Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields

Reichert, Markus A.

doi:10.1090/S0025-5718-1986-0829635-X

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ISSN 1088-6842(online) ISSN 0025-5718(print)

Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields

Author: Markus A. Reichert
Journal: Math. Comp. 46 (1986), 637-658
MSC: Primary 11G05; Secondary 11Y16, 14G25, 14K07
MathSciNet review: 829635
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Abstract: We determine equations of the modular curves ${X_1}(N)$ for and 18. Except for , these are the only existing elliptic or hyperelliptic ${X_1}(N)$ . Applying these ${X_1}(N)$ , we calculate tables of elliptic curves E over quadratic fields K with torsion groups of one of the following isomorphism types:

$\displaystyle {E_{{\operatorname{tor}}}}(K) \cong {\mathbf{Z}}/m{\mathbf{Z}},\quad m = 11,13,14,15,16\;{\text{and}}\;18.$

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