Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

Let M be an imaginary quadratic field with ring of integers Z_{;M};. Let ξ be a root of the polynomial f(x)=x⁴-2cx³+2x²+2cx+1, c∈Z_{;M};, c≠0. We consider an infinite family of octic fields K_{;c};=M(ξ)with ring of integers Z_{;K_{;c};};. Since the integral basis of K_{;c}; is not known in a parametric form, our goal is to determine all generators of the O=Z_{;M};[ξ] over Z_{;M}; (instead of Z_{;K_{;c};}; over Z_{;M};). We show that our problem reduces to solving the system of relative Pellian equations cV²-(c+2)U²=-2μ, cZ²-(c-2)U²=2μ where μ is an unit in M. We solve the system completely and find that all non-equivalent generators of the power integral basis of O over Z_{;M}; are given by α=ξ, 2ξ-2cξ²+ξ³ for |c|≥159108.