{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T14:05:01Z","timestamp":1777039501990,"version":"3.51.4"},"reference-count":19,"publisher":"American Mathematical Society (AMS)","issue":"243","license":[{"start":{"date-parts":[[2003,12,18]],"date-time":"2003-12-18T00:00:00Z","timestamp":1071705600000},"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                    We present efficient algorithms for solving Legendre equations 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                    (equivalently, for finding rational points on rational conics) and parametrizing all solutions. Unlike existing algorithms, no integer factorization is required, provided that the prime factors of the discriminant are known.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01480-1","type":"journal-article","created":{"date-parts":[[2003,4,18]],"date-time":"2003-04-18T13:09:02Z","timestamp":1050671342000},"page":"1417-1441","source":"Crossref","is-referenced-by-count":46,"title":["Efficient solution of rational conics"],"prefix":"10.1090","volume":"72","author":[{"given":"J.","family":"Cremona","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"D.","family":"Rusin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2002,12,18]]},"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. 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Number Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0022-314X","issn-type":"print"},{"key":"4","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-02945-9","volume-title":"A course in computational algebraic number theory","volume":"138","author":"Cohen, Henri","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3540556400"},{"key":"5","unstructured":"J. E. Cremona, Higher descents on elliptic curves, preprint: see \\url{http:\/\/www.maths.nott.ac.uk\/personal\/jec\/papers\/d2.ps}."},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1006\/jnth.1996.0035","article-title":"Simultaneous representation of integers by a pair of ternary quadratic forms\u2014with an application to index form equations in quartic number fields","volume":"57","author":"Ga\u00e1l, Istv\u00e1n","year":"1996","journal-title":"J. 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(N.S.)","ISSN":"https:\/\/id.crossref.org\/issn\/0273-0979","issn-type":"print"},{"key":"10","series-title":"Pure and Applied Mathematics, Vol. 30","volume-title":"Diophantine equations","author":"Mordell, L. J.","year":"1969"},{"issue":"3-4","key":"11","doi-asserted-by":"publisher","first-page":"535","DOI":"10.5486\/pmd.2000.2410","article-title":"On Legendre\u2019s equation over number fields","volume":"56","author":"Pohst, Michael E.","year":"2000","journal-title":"Publ. Math. Debrecen","ISSN":"https:\/\/id.crossref.org\/issn\/0033-3883","issn-type":"print"},{"issue":"5","key":"12","doi-asserted-by":"publisher","first-page":"702","DOI":"10.1109\/TIT.1987.1057350","article-title":"An efficient solution of the congruence \ud835\udc65\u00b2+\ud835\udc58\ud835\udc66\u00b2=\ud835\udc5a (mod \ud835\udc5b)","volume":"33","author":"Pollard, John M.","year":"1987","journal-title":"IEEE Trans. Inform. Theory","ISSN":"https:\/\/id.crossref.org\/issn\/0018-9448","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1016\/S0747-7171(86)80027-X","article-title":"On the number of divisions of the Euclidean algorithm applied to Gaussian integers","volume":"2","author":"Rolletschek, Heinrich","year":"1986","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"key":"14","unstructured":"D. Simon, \u00c9quations dans les corps de nombres et discriminants minimaux, Ph.D. thesis, Universit\u00e9 Bordeaux I, 1998."},{"key":"15","series-title":"London Mathematical Society Student Texts","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9781107359994","volume-title":"The algorithmic resolution of Diophantine equations","volume":"41","author":"Smart, Nigel P.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0521646332"},{"key":"16","first-page":"Exp. No. 13, 15","article-title":"Algorithmique dans les r\u00e9seaux de petite dimension: un point de vue affine sur la recherche des minima","author":"Vall\u00e9e, Brigitte","year":"1986"},{"key":"17","unstructured":"H.-J. Weber, Algorithmische Konstruktion hyperelliptischer Kurven mit kryptographischer Relevanz und einem Endomorphismenring echt gr\u00f6\u00dfer als Z, Ph.D. thesis, Institut f\u00fcr Experimentelle Mathematik, University of Essen, 1997."},{"key":"18","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-8176-4571-7","volume-title":"Number theory","author":"Weil, Andr\u00e9","year":"1984","ISBN":"https:\/\/id.crossref.org\/isbn\/0817631410"},{"key":"19","unstructured":"J. Wilson, Curves of genus 2 with real multiplication by a square root of 5, Ph.D. thesis, University of Oxford, 1998."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01480-1\/S0025-5718-02-01480-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01480-1\/S0025-5718-02-01480-1.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:20:46Z","timestamp":1776727246000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-02-01480-1\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,12,18]]},"references-count":19,"journal-issue":{"issue":"243","published-print":{"date-parts":[[2003,7]]}},"alternative-id":["S0025-5718-02-01480-1"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01480-1","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":[[2002,12,18]]}}}