Descent via (5,5)-isogeny on Jacobians of genus 2 curves

We describe a family of curves C of genus 2 with a maximal isotropic (Z/5)2 in J[5], where J is the Jacobian variety of C, and develop the theory required to perform descent via (5,5)-isogeny. We apply this to several examples, where it can be shown that non-reducible Jacobians have nontrivial 5-part of the Tate-Shafarevich group.