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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.


Quadratic points on modular curves with infinite Mordell–Weil group
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by Josha Box HTML | PDF
Math. Comp. 90 (2021), 321-343 Request permission


Bruin and Najman [LMS J. Comput. Math. 18 (2015), no. 1, 578–602] and Ozman and Siksek [Math. Comp. 88 (2019), no. 319, 2461–2484] have recently determined the quadratic points on each modular curve $X_0(N)$ of genus 2, 3, 4, or 5 whose Mordell–Weil group has rank 0. In this paper we do the same for the $X_0(N)$ of genus 2, 3, 4, and 5 and positive Mordell–Weil rank. The values of $N$ are 37, 43, 53, 61, 57, 65, 67, and 73.

The main tool used is a relative symmetric Chabauty method, in combination with the Mordell–Weil sieve. Often the quadratic points are not finite, as the degree 2 map $X_0(N)\to X_0(N)^+$ can be a source of infinitely many such points. In such cases, we describe this map and the rational points on $X_0(N)^+$, and we specify the exceptional quadratic points on $X_0(N)$ not coming from $X_0(N)^+$. In particular, we determine the $j$-invariants of the corresponding elliptic curves and whether they are ${\mathbb {Q}}$-curves or have complex multiplication.

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Additional Information
  • Josha Box
  • Affiliation: Mathematics Institute, University of Warwick, CV4 7AL, United Kingdom
  • Email:
  • Received by editor(s): June 27, 2019
  • Received by editor(s) in revised form: February 3, 2020, and March 4, 2020
  • Published electronically: August 13, 2020
  • Additional Notes: During the work on this article, the author was supported by an EPSRC DTP studentship.
  • © Copyright 2020 American Mathematical Society
  • Journal: Math. Comp. 90 (2021), 321-343
  • MSC (2010): Primary 11G05, 14G05, 11G18
  • DOI:
  • MathSciNet review: 4166463