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Nonexistence of elliptic curves having everywhere good reduction and cubic discriminant. (English) Zbl 0991.11029
The author proves that if $$p$$ is a prime number such that $$p\equiv 3(4)$$ and $$p\not=3,11$$, then there is no elliptic curve defined over $$K={\mathbb Q} (\sqrt{3p})$$ with everywhere good reduction over $$K$$ and whose discriminant is a cube in $$K$$.

##### MSC:
 11G05 Elliptic curves over global fields
##### Keywords:
elliptic curves; good reduction
Full Text:
##### References:
 [1] Fröhlich, A., and Taylor, M. J.: Algebraic number theory. Cambridge Stud. Adv. Math., 27 , Cambridge Univ. Press, Cambridge (1991). [2] Kagawa, T.: Determination of elliptic curves with everywhere good reduction over real quadratic fields $$\textbf{Q}(\sqrt{3p})$$. Acta\hphantom. Arith. (to appear). · Zbl 0977.11024 [3] Lang, S.: Algebraic Number Theory. 2nd ed., Grad. Texts in Math., 110 , Springer, Berlin-Heidelberg-New York (1994). · Zbl 0811.11001
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