Prime Torsion in the Brauer Group of an Elliptic Curve
We determine generators and relations in the $d$-torsion of the Brauer group of an elliptic curve, provided that the $d$-torsion of the elliptic curve itself is rational over the base field, for any integer $d \geq 2$. For $q$ any odd prime, we further give an algorithm to explicitly calculate generators and relations of the $q$-torsion of the Brauer group of an elliptic curve over any base field of characteristic different from two, three, and $q$, containing a primitive $q$-th root of unity. These generators are symbol algebras and the relations arise as their tensor products. We deduce an upper bound on the symbol length of the prime torsion of $\text{Br}(E)/\text{Br}(k)$.