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Determination of elliptic curves with everywhere good reduction over real quadratic fields \({\mathbb Q}(\sqrt{3p})\). (English) Zbl 0977.11024
Continuing research of his own [Arch. Math. 73, 25-32 (1998; Zbl 0955.11015)] and of M. Kida [Math. Comput. 68, 1679-1685 (1999; Zbl 0930.11037)] it is proved that there exist up to isomorphism over \(k= \mathbb{Q}(\sqrt{33})\) six elliptic curves with everywhere good reduction over \(k\). In particular, there is exactly one \(k\)-isogeny class of such curves.
Further, there are no elliptic curves with everywhere good reduction over \(\mathbb{Q}(\sqrt{m})\) if \(m=57, 69\) and \(93\). Hence the only real quadratic fields \(k= \mathbb{Q}(\sqrt{m})\) such that \(0 < m <100\), \(m\equiv 0 \pmod {3}\), and the class number of \(k\) is prime to \(6\) over which there are elliptic curves with everywhere good reduction, are \(m=6, 33\).

MSC:
11G05 Elliptic curves over global fields
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