Finding rational points on elliptic curves using 6-descent and 12-descent

We explain how recent work on 3-descent and 4-descent for elliptic curves over Q can be combined to search for generators of the Mordell–Weil group of large height. As an application we show that every elliptic curve of prime conductor in the Stein–Watkins database has rank at least as large as predicted by the conjecture of Birch and Swinnerton-Dyer.