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<=24h periods. Both the combined ground-based data and the Gaia measurements unambiguously point to a light curve period equal to the orbital period of Dysnomia, P=15.8d, with a light curve amplitude of dm~0.03mag, i.e. the rotation of Eris is tidally locked. Assuming that Dysnomia has a collisional origin, calculations with a simple tidal evolution model show that Dysnomia has to be relatively massive (mass ratio of q=0.01-0.03) and large (radius of R_s_>=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.