{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:28:04Z","timestamp":1776842884951,"version":"3.51.2"},"reference-count":22,"publisher":"American Mathematical Society (AMS)","issue":"267","license":[{"start":{"date-parts":[[2009,10,29]],"date-time":"2009-10-29T00:00:00Z","timestamp":1256774400000},"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                    The algorithm we develop outputs the order and the structure, including generators, of the\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u2113\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -Sylow subgroup of the group of rational points of an elliptic curve defined over a finite field. To do this, we do not assume any knowledge of the group order. We are able to choose points in such a way that a linear number of successive\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u2113\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -divisions leads to generators of the subgroup under consideration. After the computation of a couple of polynomials, each division step relies on finding rational roots of polynomials of degree\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u2113\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We specify in complete detail the case\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"script l equals 3\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>\n                              \u2113\n                              \n                            <\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\ell =3<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , when the complexity of each trisection is given by the computation of cubic roots in finite fields.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02201-1","type":"journal-article","created":{"date-parts":[[2009,4,27]],"date-time":"2009-04-27T13:47:33Z","timestamp":1240840053000},"page":"1767-1786","source":"Crossref","is-referenced-by-count":11,"title":["Computing the \u2113-power torsion of an elliptic curve over a finite field"],"prefix":"10.1090","volume":"78","author":[{"given":"J.","family":"Miret","sequence":"first","affiliation":[]},{"given":"R.","family":"Moreno","sequence":"additional","affiliation":[]},{"given":"A.","family":"Rio","sequence":"additional","affiliation":[]},{"given":"M.","family":"Valls","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,10,29]]},"reference":[{"key":"1","series-title":"Foundations of Computing Series","isbn-type":"print","volume-title":"Algorithmic number theory. 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(Basel)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-889X","issn-type":"print"},{"key":"7","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"},{"key":"8","unstructured":"D. Kohel, Endomorphism rings of elliptic curves over finite fields, Ph.D. thesis, University of California at Berkeley, 1996."},{"key":"9","unstructured":"LiDIA-Group, LiDIA Manual: A library for computational number theory, Tech. Univ. Darmstadt, 2001. Available from \\url{ftp.informatik.tu-darmstadt.de\/pub\/TI\/systems\/LiDIA}."},{"key":"10","unstructured":"Magma Group, Handbook of Magma functions, J. Canon and W. Bosma, eds. Available from \\url{http:\/\/magma.maths.usyd.edu.au\/}."},{"key":"11","doi-asserted-by":"crossref","unstructured":"J. Miret, R. Moreno, and A. Rio, Generalization of V\u00e9lu\u2019s formulae for isogenies, Proceedings of the Primeras Jornadas de Teor\u00eda de N\u00fameros (Vilanova i la Geltr\u00fa, 2005) Publ. Mat. (2007), Vol. Extra, pp. 147\u2013163.","DOI":"10.5565\/PUBLMAT_PJTN05_07"},{"issue":"249","key":"12","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1090\/S0025-5718-04-01640-0","article-title":"Determining the 2-Sylow subgroup of an elliptic curve over a finite field","volume":"74","author":"Miret, J.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"255","DOI":"10.5802\/jtnb.143","article-title":"Calcul du nombre de points sur une courbe elliptique dans un corps fini: aspects algorithmiques","volume":"7","author":"Morain, Fran\u00e7ois","year":"1995","journal-title":"J. Th\\'{e}or. 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