Galois Theory, discriminants and torsion subgroups of elliptic curves
  We find a tight relationship between the torsion subgroup and the image of the mod 2 Galois representation associated to an elliptic curve defined over the rationals. This is shown using some characterizations for the squareness of the discriminant of the elliptic curve. 
Irene Garcia-Selfa, Enrique Gonzalez-Jimenez, Jose M. Tornero