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On the rank of elliptic curves arising from Pythagorean quadruplets, II

Volume 163 / 2021

Dustin Moody, Arman Shamsi Zargar Colloquium Mathematicum 163 (2021), 189-196 MSC: Primary 14H52; Secondary 11G05, 11D25. DOI: 10.4064/cm8101-3-2020 Published online: 15 June 2020

Abstract

By a Pythagorean quadruplet $(a,b,c,d)$, we mean an integer solution to the quadratic equation $a^2+b^2=c^2+d^2$. We use Pythagorean quadruplets to construct infinite families of elliptic curves with rank as high as possible, including one family with rank at least $4$. We also find a few examples of these curves with rank 8. The families of curves we construct have trivial and $\mathbb Z/2\mathbb Z$ torsion subgroups in general. Previous work of the second named author has studied similar Pythagorean quadruplet elliptic curve families with torsion subgroup $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$.

Authors

  • Dustin MoodyComputer Security Division
    National Institute of Standards
    and Technology
    100 Bureau Drive
    Gaithersburg, MD 20899-8930, U.S.A.
    e-mail
  • Arman Shamsi ZargarDepartment of Mathematics
    and Applications
    Faculty of Science
    University of Mohaghegh Ardabili
    Ardabil 56199-11367, Iran
    e-mail

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