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

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## Details

### 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 *y*^{2} = *x*^{3} + *d* is called a Mordell curve. This paper is concerned with Mordell curves for which *d* = *k*^{2}; *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 *y*^{2} = *x*^{3} + *k*^{2} 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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## Cite this paper

Choudhry, A., & Zargar A. S.(2020). A parametrised family of Mordell curves with a rational point of order 3. Notes on Number Theory and Discrete Mathematics, 26(1), 40-44, doi: 10.7546/nntdm.2020.26.1.40-44.