  The formulas for local root numbers of abelian varieties of dimension one are
known. In this paper we treat the simplest unknown case in dimension two by
considering a curve of genus 2 defined over a $5$-adic field such that the
inertia acts on the first $\ell$-adic cohomology group through the largest
possible finite quotient, isomorphic to $C_5\rtimes C_8$. We give a few
criteria to identify such curves and prove a formula for their local root
numbers in terms of invariants associated to a Weierstrass equation.
