I am recently troubled with a computational detail in Bushnell and Henniart's book, "The local Langlands conjecture for Gl(2)". Let $(\mathfrak{A},n,\alpha)$ be a simple stratum, and define $K_\mathfrak{A}$ as the group $\{ g \in G \mid g\mathfrak{A}g^{-1} = \mathfrak{A} \} $, and it can be proved that $K_\mathfrak{A}=F[\alpha]^\times U_\mathfrak{A}$, again we define the group $J_\alpha=F[\alpha]^\times U_\mathfrak{A}^{\left[\frac{n}{2}\right]+1}$

On page 173 of the book, in order to find out of the dimension of the representation $\Xi$ in the cuspidal inducing datum $(\mathfrak{A},\Xi)$, we need to compute the index $(K_\mathfrak{A}:J_{\alpha})$, but I cannot dope out a method. I would appreciate a lot if you could tell me an approach of how to compute it!