The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceSn of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapSn is viewed as a subset of the moduli spaceA2 of principally polarized abelian surfaces. On the other hand the Humbert surfaceHΔ of invariant Δ is defined as a subvariety ofA2(C), the set of C-valued points ofA2. The purpose of this paper is to releaseSn withHΔ.