Symbol Length in Brauer Groups of Elliptic Curves

Let $\ell$ be an odd prime integer, and let $K$ be a field of characteristic not $2,3$ and coprime to $\ell$ containing a primitive $\ell$-th root of unity. For an elliptic curve $E$ over $K$, we consider the standard Galois representation \[\rho_{E,\ell}: \text{Gal}(\overline{K}/K) \rightarrow \mathrm{GL}_2(\mathbb{F}_{\ell}),\] and denote the fixed field of its kernel by $L$. The Brauer group $\text{Br}(E)$ of an elliptic curve $E$ is an important invariant. A theorem by Merkurjev and Suslin implies that every element of the $\ell$-torsion $\mathbin{_{\ell}\text{Br}(E)}$ can be written as a product of symbol algebras. We use an explicit computation of the Brauer group to show that if $\ell \nmid [L:K]$ and $[L:K] > 2$, then the symbol length of $\mathbin{_{\ell}\text{Br}(E)} / \mathbin{_{\ell}\text{Br}(K)}$ is bounded above by $[L:K]-1$. Under the additional assumption that $\text{Gal}(L/K)$ contains an element of order $d > 1$, the symbol length of $\mathbin{_{\ell}\text{Br}(E)} / \mathbin{_{\ell}\text{Br}(K)}$ is bounded above by $(1-\frac{1}{d})[L:K]$. In particular, these bounds hold for all CM elliptic curves, in which case we have an upper bound of $\ell + 1$ on the symbol length. We provide an algorithm implemented in SageMath to compute these symbols explicitly over number fields.