A family of elliptic curves of rank ≥ 5 over ℚ(m)

Arman Shamsi Zargar and Naser Zamani
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 4, Pages 24–29
DOI: 10.7546/nntdm.2019.25.4.24-29
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Authors and affiliations

Arman Shamsi Zargar
Department of Mathematics and Applications, Faculty of Science
University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran

Naser Zamani
Department of Mathematics and Applications, Faculty of Science
University of Mohaghegh Ardabili, Ardabil 56199-11367, Iran

Abstract

We construct a subfamily of elliptic curves E(r,s):(y-s)(y+s)=x(x-r)(x+r) with r=2(m^4-2m^3+2m^2+2m+1), s=2(m^2-2m-1)(m^2+1)^2, and show that its rank is at least five over \Bbb Q(m). This improves the previous results on the rank of the curves E(r,s) over \Bbb Q(m).

Keywords

  • Elliptic curves
  • Independence
  • Rank
  • Torsion subgroup

2010 Mathematics Subject Classification

  • 11G05
  • 14H52

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Cite this paper

APA

Zargar, A. S. & Zamani, N. (2019). A family of elliptic curves of rank ≥ 5 over ℚ(m). Notes on Number Theory and Discrete Mathematics, 25(4), 24-29, doi: 10.7546/nntdm.2019.25.4.24-29.

Chicago

Zargar, Arman Shamsi and Naser Zamani. “A Family of Elliptic Curves of Rank ≥ 5 over ℚ(m).” Notes on Number Theory and Discrete Mathematics 25, no. 4 (2019): 24-29, doi: 10.7546/nntdm.2019.25.4.24-29.

MLA

Zargar, Arman Shamsi and Naser Zamani. “A Family of Elliptic Curves of Rank ≥ 5 over ℚ(m).” Notes on Number Theory and Discrete Mathematics 25.4 (2019): 24-29. Print, doi: 10.7546/nntdm.2019.25.4.24-29.

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