On Hopkins' Picard groups for the prime 3 and chromatic level 2
  We give a calculation of Picard groups of K(2)-local invertible spectra and of E(2)-local invertible spectra, both at the prime 3. The main contribution of this paper is to calculation the subgroup of invertible spectra with the same Morava module as a sphere. 
Paul Goerss, Hans-Werner Henn, Mark Mahowald, and Charles Rezk
math.AT