Two-coverings of Jacobians of curves of genus 2

Given a curve C of genus 2 defined over a field k of characteristic different from 2, with a Jacobian variety J, we show that the two-coverings corresponding to elements of a large subgroup of H-1(Gal(k(s)/k), J[2](k(s))) (containing the Selmer group when k is a global field) can be embedded as an intersection of 72 quadrics in P-k(15), just as the Jacobian J itself. Moreover, we actually give explicit equations for the models of these twists in the generic case, extending the work of Gordon and Grant which applied only to the case when all Weierstrass points are rational. In addition, we describe elegant equations of the Jacobian itself, and answer a question of Cassels and Flynn concerning a map from the Kummer surface in P-3 to the desingularized Kummer surface in P-5.