{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:03:50Z","timestamp":1776801830047,"version":"3.51.2"},"reference-count":15,"publisher":"American Mathematical Society (AMS)","issue":"301","license":[{"start":{"date-parts":[[2016,12,9]],"date-time":"2016-12-09T00:00:00Z","timestamp":1481241600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Let\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    be an optimal elliptic curve defined 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                    . The\n                    <italic>critical subgroup<\/italic>\n                    of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is defined by Mazur and Swinnerton-Dyer as the subgroup of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E left-parenthesis double-struck upper Q right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>E<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi mathvariant=\"double-struck\">Q<\/mml:mi>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">E(\\mathbb {Q})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    generated by traces of branch points under a modular parametrization of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . We prove that for all rank two elliptic curves with conductor smaller than 1000, the critical subgroup is torsion. First, we define a family of\n                    <italic>critical polynomials<\/italic>\n                    attached to\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper E\">\n                        <mml:semantics>\n                          <mml:mi>E<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">E<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and develop two algorithms to compute such polynomials. We then give a sufficient condition for the critical subgroup to be torsion in terms of the factorization of critical polynomials. Finally, a table of critical polynomials is obtained for all elliptic curves of rank two and conductor smaller than 1000, from which we deduce our result.\n                  <\/p>","DOI":"10.1090\/mcom3057","type":"journal-article","created":{"date-parts":[[2015,3,25]],"date-time":"2015-03-25T14:30:50Z","timestamp":1427293850000},"page":"2499-2514","source":"Crossref","is-referenced-by-count":0,"title":["Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves"],"prefix":"10.1090","volume":"85","author":[{"given":"Hao","family":"Chen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,12,9]]},"reference":[{"issue":"2","key":"1","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1007\/s00208-002-0390-9","article-title":"Weierstrass points on \ud835\udc4b\u2080(\ud835\udc5d) and supersingular \ud835\udc57-invariants","volume":"325","author":"Ahlgren, Scott","year":"2003","journal-title":"Math. 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Stein et al., Sage Mathematics Software (Version 6.4), The Sage Development Team, 2014, http:\/\/www.sagemath.org."},{"key":"14","unstructured":"[Ste] William Stein, Algebraic number theory, a computational approach, \\url{https:\/\/github.com\/williamstein\/ant}."},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1016\/j.aim.2005.05.019","article-title":"Defining equations of modular curves","volume":"204","author":"Yang, Yifan","year":"2006","journal-title":"Adv. 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