Explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields. (English) Zbl 0605.14028

The author lists elliptic curves \(E\) defined over a quadratic field \(K\) (but not defined over \(\mathbb Q\) such that the torsion part of the group of \(K\)-rational points is cyclic of order \(N\), where \(N=11, 13, 14, 15, 16\) or 18. For each of these curves the \(j\)-invariant and its prime decomposition is also computed. The computations use explicit equations, which the author finds, for the modular curves \(X_ 1(N)\) for the above values of \(N\).
Reviewer: B. Singh


11G05 Elliptic curves over global fields
11Y16 Number-theoretic algorithms; complexity
14G25 Global ground fields in algebraic geometry
14H52 Elliptic curves
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