Fusion Categories Associated to Subfactors with Index $3+\sqrt{5}
We classify fusion categories which are Morita equivalent to even parts of subfactors with index $3+\sqrt{5} and module categories over these fusion categories. For the fusion category \mathcal{C} which is the even part of the self dual $3^\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z} subfactor, we show that there are $30$ simple module categories over \mathcal{C} there are no other fusion categories in the Morita equivalence class; and the order of the Brauer Picard group is $360$ The proof proceeds indirectly by first describing the Brauer Picard groupoid of a \mathbb{Z}/3\mathbb{Z} equivariantization \mathcal{C}\mathbb{Z}/3\mathbb{Z} (which is the even part of the $4442$ subfactor) We show that that there are exactly three other fusion categories in the Morita equivalence class of \mathcal{C}\mathbb{Z}/3\mathbb{Z} which are all \mathbb{Z}/3\mathbb{Z} graded extensions of \mathcal{C} Each of these fusion categories admits $20$ simple module categories, and their Brauer Picard group is \mathcal{S}_3 We also show that there are exactly five fusion categories in the Morita equivalence class of the even parts of the $3^\mathbb{Z}/4\mathbb{Z} subfactor; each admits $7$ simple module categories; and the Brauer Picard group is \mathbb{Z}/2\mathbb{Z}