Generators for the elliptic curve y 2 =x 3 -nx
Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 403-416.

Soit E la courbe elliptique définie par y 2 =x 3 -nxn est un entier strictement positif. En 2007, Duquesne a démontré que, pour k entier, si n=(2k 2 -2k+1)(18k 2 +30k+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)[k,l].

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 [k,l].

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DOI : https://doi.org/10.5802/jtnb.769
@article{JTNB_2011__23_2_403_0,
     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},
     doi = {10.5802/jtnb.769},
     mrnumber = {2817937},
     zbl = {1228.11081},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/}
}
Yasutsugu Fujita; Nobuhiro Terai. Generators for the elliptic curve $y^2=x^3-nx$. Journal de Théorie des Nombres de Bordeaux, Tome 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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