A parametrised family of Mordell curves with a rational point of order 3

Ajai Choudhry and Arman Shamsi Zargar
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 40–44
DOI: 10.7546/nntdm.2020.26.1.40-44
Full paper (PDF, 115 Kb)

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Authors and affiliations

Ajai Choudhry
13/4 A Clay Square, Lucknow – 226001, India

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

Abstract

An elliptic curve defined by an equation of the type y² = x³ + d is called a Mordell curve. This paper is concerned with Mordell curves for which d = k²; k ∈ ℤ; k ≠ 1. The point (0, k) on such curves is of order 3 and the torsion subgroup of the group of rational points on such Mordell curves is necessarily ℤ/3ℤ. We obtain a parametrised family of Mordell curves y² = x³ + k² such that the rank of each member of the family is at least three. Some elliptic curves of the family have ranks 4 and 5.

Keywords

• Mordell curves
• Rank of elliptic curves

2010 Mathematics Subject Classification

• 11D25
• 11G05

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