Elliptic curves over quadratic fields with fixed torsion subgroup and positive rank

Abstract

For each of the torsion groups Z/2Z⊕Z/10Z,Z/2Z⊕Z/12Z,Z/15Z we find the quadratic field with the smallest absolute value of its discriminant such that there exists an elliptic curve with that torsion and positive rank. For the torsion groups Z/11Z, Z/14Z we solve the analogous problem after assuming the Parity conjecture.