TY - JOUR A1 - Watkins, M A1 - Donnelly, S A1 - Elkies, N A1 - Fisher, T A1 - Granville, A A1 - Rogers, N JF - Publications mathématiques de Besançon: Algèbre et Théorie des Nombres N1 - This version is the version of record. For information on re-use, please refer to the publisher?s terms and conditions. SP - 63 N2 - 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. IS - 2 EP - 98 UR - http://pmb.univ-fcomte.fr/2014_en.html ID - discovery10038649 TI - Ranks of quadratic twists of elliptic curves Y1 - 2014/11/27/ AV - public ER -