  Recently superconductivity has been discovered at around 200~K in a hydrogen
sulfide system and around 260~K in a lanthanum hydride system, both under
pressures of about 200 GPa. These record-breaking transition temperatures bring
within reach the long-term goal of obtaining room temperature
superconductivity. We have used first-principle calculations based on density
functional theory (DFT) along with Migdal-Eliashberg theory to investigate the
electron-phonon mechanism for superconductivity in the $Fm\bar{3}m$ phase
proposed for the LaH$_{10}$ superconductor. We show that the very high
transition temperature $T_c$ results from a highly optimized electron-phonon
interaction that favors coupling to high frequency hydrogen phonons. Various
superconducting properties are calculated, such as the energy gap, the isotope
effect, the specific heat jump at $T_c$, the thermodynamic critical field and
the temperature-dependent penetration depth. However, our main emphasis is on
the finite frequency optical properties, measurement of which may allow for an
independent determination of $T_c$ and also a confirmation of the mechanism for
superconductivity.
