Genus Two Curves with Many Elliptic Subcovers

We determine all genus 2 curves, defined over ?, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ?₂. For each component, we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to ?-isomorphism) defined over ?, which have degree 2 and 3 elliptic subcovers also defined over ?.