Realizing alternating groups as monodromy groups of genus one covers

We prove that if n greater than or equal to 4, a generic Riemann surface of genus 1 admits a meromorphic function (i.e., an analytic branched cover of 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.