Elliptic Factors in Jacobians of Hyperelliptic Curves with Certain Automorphism Groups

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We decompose the Jacobian variety of hyperelliptic curves up to genus 10, defined over an algebraically closed field of characteristic zero, with reduced automorphism group A 4 , S 4 , or A 5 . Among those curves is a genus 4 curve with Jacobian variety isogenous to E 2 1 × E 2 2 and a genus 5 curve with Jacobian variety isogenous to E 5 for E and E i elliptic curves. These types of results have some interesting consequences to questions of ranks of elliptic curves and ranks of their twists.