  Transcendental Brauer elements are notoriously difficult to compute. Work of
Wittenberg, and later, Ieronymou, gives a method for computing 2-torsion
transcendental classes on surfaces that have a genus 1 fibration with rational
2-torsion in the Jacobian fibration. We use ideas from a descent paper of
Poonen and Schaefer to remove this assumption on the rational 2-torsion.
