@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.}
}