Invertible objects in Franke's comodule categories

We study the Picard group of Franke's category of quasi-periodic E₀E-comodules for E a 2-periodic Landweber exact cohomology theory of height n such as Morava E-theory, showing that for 2p-2 > n²+n, this group is infinite cyclic, generated by the suspension of the unit. This is analogous to, but independent of, the corresponding calculations by Hovey and Sadofsky in the E-local stable homotopy category. We also give a computation of the Picard group of Iₙ-complete quasi-periodic E₀E-comodules when E is Morava E-theory, as studied by Barthel-Schlank-Stapleton for 2p-2 ≥ n² and p-1 n, and compare this to the Picard group of the $K(n)$-local stable homotopy category, showing that they agree up to extension.