{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:41:35Z","timestamp":1776724895138,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"227","license":[{"start":{"date-parts":[[2000,2,15]],"date-time":"2000-02-15T00:00:00Z","timestamp":950572800000},"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":"<p>\n                    In this paper we describe an algorithm for computing the rank of an elliptic curve defined over a real quadratic field of class number one. This algorithm extends the one originally described by Birch and Swinnerton-Dyer for curves over\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"double-struck upper Q\">\n                        <mml:semantics>\n                          <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                            <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\mathbb {Q}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Several examples are included.\n                  <\/p>","DOI":"10.1090\/s0025-5718-99-01055-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1187-1200","source":"Crossref","is-referenced-by-count":9,"title":["Computing the rank of elliptic curves over real quadratic number fields of class number 1"],"prefix":"10.1090","volume":"68","author":[{"given":"J.","family":"Cremona","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"P.","family":"Serf","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,2,15]]},"reference":[{"key":"1","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1515\/crll.1963.212.7","article-title":"Notes on elliptic curves. I","volume":"212","author":"Birch, B. J.","year":"1963","journal-title":"J. Reine Angew. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0075-4102","issn-type":"print"},{"key":"2","unstructured":"J. E. Cremona, Algorithms for Modular Elliptic Curves (Second Edition), Cambridge University Press, 1997."},{"key":"3","unstructured":"J. E. Cremona, Classical invariants and elliptic curves, preprint, 1996."},{"key":"4","series-title":"The Wadsworth \\& Brooks\/Cole Mathematics Series","isbn-type":"print","volume-title":"Holomorphic Hilbert modular forms","author":"Garrett, Paul B.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0534103448"},{"key":"5","unstructured":"H. Graf, Konstruktion elliptischer Kurven hohen Ranges \u00fcber quadratischen Zahlk\u00f6rpern der Klassenzahl eins, Diploma thesis, Universit\u00e4t des Saarlandes, Saarbr\u00fccken, 1995."},{"key":"6","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5119-2","volume-title":"Elliptic curves","volume":"111","author":"Husemoller, Dale","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0387963715"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"379","DOI":"10.2307\/2045480","article-title":"A family of semistable elliptic curves with large Tate-Shafarevitch groups","volume":"89","author":"Kramer, Kenneth","year":"1983","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"8","unstructured":"P. Serf, The rank of elliptic curves over real quadratic number fields of class number 1, PhD thesis, Universit\u00e4t des Saarlandes, Saarbr\u00fccken, 1995."},{"issue":"4","key":"9","doi-asserted-by":"publisher","first-page":"1501","DOI":"10.1216\/rmjm\/1181072159","article-title":"Infinite descent on elliptic curves","volume":"25","author":"Siksek, Samir","year":"1995","journal-title":"Rocky Mountain J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0035-7596","issn-type":"print"},{"key":"10","series-title":"Undergraduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4252-7","volume-title":"Rational points on elliptic curves","author":"Silverman, Joseph H.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0387978259"},{"key":"11","unstructured":"J. Tate, Rational points on elliptic curves, Lectures held at Haverford College, 1961."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01055-8\/S0025-5718-99-01055-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01055-8\/S0025-5718-99-01055-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:06:00Z","timestamp":1776722760000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1999-68-227\/S0025-5718-99-01055-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,2,15]]},"references-count":11,"journal-issue":{"issue":"227","published-print":{"date-parts":[[1999,7]]}},"alternative-id":["S0025-5718-99-01055-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01055-8","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":[[1999,2,15]]}}}