Nanoporous carbon-based supercapacitors are established energy storage devices, that complement Li-ion batteries in high-power applications. So far, the study of these systems through simulations required drastic simplification of the electrolyte and/or of the electrode. In this work, we use state-of-the-art constant potential classical molecular dynamics simulations, where both the electrode and the electrolyte are polarizable, to study complex carbide-derived carbons with 1-ethyl-3-methylimidazolium bis(trifluoromethane)sulfonyl imide ionic liquid electrolyte. We show that using this set-up allows to obtain integral capacitances (i.e. the main property to characterize supercapacitors) in quantitative agreement with experiments. The simulations, performed in a range of applied potentials of 1V to 4V, then provide microscopic details on the charging mechanism, revealing that it is not symmetrical between the positive and negative electrodes. A pure exchange of ions is observed on the positive electrode, while an ion exchange combined with cation adsorption takes place on the negative one. Our work shows that accounting for the coupling of electrode and electrolyte polarization improves the accuracy of the results, opening the way toward a more quantitative comparison with experiments.