  The computation of the field of moduli of a closed Riemann surface seems to
be a very difficult problem and even more difficult is to determine if the
field of moduli is a field of definition. In this paper we consider the family
of closed Riemann surfaces of genus five admitting a group of conformal
automorphisms isomorphic to ${\mathbb Z}_{2}^{4}$. These surfaces are
non-hyperelliptic ones and turn out to be the highest branched abelian covers
of the orbifolds of genus zero and five cone points of order two. We compute
the field of moduli of these surfaces and we prove that they are fields of
definition. This result is in contrast with the case of the highest branched
abelian covers of the orbifolds of genus zero and six cone points of order two
as there are cases for which the above property fails.
