Relative Brauer groups of torsors of period two

We consider the problem of computing the relative Brauer group of a torsor of period two under an elliptic curve. We show how this problem can be reduced to finding a set of generators for the group of rational points on the elliptic curve. This extends work of Haile and Han to the case of torsors with unequal period and index. Our results also apply to torsors under higher dimensional abelian varieties. Several numerical examples are given.