  The ordered L1$_0$ FeNi phase (tetrataenite) is recently considered as a
promising candidate for the rare-earth free permanent magnets applications. In
this work we calculate several characteristics of the L1$_0$ FeNi, where most
of the results come form the fully relativistic full potential FPLO method with
the generalized gradient approximation (GGA). A special attention deserves the
summary of the magnetocrystalline anisotropy energies (MAE's), the full
potential calculations of the anisotropy constant $K_3$, and the combined
analysis of the Fermi surface and three-dimensional $\mathbf{k}$-resolved MAE.
Other calculated parameters presented in this article are the magnetic moments
$m_{s}$ and $m_{l}$, magnetostrictive coefficient $\lambda_{001}$, bulk modulus
B$_0$, and lattice parameters. The MAE's summary shows rather big discrepancies
between the experimental MAE's from literature and also between the calculated
MAE's.
