  We prove that the moduli space A_3(1,1,4) of polarized abelian threefolds
with polarization of type (1,1,4) is unirational. By a result of Birkenhake and
Lange this implies the unirationality of the isomorphic moduli space
A_3(1,4,4). The result is based on the study the Hurwitz space H_{4,n}(Y) of
quadruple coverings of an elliptic curve Y simply branched in n points. We
prove the unirationality of its codimension one subvariety H^{0}_{4,A}(Y) which
parametrizes quadruple coverings \pi:X --> Y with Tschirnhausen modules
isomorphic to A^{-1}, where A\in Pic^{n/2}Y, and for which \pi^*:J(Y)--> J(X)
is injective. This is an analog of the result of Arbarello and Cornalba that
the Hurwitz space H_{4,n}(P^1) is unirational.
