  The Crab supernova remnant has been observed regularly with the stereoscopic
system of 5 imaging air Cherenkov telescopes that was part of the High Energy
Gamma Ray Astronomy (HEGRA) experiment. In total, close to 400 hours of useful
data have been collected from 1997 until 2002. The spectrum extends up to
energies of 80 TeV and is well matched by model calculations in the framework
of inverse Compton scattering of various seed photons in the nebula including
for the first time a recently detected compact emission region at
mm-wavelengths. The observed indications for a gradual steepening of the energy
spectrum in data is expected in the inverse Compton emission model.The average
magnetic field in the emitting volume is determined to be
$(161.6\pm0.8mathrm{stat}\pm18_\mathrm{sys}) \mu$G. The presence of protons in
the nebula is not required to explain the observed flux and upper limits on the
injected power of protons are calculated being as low as 20 % of the total spin
down luminosity for bulk Lorentz factors of the wind in the range of
$10^4-10^6$.The position and size of the emission region have been studied over
a wide range of energies. The position is shifted by 13\arcsec to the west of
the pulsar with a systematic uncertainty of 25\arcsec. No significant shift in
the position with energy is observed. The size of the emission region is
constrained to be less than 2\arcmin at energies between 1 and 10 TeV. Above 30
TeV the size is constrained to be less than 3\arcmin.No indications for pulsed
emission has been found and upper limits in differential bins of energy have
been calculated reaching typically 1-3 % of the unpulsed component.
