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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


Computing the Mazur and Swinnerton-Dyer critical subgroup of elliptic curves
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by Hao Chen PDF
Math. Comp. 85 (2016), 2499-2514


Let $E$ be an optimal elliptic curve defined over $\mathbb {Q}$. The critical subgroup of $E$ is defined by Mazur and Swinnerton-Dyer as the subgroup of $E(\mathbb {Q})$ generated by traces of branch points under a modular parametrization of $E$. 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 critical polynomials attached to $E$ 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.
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Additional Information
  • Hao Chen
  • Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98115
  • Email:
  • Received by editor(s): January 17, 2015
  • Received by editor(s) in revised form: March 1, 2015, and March 18, 2015
  • Published electronically: December 9, 2015
  • © Copyright 2015 Hao Chen
  • Journal: Math. Comp. 85 (2016), 2499-2514
  • MSC (2010): Primary 11G05
  • DOI:
  • MathSciNet review: 3511290