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Abstract CompOSE (CompStar Online Supernovae Equations of State) is an online repository of equations of state (EoS) for use in nuclear physics and astrophysics, e.g., in the description of compact stars or the simulation of core-collapse supernovae and neutron-star mergers, see . The main services, offered via the website, are: a collection of data tables in a flexible and easily extendable data format for different EoS types and related physical quantities with extensive documentation and referencing; software for download to extract and to interpolate these data and to calculate additional quantities; webtools to generate EoS tables that are customized to the needs of the users and to illustrate dependencies of various EoS quantities in graphical form. This manual is an update of previous versions that are available on the CompOSE website, at , and that was originally published in the journal “Physics of Particles and Nuclei” with . It contains a detailed description of the service, containing a general introduction as well as instructions for potential contributors and for users. Short versions of the manual for EoS users and providers will also be available as separate publications. Graphical Abstract 

Aims . In this work, we study the structure of neutron stars under the effect of a poloidal magnetic field and determine the limiting largest magnetic field strength that induces a deformation such that the ratio between the polar and equatorial radii does not exceed 2%. We consider that, under these conditions, the description of magnetic neutron stars in the spherical symmetry regime is still satisfactory. Methods . We described different compositions of stars (nucleonic, hyperonic, and hybrid) using three state-of-the-art relativistic mean field models (NL3 ω ρ , MBF, and CMF, respectively) for the microscopic description of matter, all in agreement with standard experimental and observational data. The structure of stars was described by the general relativistic solution of both Einstein’s field equations assuming spherical symmetry and Einstein-Maxwell’s field equations assuming an axi-symmetric deformation. Results . We find a limiting magnetic moment on the order of 2 × 10 31 Am 2 , which corresponds to magnetic fields on the order of 10 16 G at the surface and 10 17 G at the center of the star, above which the deformation due to the magnetic field is above 2%, and therefore not negligible. We show that the intensity of the magnetic field developed in the star depends on the equation of state (EoS), and, for a given baryonic mass and fixed magnetic moment, larger fields are attained with softer EoS. We also show that the appearance of exotic degrees of freedom, such as hyperons or a quark core, is disfavored in the presence of a very strong magnetic field. As a consequence, a highly magnetized nucleonic star may suffer an internal conversion due to the decay of the magnetic field, which could be accompanied by a sudden cooling of the star or a gamma ray burst.