  The representative cuprate, La$_{2-x}$M$_x$CuO$_4$, with M = Sr and $x = 1/8$
is studied via first-principles calculations in the high-temperature tetragonal
(HTT), low-temperature orthorhombic (LTO), and low-temperature
less-orthorhombic (LTLO) structures. By suppressing the magnetism and
superconductivity, the LTLO phase, which has not been observed for
La$_{2-x}$Sr$_x$CuO$_4$, is found to be the ground state, where the structural
phase transitions, HTT$\rightarrow$LTO$\rightarrow$LTLO, can be understood via
phonon instability. While the La-O composition is identified to be responsible
for the phonon softening, the superconducting CuO$_2$ layer is dynamically
stable. The LTLO phase, which can exhibit a $\sim$20 meV splitting in the
density of states, is proposed to have an intimate relationship with the
observed pseudogap and the charge density wave giving the stripe. We argue that
at low temperatures, the superconducting LTO La$_{1.875}$Sr$_{0.125}$CuO$_4$
competes with the phonon-preferred LTLO phase by spontaneously forming the
Cooper pairs, resulting in suppressing the stripe. Therefore, the revealed LTLO
phase is indispensable for understanding La$_{2-x}$Sr$_x$CuO$_4$.
