Symbol Length in Brauer Groups of Elliptic Curves

Let ℓ \ell be an odd prime, and let alttext="upper K"> K encoding="application/x-tex">K a field of characteristic not alttext="2"> 2 encoding="application/x-tex">2 , alttext="3"> 3 encoding="application/x-tex">3 or containing primitive -th root unity. For elliptic curve E"> E encoding="application/x-tex">E over we consider the standard Galois representation ρ / stretchy="false">) stretchy="false">→<!-- → <mml:mi>L F encoding="application/x-tex">\rho _{E,\ell }: Gal(\overline {K}/K) \rightarrow GL_2(\mathbb {F}_{\ell }), \] denote fixed its kernel by L"> encoding="application/x-tex">L . Recently, last author gave algorithm to compute elements in Brauer group explicitly, deducing bound alttext="2 plus 1 minus right-parenthesis"> + 1 −<!-- − encoding="application/x-tex">2(\ell +1)(\ell -1) on symbol length alttext="Subscript B r /> B r encoding="application/x-tex">_{\ell }Br(E) / _{\ell }Br(K) More precisely, is bounded above left-bracket right-bracket"> stretchy="false">[ stretchy="false">] encoding="application/x-tex">2[L:K] We improve this alttext="left-bracket right-bracket 1"> encoding="application/x-tex">[L:K]-1 if does-not-divide ∤<!-- ∤ encoding="application/x-tex">\ell \nmid [L:K] Under additional assumption that encoding="application/x-tex">Gal(L/K) contains element order alttext="d greater-than d &gt; encoding="application/x-tex">d &gt; 1 further reduce it alttext="left-parenthesis StartFraction Over d EndFraction encoding="application/x-tex">(1-\frac {1}{d})[L:K] In particular, these bounds hold for all curves with complex multiplication, which case deduce general + provide implemented SageMath symbols explicitly number fields.