  We study the large $N$ limit of the refined topologically twisted index of
three-dimensional $\mathcal{N} = 2$ gauge theories with a holographic dual,
which is relevant for counting the number of microstates of rotating,
magnetically charged and twisted AdS$_4$ black holes. As a difference with
previous computations, we perform a direct saddle point analysis of the
relevant matrix model. Among other things we explicitly identify the
distribution of gauge magnetic fluxes that contributes at large $N$. The index
exhibits a large $N$ factorized form in agreement with the supergravity
expectations based on gravitational blocks and correctly reproduces the entropy
of magnetically charged rotating AdS$_4 \times S^7$ black holes.
