The slice spectral sequence of a $C_4$-equivariant height-4 Lubin-Tate theory

We completely compute the slice spectral sequence of the $C_4$-spectrum $BP^{((C_4))}langle 2 rangle$. After periodization and $K(4)$-localization, this spectrum is equivalent to a height-4 Lubin-Tate theory $E_4$ with $C_4$-action induced from the Goerss-Hopkins-Miller theorem. In particular, our computation shows that $E_4^{hC_{12}}$ is 384-periodic.