  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).
