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Genus two curves covering elliptic curves have been the object of study ofmany articles. For a fixed degree $n$ the subloci of the moduli space $\mathcalM_2$ of curves having a degree $n$ elliptic subcover has been computed for$n=3, 5$ and discussed in detail for $n$ odd; see \cite{Sh1, SV2, Fr, FK}. Whenthe 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 curveshaving a degree 4 elliptic subcover.

Degree 4 coverings of elliptic curves by genus 2 curves