We experimentally study the magnetic properties of spin-1 Bose-Einstein condensate of Sodium at equilibrium. In this system the atoms can occupy any of the three Zeeman states. We measure the spin state of the system as a function of the applied magnetic field and of the magnetization of the atomic sample. We find that our measurements reproduce well the mean-field prediction, and we identify two magnetic phases expressing the competition between the antiferromagnetic inter-particle interactions and the effect of the magnetic field. In a second part we focus on the properties of condensates of very low magnetization under a weak magnetic field. In these conditions, the symmetry of the system manifests itself in huge spin fluctuations. This phenomenon is not explainable by a naive mean-field theory and we develop a statistical approach to describe the spin state of the condensate. We measure the spin fluctuations and are able from their analysis to infer the temperature characterizing the spin degree of freedom of the condensate. We find that this temperature differs from the temperature of the thermal fraction surrounding the condensate. We interpret this difference as a consequence of the weak coupling between these two systems.