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

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.