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The extraordinary energetic activity of magnetars is usually explained interms of dissipation of a huge internal magnetic field of the order of$10^{15-16}$G. How such a strong magnetic field can originate during theformation of a neutron star is still subject of active research. An importantrole can be played by fast rotation: if magnetars are born as millisecondrotators dynamo mechanisms may efficiently amplify the magnetic field inheritedfrom the progenitor star during the collapse. In this case, the combination ofrapid rotation and strong magnetic field determine the right physical conditionnot only for the development of a powerful jet driven explosion, manifesting asa gamma ray burst, but also for a copious gravitational waves emission. Strongmagnetic fields are indeed able to induce substantial quadrupolar deformationsin the star. In this paper we analyze the joint effect of rotation andmagnetization on the structure of a polytropic and axisymmetric neutron star,within the ideal magneto-hydrodynamic regime. We will consider either purelytoroidal or purely poloidal magnetic field geometries. Through the sampling ofa large parameter space, we generalize previous results in literature,inferring new quantitative relations that allow for a parametrization of theinduced deformation, that takes into account also the effects due to thestellar compactness and the current distribution. Finally, in the case ofpurely poloidal field, we also discuss how different prescriptions on thesurface charge distribution (a gauge freedom) modify the properties of thesurrounding electrosphere and its physical implications.

General relativistic models for rotating magnetized neutron stars in conformally flat spacetime