2-torsion field (reviewed)

The 2-torsion field of an abelian variety is the minimal field extension over which the 2-torsion subgroup of its Mordell-Weil group is rational; it is a Galois extension of the field over which the abelian variety is defined.

The 2-torsion field of the Jacobian of a hyperelliptic curve of the form $y^2=f(x)$ is the same as the splitting field of the polynomial $f(x)$.

For abelian varieties over number fields, the 2-torsion field is specified by giving a number field of minimal degree and absolute discriminant whose Galois closure is equal to the 2-torsion field.