The Brown-Comenetz dual of the K(2)-local sphere at the prime 3

We calculate the homotopy type of the Brown-Comenetz dual $I_2$ of the K(2)-local sphere at the prime 3 and show that there is a twisting by a non-trivial element $P$ in the exotic part of the Picard group. We give a complete characterization of $P$ as well. The main technique is to give a sequence of calculations of the homotopy groups of elements of the Picard group after smashing with the Smith-Toda complex V(1).