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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, ourcomputation shows that $E_4^{hC_{12}}$ is 384-periodic.

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