The following tables contain data on the commensurability classes of all manifolds in the Callahan-Hildebrand-Weeks census of cusped hyperbolic manifolds with up to $7$ tetrahedra, and for complements of hyperbolic knots and links up to 12 crossings, supplied by Morwen Thistlethwaite.
There are 8557 hyperbolic links and 4929 manifolds in the cusped censi, giving a total of 13486 entries in the tables comm_data, comm_grouped, and comm_grouped_cd.
README (information on file formats)
comm_grouped (manifolds grouped into commensurability classes)
comm_grouped_cd (manifolds grouped into commensurability classes, including cusp densities)
comm_data (basic data on commensurators)
link_fields (information on fields arising in these tables)
The algorithms used to produce these tables are described in the paper
Commensurators of cusped hyperbolic manifolds, by Heard, Goodman and Hodgson (available here).