  We address the degeneracy of the ground state multiplet on the 5$d^1$
Re$^{6+}$ ion in double perovskite Ba$_{2}$MgReO$_{6}$ using a combination of
specific heat measurements and density functional calculations. For
Ba$_{2}$MgReO$_{6}$, two different ground state multiplets have previously been
proposed - a quartet (with degeneracy $N$=4) [1] and a doublet ($N$=2) [2].
Here we employ two independent methods for the estimation of phonon
contribution in heat capacity data to obtain the magnetic entropy $S_{mag}$,
which reflects the degeneracy of the ground state multiplet $N$ through
$S_{mag}=R$ln$N$. In both cases, we obtain a better fit to $S_{mag}=R$ln2
indicating evidence of $N$=2 degeneracy in the range from 2 to 120~K. The
detailed nature of the ground state multiplet in Ba$_{2}$MgReO$_{6}$ remains an
open question.
