In this work we study the e ects of strong magnetic elds on hybrid stars by using a
full general-relativity approach, solving the coupled Maxwell-Einstein equation in a self-
consistent way. The magnetic eld is assumed to be axi-symmetric and poloidal. We take
into consideration the anisotropy of the energy-momentum tensor due to the magnetic
eld, magnetic eld e ects on equation of state, the interaction between matter and the
magnetic eld (magnetization), and the anomalous magnetic moment of the hadrons. The
equation of state used is an extended hadronic and quark SU(3) non-linear realization of
the sigma model that describes magnetized hybrid stars containing nucleons, hyperons
and quarks. According to our results, the e ects of the magnetization and the magnetic
eld on the EoS do not play an important role on global properties of these stars. On the
other hand, the magnetic eld causes the central density in these objects to be reduced,
inducing major changes in the populated degrees of freedom and, potentially, converting
a hybrid star into a hadronic star.