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Elliptic curves with good reduction away from 3. (English) Zbl 0632.14030

The author gives a complete list of the (eighteen) isomorphism classes of elliptic curves defined over the field \({\mathbb{Q}}(\sqrt{-3})\) with good reduction outside (the prime dividing) 3. The method relies, via the criterion of Néron-Ogg-Shafarevich, on the study of the quartic equations giving the points of order 3 on an elliptic curve in Weierstraß form, on Minkowski bounds for discriminants - and ultimately on a computer search. Among these classes, four come from curves defined over \({\mathbb{Q}}\), and none has actually good reduction everywhere [for a generalization of the latter fact to non-principal imaginary quadratic fields, see H. Ishii, Jap. J. Math., New Ser. 12, 45-52 (1986; Zbl 0613.14021)].
Reviewer: D.Bertrand

MSC:

14H45 Special algebraic curves and curves of low genus
14H52 Elliptic curves
14-04 Software, source code, etc. for problems pertaining to algebraic geometry
11R16 Cubic and quartic extensions
11R23 Iwasawa theory

Citations:

Zbl 0613.14021
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References:

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