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

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