Degree 4 coverings of elliptic curves by genus 2 curves

Genus two curves covering elliptic curves have been the object of study of many articles. For a fixed degree $n$ the subloci of the moduli space $\mathcal M_2$ofcurveshavingadegree$n$ellipticsubcoverhasbeencomputedfor$n=3, 5$anddiscussedindetailfor$n$ odd; see \cite{Sh1, SV2, Fr, FK}. When the degree of the cover is even the case in general has been treated in \cite{PRS}. In this paper we compute the sublocus of ${\mathcal{M}}_2$ of curves having a degree 4 elliptic subcover.