  We calculate exactly the first cumulants of the integrated current and of the
activity (which is the total number of changes of configurations) of the
symmetric simple exclusion process (SSEP) on a ring with periodic boundary
conditions. Our results indicate that for large system sizes the large
deviation functions of the current and of the activity take a universal scaling
form, with the same scaling function for both quantities. This scaling function
can be understood either by an analysis of Bethe ansatz equations or in terms
of a theory based on fluctuating hydrodynamics or on the macroscopic
fluctuation theory of Bertini, De Sole, Gabrielli, Jona-Lasinio and Landim.
