of the Projective line over K. Our construction is based on a remark in a paper by Howe. Later, we describe an obstruction to perusing the same technique over Q.