  Motivated by the recently discovered high-$T_c$ superconductor
La$_3$Ni$_2$O$_7$, we comprehensively study this system using density
functional theory and random phase approximation calculations. At low
pressures, the Amam phase is stable, containing the Y$^{2-}$ mode distortion
from the Fmmm phase, while the Fmmm phase is unstable. Because of small
differences in enthalpy and a considerable Y$^{2-}$ mode amplitude, the two
phases may coexist in the range between 10.6 and 14 GPa, beyond which the Fmmm
phase dominates. In addition, the magnetic stripe-type spin order with
wavevector ($\pi$, 0) was stable at the intermediate region. Pairing is induced
in the $s_{\pm}$-wave channel due to partial nesting between the {\bf M}=$(\pi,
\pi)$ centered pockets and portions of the Fermi surface centered at the {\bf
X}=$(\pi, 0)$ and {\bf Y}=$(0, \pi)$ points. This resembles results for
iron-based superconductors but has a fundamental difference with iron pnictides
and selenides. Moreover, our present efforts also suggest that
La$_3$Ni$_2$O$_7$ is qualitatively different from infinite-layer nickelates and
cuprate superconductors.
