Prime torsion in the Brauer group of an elliptic curve

The Brauer group is an invariant in algebraic geometry and number theory, that can be associated to a field, variety, or scheme. Let k be a field of characteristic different from 2 or 3, and let E be an elliptic curve over k. The Brauer group of E is a torsion abelian group with elements given by Morita equivalence classes of central simple algebras over the function field k(E). The Merkurjev-Suslin theorem implies that any such element can be described by a tensor product of symbol algebras. We give a description of elements in the d-torsion of the Brauer group of E in terms of these tensor products, provided that the d-torsion of E is k-rational and k contains a primitive d-th root of unity. Furthermore, if d = q is a prime, we give an algorithm to compute the q-torsion of the Brauer group over any field k of characteristic different from 2,3, and q containing a primitive q-th root of unity.