  We give an algorithm to explicitly determine all elements of the $q$-torsion
(for $q$ an odd prime) of the Brauer group of an elliptic curve over any base
field of characteristic different from $q$, containing a primitive $q$-th root
of unity. These elements of the Brauer group are given as tensor products of
symbol algebras over the function field of the elliptic curve. We give
sufficient conditions to determine if the Brauer classes that arise are
trivial. Using our algorithm, we derive an upper bound on the symbol length of
the prime torsion of $\mathrm{Br}(E)/\mathrm{Br}(k)$.
