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

• [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
• Volume: 23, Issue: 2, page 403-416
• ISSN: 1246-7405

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## Abstract

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Let $E$ be an elliptic curve given by ${y}^{2}={x}^{3}-nx$ with a positive integer $n$. Duquesne in 2007 showed that if $n=\left(2{k}^{2}-2k+1\right)\left(18{k}^{2}+30k+17\right)$ 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 $n=n\left(k,l\right)$ in $ℤ\left[k,l\right]$.

## How to cite

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Fujita, Yasutsugu, and Terai, Nobuhiro. "Generators for the elliptic curve $y^2=x^3-nx$." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 403-416. <http://eudml.org/doc/219816>.

@article{Fujita2011,
abstract = {Let $E$ be an elliptic curve given by $y^2=x^3-nx$ with a positive integer $n$. Duquesne in 2007 showed that if $n=(2k^2-2k+1)(18k^2+30k+17)$ 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 $\mathbb\{Z\}[k,l]$.},
affiliation = {College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba 275–8576 Japan; Division of General Education Ashikaga Institute of Technology 268-1 Omae, Ashikaga, Tochigi 326–8558 Japan},
author = {Fujita, Yasutsugu, Terai, Nobuhiro},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {elliptic curves},
language = {eng},
month = {6},
number = {2},
pages = {403-416},
publisher = {Société Arithmétique de Bordeaux},
title = {Generators for the elliptic curve $y^2=x^3-nx$},
url = {http://eudml.org/doc/219816},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Fujita, Yasutsugu
AU - Terai, Nobuhiro
TI - Generators for the elliptic curve $y^2=x^3-nx$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 403
EP - 416
AB - Let $E$ be an elliptic curve given by $y^2=x^3-nx$ with a positive integer $n$. Duquesne in 2007 showed that if $n=(2k^2-2k+1)(18k^2+30k+17)$ 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 $\mathbb{Z}[k,l]$.
LA - eng
KW - elliptic curves
UR - http://eudml.org/doc/219816
ER -

## References

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10. M. Krir, À propos de la conjecture de Lang sur la minoration de la hauteur de Néron-Tate pour les courbes elliptiques sur $ℚ$. Acta Arith. 100 (2001), 1–16. Zbl0981.11021MR1864622
11. S. Siksek, Infinite descent on elliptic curves. Rocky Mountain J. Math. 25:4 (1995), 1501–1538. Zbl0852.11028MR1371352
12. J. H. Silverman, The arithmetic of elliptic curves. Springer-Verlag, 1986. Zbl1194.11005MR817210
13. J. H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), 339–358. Zbl0656.14016MR942161
14. J. H. Silverman, The advanced topics in the arithmetic of elliptic curves. Springer-Verlag, 1994. Zbl0911.14015MR1312368
15. J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil. In Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer-Verlag, 1975, 33–52. Zbl1214.14020MR393039

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