  Ultrathin L1$_0$ films with different $c$-axis orientations relative to the
film plane are promising candidates for data storage materials. In this work,
within the framework of density functional theory, we calculated the magnetic
properties of ultrathin L1$_0$ (111) and (010) films with thicknesses ranging
from 4 to 16 atomic monolayers (from about 0.8 to 3.5~nm). The highest average
magnetic moments are observed for the thinnest films considered, and with
increasing film thickness, the values converge towards the magnetic moment for
bulk. The observed increase comes mainly from enhanced moments in the two
atomic monolayers closest to the surface of the films. The easy axis of
magnetization of (111) films prefers an alignment close to the tetragonal axis,
an example of tilted magnetic anisotropy. The 6-monolayer (111) film (about
1.3~nm thick) inclines the easy axis of magnetization of about 45{\deg} to the
film plane, which can find use in applications. The (010) films show an
in-plane easy magnetization axis in a unique L1$_0$ tetragonal direction. This
is an unusual type of in-plane anisotropy, as the particular direction
preference is very strong. The computational results encourage further
experimental studies of L1$_0$ systems with tilted and in-plane fixed magnetic
anisotropy.
