For the ring spectrum \(E_*=\mathbb{F}_p\), the \(E_{\infty}\)-ring \(A=\mathbb{S}_{W(\mathbb{F}_q)}[z]\) and the \(E_1\)-\(A\)-algebra \(B=\mathcal{O}_L\), the ring of integers in finite extension of \(\mathbb{Q}_p\), the author defines the spectra \(TC^-(B/A)[z]\) and \(TP(B/A)[z]\) by using the topological Hochschild homology \(THH(B/A)\). Then the author uses a strongly convergent, multiplicative approximate homotopy fixed point spectral sequence converging to \(E_*(TC^-(B/A)[z])\) for studying complex orientations \(MU\to TC^-(B/A)[z]\) and \(MU\to TP(B/A)[z]\) and the corresponding formal group laws (Theorem 1.1 and Corollary 1.2). In particular, the author proves that the complex oriented spectrum \(\mathbb{S}/p \otimes TC^-(B/A)[z]\) carries a \(p\)-height one graded formal group law \(F\) and presents a formula for the \(p\)-series \([p]_F\) containing the Eisenstein polynomial. Similarly for \(\mathbb{S}/p \otimes TP(B/A)[z]\).