  We consider the loci of d-elliptic curves in $M_2$, and corresponding loci of
d-elliptic surfaces in $A_2$. We show how a description of these loci as
quotients of a product of modular curves can be used to calculate cohomology of
natural local systems on them, both as mixed Hodge structures and $\ell$-adic
Galois representations. We study in particular the case d=2, and compute the
Euler characteristic of the moduli space of n-pointed bi-elliptic genus 2
curves in the Grothendieck group of Hodge structures.
