%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.
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