Second isogeny descents and the Birch and Swinnerton-Dyer conjectural formula

Abstract Let φ : E → E ′ be an isogeny of prime degree l between elliptic curves defined over a number field. We describe how to perform φ-descents on the nontrivial elements in the Shafarevich–Tate group of E ′ which are killed by the dual isogeny φ ′ . This makes computation of l-Selmer groups of elliptic curves admitting an l-isogeny over Q feasible for l = 5 , 7 in cases where a φ-descent on E is insufficient and a full l-descent would be infeasible. As an application we complete the verification of the full Birch and Swinnerton-Dyer conjectural formula for all elliptic curves over Q of rank zero or one and conductor less than 5000.