{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,24]],"date-time":"2026-01-24T02:32:08Z","timestamp":1769221928526,"version":"3.49.0"},"reference-count":8,"publisher":"Wiley","license":[{"start":{"date-parts":[[2010,2,1]],"date-time":"2010-02-01T00:00:00Z","timestamp":1264982400000},"content-version":"unspecified","delay-in-days":2953,"URL":"https:\/\/www.cambridge.org\/core\/terms"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["LMS J. Comput. Math."],"published-print":{"date-parts":[[2002]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper concerns the existence and algorithmic determination of minimal models for curves of genus 1, given by equations of the form y2 = Q(x), where Q(x) has degree 4. These models are used in the method of 2-descent for computing the rank of an elliptic curve. The results described here are complete for unramified extensions of Q2 and Q3, and for all p-adic fields for p greater than or equal to 5. The primary motivation for this work was to complete the results of Birch and Swinnerton-Dyer, which are incomplete in the case of Q2. The results in this case (when applied to 2-coverings of elliptic curves over Q) yield substantial improvements in the running times of the 2-descent algorithm implemented in the program mwrank. The paper ends with a section on implementation and examples, and an appendix gives constructive proofs in sufficient detail to be used for implementation.<\/jats:p>","DOI":"10.1112\/s1461157000000760","type":"journal-article","created":{"date-parts":[[2013,8,6]],"date-time":"2013-08-06T07:42:12Z","timestamp":1375774932000},"page":"220-243","source":"Crossref","is-referenced-by-count":12,"title":["Minimal Models for 2-coverings of Elliptic Curves"],"prefix":"10.1112","volume":"5","author":[{"given":"Michael","family":"Stoll","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"John E.","family":"Cremona","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"311","published-online":{"date-parts":[[2010,2,1]]},"reference":[{"key":"S1461157000000760_ref006","article-title":"\u2018Computing the rank of elliptic curves over real quadratic fields of class number 1\u2019","volume":"68","author":"Cremona","journal-title":"Math. Comp"},{"key":"S1461157000000760_ref005","unstructured":"5 Cremona J. E. , \u2018mwrank\u2019 and other programs for elliptic curves over \u211a, http:\/\/www.maths.nott.ac.uk\/personal\/jec\/ftp\/progs."},{"key":"S1461157000000760_ref003","volume-title":"Algorithms for modular elliptic curves","author":"Cremona"},{"key":"S1461157000000760_ref001","article-title":"\u2018The rank of elliptic curves\u2019","volume":"44","author":"Brumer","journal-title":"Duke Math. J."},{"key":"S1461157000000760_ref002","article-title":"Notes on elliptic curves, I","volume":"212","author":"Birch","journal-title":"J.Reine Angew. Math."},{"key":"S1461157000000760_ref004","article-title":"Classical invariants and 2-descent on elliptic curves\u2019","volume":"31","author":"Cremona","journal-title":"J.Symbolic Comput"},{"key":"S1461157000000760_ref007","volume":"106","author":"Silverman","journal-title":"The arithmetic of elliptic curves"},{"key":"S1461157000000760_ref008","unstructured":"8 Serf P. : \u2018The rank of elliptic curves over real quadratic number fields of class number 1\u2019, Thesis, Universit\u00e4t des Saarlandes, 1995."}],"container-title":["LMS Journal of Computation and Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.cambridge.org\/core\/services\/aop-cambridge-core\/content\/view\/S1461157000000760","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,6,7]],"date-time":"2019-06-07T19:48:52Z","timestamp":1559936932000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.cambridge.org\/core\/product\/identifier\/S1461157000000760\/type\/journal_article"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002]]},"references-count":8,"alternative-id":["S1461157000000760"],"URL":"https:\/\/doi.org\/10.1112\/s1461157000000760","relation":{},"ISSN":["1461-1570"],"issn-type":[{"value":"1461-1570","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002]]}}}