We perform a linear analysis of the stability of a magnetized relativistic non-rotating cylindrical flow in the approximation of zero thermal pressure, considering only the |m| = 1 mode. We find that there are two modes of instability: Kelvin–Helmholtz and current driven. The Kelvin–Helmholtz mode is found at low magnetizations and its growth rate depends very weakly on the pitch parameter. The current-driven modes are found at high magnetizations and the value of the growth rate and the wavenumber of the maximum increase as we decrease the pitch parameter. In the relativistic regime the current-driven mode is split in two branches, the branch at high wavenumbers is characterized by the eigenfunction concentrated in the jet core; the branch at low wavenumbers is instead characterized by the eigenfunction that extends outside the jet velocity shear region.