The rotational states of the members in the dwarf planet - satellite systems in the transneptunian region are determined by the formation conditions and the tidal interaction between the components, and these rotational characteristics are the prime tracers of their evolution. Previously a number of authors claimed highly diverse values for the rotation period for the dwarf planet Eris, ranging from a few hours to a rotation (nearly) synchronous with the orbital period (15.8d) of its satellite, Dysnomia. In this letter we present new light curve data of Eris, taken with ~1-2m-class ground based telescopes, and with the TESS and Gaia space telescopes. TESS data could not provide a well-defined light curve period, but could constrain light curve variations to a maximum possible light curve amplitude of dm<=0.03mag (1-sigma) for P=300km) to slow down Eris to synchronized rotation. These simulations also indicate that - assuming tidal parameters usually considered for transneptunian objects - the density of Dysnomia should be 1.8-2.4g/cm^3^, an exceptionally high value among similarly sized transneptunian objects, putting important constraints on the formation conditions.