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