Critical points of Eisenstein series

For any even integer $k\ge 4$, let ${\text{E}}_k$ be the normalized Eisenstein series of weight $k$ for ${\text{SL}}_2(\mathbb{Z})$. Also let $\text{D}$ be the closure of the standard fundamental domain of the Poincar'e upper half plane modulo ${\text{SL}}_2(\mathbb{Z})$. F.~K.~C.~Rankin and H. P. F. Swinnerton-Dyer showed that all zeros of ${\text{E}}_k$ in $\text{D}$ are of modulus one. In this article, we study the critical points of ${\text{E}}_k$, that is to say the zeros of the derivative of ${\text{E}}_k$. We show that they are simple. We count those belonging to $\text{D}$, prove that they are located on the two vertical edges of $\text{D}$ and produce explicit intervals that separate them. We then count those belonging to $\gamma \text{D}$, for any $\gamma \in \SL_2(\Z)$.