## 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

### MSC:

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

### Keywords:

elliptic curves; torsion; j-invariant
Full Text: