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We determine equations of the modular curves X1(N) for N = 11, 13, 14, 15, 16, 17 and 18. Except for N = 17, these are the only existing elliptic or hyperelliptic X1(N). Applying these X1(N), we calculate tables of elliptic curves E over quadratic fields K with torsion groups of one of the following isomorphism types: Etor(K) -Z/mZ, m = 11, 13, 14,15, 16 and 18.

Explicit Determination of Nontrivial Torsion Structures of Elliptic Curves Over Quadratic Number Fields