{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:13:31Z","timestamp":1776802411670,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"307","license":[{"start":{"date-parts":[[2017,12,21]],"date-time":"2017-12-21T00:00:00Z","timestamp":1513814400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n Let\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be an elliptic curve over a number field\n \n \n \n K<\/mml:mi>\n K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Descent calculations on\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n can be used to find upper bounds for the rank of the Mordell-Weil group, and to compute covering curves that assist in the search for generators of this group. The general method of\n \n \n \n 4<\/mml:mn>\n 4<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -descent, developed in the PhD theses of Siksek, Womack and Stamminger, has been implemented in Magma (when\n \n \n \n \n K<\/mml:mi>\n =<\/mml:mo>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n K=\\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) and works well for elliptic curves with sufficiently small discriminant. By extending work of Bremner and Cassels, we describe the improvements that can be made when\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has a rational\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -torsion point. In particular, when\n \n \n \n E<\/mml:mi>\n E<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n has full rational\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -torsion, we describe a method for\n \n \n \n 8<\/mml:mn>\n 8<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -descent that is practical for elliptic curves\n \n \n \n \n E<\/mml:mi>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n \n Q<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n E\/\\mathbb {Q}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with large discriminant.\n <\/p>","DOI":"10.1090\/mcom\/3163","type":"journal-article","created":{"date-parts":[[2016,4,14]],"date-time":"2016-04-14T12:47:00Z","timestamp":1460638020000},"page":"2493-2518","source":"Crossref","is-referenced-by-count":5,"title":["Higher descents on an elliptic curve with a rational 2-torsion point"],"prefix":"10.1090","volume":"86","author":[{"given":"Tom","family":"Fisher","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,12,21]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1515\/crll.1965.218.79","article-title":"Notes on elliptic curves. II","volume":"218","author":"Birch, B. J.","year":"1965","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"issue":"3-4","key":"2","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"165","key":"3","doi-asserted-by":"publisher","first-page":"257","DOI":"10.2307\/2007578","article-title":"On the equation \ud835\udc4c\u00b2=\ud835\udc4b(\ud835\udc4b\u00b2+\ud835\udc5d)","volume":"42","author":"Bremner, A.","year":"1984","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"4","doi-asserted-by":"publisher","first-page":"180","DOI":"10.1515\/crll.1965.217.180","article-title":"Arithmetic on curves of genus 1. VIII. On conjectures of Birch and Swinnerton-Dyer","volume":"217","author":"Cassels, J. W. S.","year":"1965","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"5","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781139172530","volume-title":"Lectures on elliptic curves","volume":"24","author":"Cassels, J. W. S.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0521415179"},{"key":"6","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1515\/crll.1998.001","article-title":"Second descents for elliptic curves","volume":"494","author":"Cassels, J. W. S.","year":"1998","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"7","unstructured":"[Cr] J. E. Cremona, Higher descents on elliptic curves, notes for a talk, 1997, \\url{http:\/\/homepages.warwick.ac.uk\/ masgaj\/papers\/d2.ps}"},{"issue":"243","key":"8","doi-asserted-by":"publisher","first-page":"1417","DOI":"10.1090\/S0025-5718-02-01480-1","article-title":"Efficient solution of rational conics","volume":"72","author":"Cremona, J. E.","year":"2003","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"6","key":"9","doi-asserted-by":"publisher","first-page":"673","DOI":"10.1016\/j.jsc.2008.09.004","article-title":"On the equivalence of binary quartics","volume":"44","author":"Cremona, J. E.","year":"2009","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"6","key":"10","doi-asserted-by":"publisher","first-page":"763","DOI":"10.2140\/ant.2010.4.763","article-title":"Minimisation and reduction of 2-, 3- and 4-coverings of elliptic curves","volume":"4","author":"Cremona, John E.","year":"2010","journal-title":"Algebra Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/1937-0652","issn-type":"print"},{"key":"11","unstructured":"[D] A. Dujella, High rank elliptic curves with prescribed torsion, \\url{http:\/\/web.math.pmf.unizg.hr\/ duje\/tors\/tors.html}"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/S0022-314X(02)00038-0","article-title":"The Cassels-Tate pairing and the Platonic solids","volume":"98","author":"Fisher, Tom A.","year":"2003","journal-title":"J. Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"13","isbn-type":"print","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/978-3-540-79456-1_8","article-title":"Some improvements to 4-descent on an elliptic curve","author":"Fisher, Tom","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540794554"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"385","DOI":"10.4064\/aa-77-4-385-404","article-title":"Explicit 4-descents on an elliptic curve","volume":"77","author":"Merriman, J. R.","year":"1996","journal-title":"Acta Arith.","ISSN":"https:\/\/id.crossref.org\/issn\/0065-1036","issn-type":"print"},{"key":"15","unstructured":"[Sik] S. Siksek, Descent on curves of genus 1, PhD thesis, University of Exeter, 1995."},{"key":"16","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-1920-8","volume-title":"The arithmetic of elliptic curves","volume":"106","author":"Silverman, Joseph H.","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/0387962034"},{"issue":"251","key":"17","doi-asserted-by":"publisher","first-page":"1531","DOI":"10.1090\/S0025-5718-05-01729-1","article-title":"Solving quadratic equations using reduced unimodular quadratic forms","volume":"74","author":"Simon, Denis","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"18","unstructured":"[S2] D. Simon, Quadratic equations in dimensions 4, 5 and more, preprint 2005."},{"key":"19","unstructured":"[St] S. Stamminger, Explicit 8-descent on elliptic curves, PhD thesis, International University Bremen, 2005."},{"issue":"3","key":"20","doi-asserted-by":"publisher","first-page":"707","DOI":"10.1112\/jlms\/jds063","article-title":"2\u207f-descent on elliptic curves for all \ud835\udc5b","volume":"87","author":"Swinnerton-Dyer, Peter","year":"2013","journal-title":"J. Lond. Math. Soc. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0024-6107","issn-type":"print"},{"key":"21","first-page":"63","article-title":"Ranks of quadratic twists of elliptic curves","author":"Watkins, Mark","year":"2015"},{"key":"22","unstructured":"[Wo] T. Womack, Explicit descent on elliptic curves, PhD thesis, University of Nottingham, 2003."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03163-4\/S0025-5718-2016-03163-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03163-4\/S0025-5718-2016-03163-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:18:41Z","timestamp":1776799121000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-307\/S0025-5718-2016-03163-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,21]]},"references-count":22,"journal-issue":{"issue":"307","published-print":{"date-parts":[[2017,9]]}},"alternative-id":["S0025-5718-2016-03163-4"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3163","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2016,12,21]]}}}