A study of fully developed plane turbulent channel flow subject to spanwise system rotation through direct numerical simulations is presented. In order to study both the influence of the Reynolds number and spanwise rotation on channel flow, the Reynolds number $Re = U_b h/\nu$ is varied between 5000 and $31\,600$ and the rotation number $Ro = 2 \Omega h/U_b$ is varied between 0 and 2.7, where ... [Show full abstract] $U_b$ is the mean bulk velocity, $h$ the channel half gap and $\Omega$ the system rotation rate. The mean streamwise velocity profile displays also at higher $Re$ the characteristic linear part with a slope near to $2 \Omega$ and a corresponding linear part in the profiles of the production and dissipation rate of turbulent kinetic energy appears. With increasing $Ro$ a distinct unstable side with large spanwise and wall-normal Reynolds stresses and a stable side with much weaker turbulence develops in the channel. The flow starts to relaminarize on the stable side of the channel and persisting turbulent-laminar patterns appear at higher $Re$. If $Ro$ is further increased the flow on the stable becomes laminar-like while at yet higher $Ro$ the whole flow relaminarizes, although the calm periods might be disrupted by repeating bursts of turbulence, as explained by Brethouwer (2016). The influence of the Reynolds number is notable, in particular on the stable side of the channel where velocity fluctuations are stronger at higher $Re$. Visualizations and spectra show that at lower $Ro$ large counter rotating streamwise roll cells develop on the unstable side. These become less noticeable and disappear when raising $Ro$, especially at higher $Re$. Large-scale structures are present on the unstable side at lower $Ro$ according to spectra, but these structures vanish if $Ro$ is raised.