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In this thesis we describe a family of Jacobian varieties of non-hyperelliptic genus 2g curves that are isogenous to a product of Jacobians of genus g curves in a specific way. For any hyperelliptic genus g curve C we construct a 2-parameter family of hyperelliptic genus g curves H with J(H)[2] isomorphic to J(C)[2], and a generically non-hyperelliptic curve A such that there is an isogeny from J(C) J(H) to J(A) whose kernel is the graph of the isomorphism taking J(H)[2] to J(C)[2]. This is accomplished by first showing that C can be considered as a subcover of a Galois cover of a P1 that has A and H naturally arising as subcovers and then showing the naturally occurring isogeny relations have the desired kernel. We also list some corollaries to the main result and provide a magma script to generate non-hyperelliptic genus 4 curves that have curious automorphism groups.
