  Consider genus $g$ curves that admit degree $d$ covers to elliptic curves
only branched at one point with a fixed ramification type. The locus of such
covers forms a one parameter family $Y$ that naturally maps into the moduli
space of stable genus $g$ curves $\bar{\mathcal M}_{g}$. We study the geometry
of $Y$, and produce a combinatorial method by which to investigate its slope,
irreducible components, genus and orbifold points. As a by-product of our
approach, we find some equalities from classical number theory. Moreover, a
correspondence between our method and the viewpoint of square-tiled surfaces is
established. We also use our results to study the lower bound for slopes of
effective divisors on $\bar{\mathcal M}_{g}$.
