  We study modular fibers of elliptic differentials, which are roughly spaces
of torus-coverings over a fixed base torus. For genus 2 torus covers with fixed
degree we show, that the modular fibers F_d(1,1) are itself connected torus
covers with Veech group SL_2(Z). Using results of Eskin, Masur and Schmoll we
calculate the Euler Characteristic and the parity of the spin structure of the
quadratic differential (F_d(1,1)/(-id),q_d). We state and apply formulas for
the asymptotic quadratic growth rates of various types of geodesic segments on
a surface S "contained" in F_d(1,1). The quadratic growth rates are expressed
in terms of the SL_2(Z) orbit closure of S \in F_d(1,1) and the flat geometry
of F_d(1,1).

Geometric Topology;Dynamical Systems