@article{discovery10038649, note = {This version is the version of record. For information on re-use, please refer to the publisher's terms and conditions.}, pages = {63--98}, number = {2}, journal = {Publications math{\'e}matiques de Besan{\cc}on: Alg{\`e}bre et Th{\'e}orie des Nombres}, month = {November}, year = {2014}, title = {Ranks of quadratic twists of elliptic curves}, author = {Watkins, M and Donnelly, S and Elkies, N and Fisher, T and Granville, A and Rogers, N}, url = {http://pmb.univ-fcomte.fr/2014\%5fen.html}, abstract = {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-{\`a}-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.} }