On the Bousfield-Kan spectral sequence for $Q(2)_{(3)}$

We compute the $E_2$-term of the Bousfield-Kan spectral sequence converging to the homotopy groups of the semi-cosimplicial $E_{\infty}$ ring spectrum $Q(2)_{(3)}$. This 3-local spectrum was constructed by M. Behrens using degree 2 isogenies of elliptic curves, and its localization with respect to the 2nd Morava $K$-theory $K(2)$ is "one half" of the $K(2)$-local sphere at the prime 3. The computation in this paper uses techniques developed in the author's previous work on the Adams-Novikov spectral sequence for $Q(2)_{(3)}$, and provides another gateway to the homotopy ring $\pi_*Q(2)_{(3)}$.