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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## Details

### 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 with , and show that its rank is at least five over . This improves the previous results on the rank of the curves over .

### Keywords

- Elliptic curves
- Independence
- Rank
- Torsion subgroup

### 2010 Mathematics Subject Classification

- 11G05
- 14H52

### References

- Antoniewicz, A. (2005). On a family of elliptic curves, Univ. Iagel. Acta Math., 43, 21–32.
- Birch, B. J., Stephens, N. M. (1996). The parity of the rank of the Mordell–Weil group,Topology, 5, 295–299.
- Brown, E. A., & Myers, B. T. (2002). Elliptic curves from Mordell to Diophantus and back,Amer. Math. Monthly, 109, 639–649
- Cremona, J. mwrank program, Available online at: http://homepages.warwick.ac.uk/staff/J.E.Cremona//ftp/progs/.
- Das, P., Juyal, A., & Moody, D. (2017). Integral isosceles triangle-parallelogram and Heron triangle-rhombus pairs with a common area and common perimeter, J. Number Theory, 180, 208–218.
- Duquesne, S., Nara, T., & Shamsi Zargar, A. (2019). Generators and integral points on elliptic curves associated with simplest quartic fields, Math. Slovaca, Accepted.
- Eikenberg, E. V. (2004). Rational Points on Some Families of Elliptic Curves, PhD thesis, University of Maryland.
- Fujita, Y.,& Nara, T. (2018). The Mordell–Weil base for the elliptic curve of the form , Publ. Math. Debrecen, 92, 1-2, 79–99.
- Izadi, F., & Baghalaghdam, M. (2018). Some new families of positive-rank elliptic curves arising from Pythagorean triples, Notes Number Theory Discrete Math., 24, 3, 27–36.
- Izadi, F., & Nabardi, K. (2016). A family of elliptic curves of rank , Involve, 9, 5, 733–736.
- Izadi, F., & Shamsi Zargar, A. (2017). A note on twists of , Iran. J. Math. Sci. Inform., 12, 1, 27–34.
- Johnstone, J. A., & Spearman, B. K. (2010). Congruent number elliptic curves with rank at least three, Canad. Math. Bull., 53, 4, 661–666.
- Mestre, J.-F. (1986). Formules explicites et minorations de conducteurs de variétés algébriques, Compos. Math., 58, 209–232.
- Pari/gp, version 2.3.4, Available online at: http://pari.math.u-bordeaux.fr/
- Park, J., Poonen, B., Voight, J., & Wood, M. M. A heuristic for boundedness of ranks of elliptic curves, arXiv:1602.01431.
- Poonen, B. Heuristics for the arithmetic of elliptic curves, arXiv:1711.10112
- Sage software, Available online at: http://www.sagemath.org
- Silverman, J. H. (1983). Heights and the specialization map for families of abelian varieties, J. Reine Angew. Math., 342, 197–211.
- Tadić, P. (2012). On the family of elliptic curves , Glas. Math. Ser. III, 47, 81–93.
- Tadić, P. (2012). The rank of certain subfamilies of elliptic curve , Ann. Math. Inform., 40, 145–153.
- Zhang, Y., Peng, J. Y., & Wang, J. M. (2018). Integral triangles and trapezoids pairs with a common area and a common perimeter, Forum Geom., 18, 371–380.
- Zhang, Y., & Shamsi Zargar, A. (2019). Integral triangles and cyclic quadrilateral pairs with a common area and a common perimeter, Forum Geom., Accepted.

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## Cite this paper

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.