  Cs$_3$C$_{60}$ in the A15 structure is an antiferromagnet at ambient pressure
in contrast with other superconducting trivalent fullerides. Superconductivity
is recovered under pressure and reaches the highest critical temperature of the
family. Comparing density-functional calculations with generalized gradient
approximation to the hybrid functional HSE, which includes a suitable component
of exchange, we establish that the antiferromagnetic state of Cs$_3$C$_{60}$ is
not due to a Slater mechanism, and it is stabilized by electron correlation.
HSE also reproduces the pressure-driven metalization. Our findings corroborate
previous analyses suggesting that the properties of this compound can be
understood as the result of the interplay between electron correlations and
Jahn-Teller electron-phonon interaction.
