eprintid: 10038649 rev_number: 38 eprint_status: archive userid: 608 dir: disk0/10/03/86/49 datestamp: 2018-03-09 15:24:38 lastmod: 2020-08-01 23:44:42 status_changed: 2018-03-12 09:26:39 type: article metadata_visibility: show creators_name: Watkins, M creators_name: Donnelly, S creators_name: Elkies, N creators_name: Fisher, T creators_name: Granville, A creators_name: Rogers, N title: Ranks of quadratic twists of elliptic curves ispublished: pub divisions: UCL divisions: A01 divisions: B04 divisions: C06 note: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions. 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-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. date: 2014-11-27 date_type: published official_url: http://pmb.univ-fcomte.fr/2014_en.html oa_status: green full_text_type: pub language: eng primo: open primo_central: open_green article_type_text: Article verified: verified_manual elements_id: 1512212 lyricists_name: Granville, Andrew lyricists_id: AGRAN98 actors_name: Granville, Andrew actors_id: AGRAN98 actors_role: owner full_text_status: public publication: Publications mathématiques de Besançon: Algèbre et Théorie des Nombres number: 2 pagerange: 63-98 citation: Watkins, M; Donnelly, S; Elkies, N; Fisher, T; Granville, A; Rogers, N; (2014) Ranks of quadratic twists of elliptic curves. Publications mathématiques de Besançon: Algèbre et Théorie des Nombres (2) pp. 63-98. Green open access document_url: https://discovery.ucl.ac.uk/id/eprint/10038649/7/Watkins_et_all.pdf