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