Gluing curves of genus 1 and 2 along their 2-torsion

Let X X (resp. Y"> Y encoding="application/x-tex">Y ) be a curve of genus alttext="1"> 1 encoding="application/x-tex">1 alttext="2"> 2 encoding="application/x-tex">2 over base field alttext="k"> k encoding="application/x-tex">k whose characteristic does not equal . We give criteria for the existence Z"> Z encoding="application/x-tex">Z Jacobian is up to twist alttext="left-parenthesis 2 comma right-parenthesis"> ( , stretchy="false">) encoding="application/x-tex">(2,2,2) -isogenous products Jacobians and Moreover, we algorithms construct once equations are given. The first these based on interpolation methods involving numerical results alttext="double-struck upper C"> C encoding="application/x-tex">\mathbb {C} that proved correct general fields posteriori, whereas second involves use hyperplane sections Kummer variety desingularization isomorphic As an application, find Q"> mathvariant="double-struck">Q {Q} admits rational alttext="70"> 70 encoding="application/x-tex">70 -torsion point.