%0 Journal Article %A Watkins, M %A Donnelly, S %A Elkies, N %A Fisher, T %A Granville, A %A Rogers, N %D 2014 %F discovery:10038649 %J Publications mathématiques de Besançon: Algèbre et Théorie des Nombres %N 2 %P 63-98 %T Ranks of quadratic twists of elliptic curves %U https://discovery.ucl.ac.uk/id/eprint/10038649/ %X We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability to find a rank 8 twist, and discuss how our results here compare to some predictions of rank growth vis-à-vis conductor. Finally we explicate a heuristic of Granville, which when interpreted judiciously could predict that 7 is indeed the maximal rank in this quadratic twist family. %Z This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.