Root Numbers of 5-adic Curves of Genus Two Having Maximal Ramification

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