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

Root numbers of 5-adic curves of genus two having maximal ramification