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On Elliptic Curves with Everywhere Good Reduction over Certain Number Fields

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We prove the existence and nonexistence of elliptic curves having good reduction everywhere over certain real quadratic fields Q(m) for m≤200. These results of computations give best-possible data including structures of Mordell-Weil groups over some real quadratic fields via two-descent. We also prove similar results for the case of certain cubic fields. Especially, we give the first example of elliptic curve having everywhere good reduction over a pure cubic field using our method.

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The authors declare no conflicts of interest.

Cite this paper

*American Journal of Computational Mathematics*, Vol. 2 No. 4, 2012, pp. 358-366. doi: 10.4236/ajcm.2012.24049.

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