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Elliptic Curves Associated with Simplest Quartic Fields

Abstract : We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information has been obtained both on the structure of the Mordell-Weil group and on integral points for an infinite family of curves of rank 2. The canonical height is the main tool we used for that study.
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https://hal-lirmm.ccsd.cnrs.fr/lirmm-00145846
Contributor : Sylvain Duquesne <>
Submitted on : Friday, May 11, 2007 - 5:01:56 PM
Last modification on : Wednesday, May 27, 2020 - 2:13:57 PM
Long-term archiving on: : Wednesday, April 7, 2010 - 3:35:21 AM

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Sylvain Duquesne. Elliptic Curves Associated with Simplest Quartic Fields. Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2007, 19 (1), pp.81-100. ⟨10.5802/jtnb.575⟩. ⟨lirmm-00145846⟩

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