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This paper introduces a computational approach to classifying low rankmodular categories up to their modular data. The modular data of a modularcategory is a pair of matrices, $(S,T)$. Virtually all the numericalinformation of the category is contained within or derived from the modulardata. The modular data satisfy a variety of criteria that Bruillard, Ng,Rowell, and Wang call the admissibility criteria. Of note is the Galois groupof the $S$ matrix is an abelian group that acts faithfully on the columns ofthe eigenvalue matrix, $s = (\frac{S_{ij}}{S_{0j}})$. This gives an injectionfrom Gal$(\mathbb{Q}(S),\mathbb{Q}) \to $ Sym$_r$, where $r$ is the rank of thecategory. Our approach begins by listing all the possible abelian subgroups ofSym$_6$ and building all the possible modular data for each group. We run eachset of modular data through a series of Gr\"obner basis calculations until weeither find a contradiction or solve for the modular data.

A Computational Approach to Classifying Low Rank Modular Categories