  Aims. We aim to extend the relationship between X-ray luminosity (Lx) and
rotation period (Prot) found for main-sequence FGK stars and test whether it
also holds for early-M dwarfs, especially in the non-saturated regime (Lx
{\alpha} P_{rot}^{-2}) which corresponds to slow rotators. Methods. We use the
luminosity coronal activity indicator (Lx) of a sample of 78 early-M dwarfs
with masses in the range from 0.3 to 0.75 Msun from the HArps-N red Dwarf
Exoplanet Survey (HADES) radial velocity (RV) programme collected from ROSAT
and XMM-Newton. The determination of the rotation periods (P_{rot}) was done by
analysing time-series of high-resolution spectroscopy of the Ca ii H & K and
H{\alpha} activity indicators. Our sample principally covers the slow rotation
regime with rotation periods from 15 to 60 days. Results. Our work extends to
the low mass regime the observed trend for more massive stars showing a
continuous shift of the Lx/Lbol vs. Prot power-law towards longer rotation
period values and includes the determination, in a more accurate way, of the
value of the rotation period at which the saturation occurs (P_{sat}) for M
dwarf stars. Conclusions. We conclude that the relations between coronal
activity and stellar rotation for FGK stars also hold for early-M dwarfs in the
non-saturated regime, indicating that the rotation period is sufficient to
determine the ratio Lx/Lbol.
