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

Source abstract

We determine equations of the modular curves X 1 ( N ) {X_1}(N) for N = 11 , 13 , 14 , 15 , 16 , 17 N = 11,13,14,15,16,17 and 18. Except for N = 17 N = 17 , these are the only existing elliptic or hyperelliptic X 1 ( N ) {X_1}(N) . Applying these X 1 ( N ) {X_1}(N) , we calculate tables of elliptic curves E over quadratic fields K with torsion groups of one of the following isomorphism types: $$E tor ( K ) ≅ Z / m Z , m = 11 , 13 , 14 , 15 , 16 and 18. {E_{{\operatorname {tor}}}}(K) \cong {\mathbf {Z}}/m{\mathbf {Z}},\quad m = 11,13,14,15,16\;{\text {and}}\;18.$$