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We give a calculation of Picard groups    Pic  2    of    K  (  2  )   ‐local invertible spectra and    Pic  (   L  2   )    of    E  (  2  )   ‐local invertible spectra, both at the prime   3  . The main contribution of this paper is to calculate the subgroup    κ  2    of invertible spectra   X   with(   E  2   )   *   X  ≅    (   E  2   )   *    S  0as twisted modules over the Morava stabilizer group    G  2   .

On Hopkins’ Picard groups for the prime 3 and chromatic level 2