  Let $\ell$ be an odd prime, and let $K$ be a field of characteristic not
$2,3,$ or $\ell$ containing a primitive $\ell$-th root of unity. For an
elliptic curve $E$ over $K$, we consider the standard Galois representation
$$\rho_{E,\ell}: \text{Gal}(\overline{K}/K) \rightarrow
\text{GL}_2(\mathbb{F}_{\ell}),$$ and denote the fixed field of its kernel by
$L$. Recently, the last author gave an algorithm to compute elements in the
Brauer group explicitly, deducing an upper bound of $2(\ell+1)(\ell-1)$ on the
symbol length in $\mathbin{_{\ell}\text{Br}(E)} /
\mathbin{_{\ell}\text{Br}(K)}$. More precisely, the symbol length is bounded
above by $2[L:K]$. We improve this bound to $[L:K]-1$ if $\ell \nmid [L:K]$.
Under the additional assumption that $\text{Gal}(L/K)$ contains an element of
order $d > 1$, we further reduce it to $(1-\frac{1}{d})[L:K]$. In particular,
these bounds hold for all CM elliptic curves, in which case we deduce a general
upper bound of $\ell + 1$. We provide an algorithm implemented in SageMath to
compute these symbols explicitly over number fields.
