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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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