  We use a three-flavor Nambu--Jona-Lasinio model to study the thermodynamics
of strange quark matter under a strong magnetic field. The model Lagrangian
features flavor SU(3) four-quark interactions and six-quark interactions that
break the UA(1) symmetry. We incorporate thermomagnetic effects in the
four-quark coupling. The model predicts magnetic catalysis at low temperatures
and inverse magnetic catalysis at temperatures close to the pseudocritical
temperature of the QCD transition, in agreement with lattice QCD results. We
compute the pressure at the mean field level and obtain the magnetization of
quark matter. We adopt the recently proposed vacuum magnetic regularization
(VMR) scheme, in that divergent quark mass independent contributions are not
subtracted, thereby avoiding unphysical results for the magnetization. We
devote special attention to the renormalized magnetization, a projected
quantity that allows for direct comparisons with lattice QCD simulations. Our
results are in very good agreement with lattice data indicating a paramagnetic
behavior for quark matter.
