Realizing alternating groups as monodromy groups of genus one covers

We prove that if $n\geq 4$, a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of $\mathbb{P}^{1}$) of degree $n$ such that every branch point has multiplicity $3$ and the monodromy group is the alternating group $A_{n}$. To prove this theorem, we construct a Hurwitz space and show that it maps (generically) onto the genus one moduli space.