Generators for the elliptic curve $y^2=x^3-nx$

Yasutsugu Fujita; Nobuhiro Terai

doi:10.5802/jtnb.769

Generators for the elliptic curve

y^{2} = x^{3} - n x

Yasutsugu Fujita ¹; Nobuhiro Terai ²

¹ College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba 275–8576 Japan
² Division of General Education Ashikaga Institute of Technology 268-1 Omae, Ashikaga, Tochigi 326–8558 Japan

Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 403-416.

Abstract
Résumé

Let $E$ be an elliptic curve given by $y^{2} = x^{3} - n x$ with a positive integer $n$ . Duquesne in 2007 showed that if $n = (2 k^{2} - 2 k + 1) (18 k^{2} + 30 k + 17)$ is square-free with an integer $k$ , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of $E$ . In this paper, we generalize this result and show that the same is true for infinitely many binary forms $n = n (k, l)$ in $ℤ [k, l]$ .

Soit $E$ la courbe elliptique définie par $y^{2} = x^{3} - n x$ où $n$ est un entier strictement positif. En $2007$ , Duquesne a démontré que, pour $k$ entier, si $n = (2 k^{2} - 2 k + 1) (18 k^{2} + 30 k + 17)$ est sans facteur carré, alors deux points rationnels spécifiques peuvent toujours se compléter en un système de générateurs du groupe de Mordell-Weil associé à $E$ . Dans ce papier, nous généralisons ce résultat en le montrant pour des entiers $n = n (k, l)$ pour une infinité de formes binaires $n (k, l) \in ℤ [k, l]$ .

MR: 2817937 | Zbl: 1228.11081

DOI: 10.5802/jtnb.769

Author's affiliations:

Yasutsugu Fujita ¹; Nobuhiro Terai ²

¹ College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba 275–8576 Japan
² Division of General Education Ashikaga Institute of Technology 268-1 Omae, Ashikaga, Tochigi 326–8558 Japan

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     author = {Yasutsugu Fujita and Nobuhiro Terai},
     title = {Generators for the elliptic curve $y^2=x^3-nx$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {403--416},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {2},
     year = {2011},
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     mrnumber = {2817937},
     zbl = {1228.11081},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/}
}

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Yasutsugu Fujita; Nobuhiro Terai. Generators for the elliptic curve $y^2=x^3-nx$. Journal de théorie des nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 403-416. doi : 10.5802/jtnb.769. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/

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