  Based on the magnetar model, we have studied in detail the processes of
neutrino cooling of an electron--positron plasma generating an SGR giant flare
and the influence of the magnetar magnetic field on these processes.
Electron--positron pair annihilation and synchrotron neutrino emission are
shown to make a dominant contribution to the neutrino emissivity of such a
plasma. We have calculated the neutrino energy losses from a plasma-filled
region at the long tail stage of the SGR 0526--66, SGR 1806--20, and SGR
1900+14 giant flares. This plasma can emit the energy observed in an SGR giant
flare only in the presence of a strong magnetic field suppressing its neutrino
energy losses. We have obtained a lower bound on the magnetic field strength
and showed this value to be higher than the upper limit following from an
estimate of the magnetic dipole losses for the magnetars being analyzed in a
wide range of magnetar model parameters. Thus, it is problematic to explain the
observed energy release at the long tail stage of an SGR giant flare in terms
of the magnetar model.
