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  -