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Generators and integral points on elliptic curves associated with simplest quartic fields

In: Mathematica Slovaca, vol. 70, no. 2
Sylvain Duquesne - Tadahisa Nara - Arman Shamsi Zargar

Details:

Year, pages: 2020, 273 - 288
Keywords:
elliptic curve, simplest quartic field, Mordell-Weil group, rank, infinite family, integral points
About article:
We associate to some simplest quartic fields a family of elliptic curves that has rank at least three over $\mathbb Q(m)$. It is given by the equation \[ Em:y2=x3-36(36m4+48m2+25)(36m4-48m2+25)x. \] Employing canonical heights we show the rank is in fact at least three for all $m$. Moreover, we get a parametrized infinite family of rank at least four. Further, the integral points on the curve $Em$ are discussed and we determine all the integral points on the original quartic model when the rank is three. Previous work in this setting studied the elliptic curves associated with simplest quartic fields of ranks at most two along with their integral points (see [Duq1,Duq2]).
How to cite:
ISO 690:
Duquesne, S., Nara, T., Zargar, A. 2020. Generators and integral points on elliptic curves associated with simplest quartic fields. In Mathematica Slovaca, vol. 70, no.2, pp. 273-288. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0350

APA:
Duquesne, S., Nara, T., Zargar, A. (2020). Generators and integral points on elliptic curves associated with simplest quartic fields. Mathematica Slovaca, 70(2), 273-288. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0350
About edition:
Publisher: Mathematical Institute, Slovak Academy of Sciences, Bratislava
Published: 10. 3. 2020