  For any finite abelian group G, we study the moduli space of abelian
$G$-covers of elliptic curves, in particular identifying the irreducible
components of the moduli space. We prove that, in the totally ramified case,
the moduli space has trivial rational Picard group, and it is birational to the
moduli space M_{1,n}, where n is the number of branch points. In the particular
case of moduli of bielliptic curves, we also prove that the boundary divisors
are a basis of the rational Picard group of the admissible covers
compactification of the moduli space. Our methods are entirely
algebro-geometric.
