Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
You could try: lmfdb.org/NumberField, which is a database. If you scroll to the bottom you can enter the minimal polynomial and it will take you to the entry for that splitting field. This will have the generators for the units.
I believe that zesting does not ruin unitarity--the various morphisms that get changed are in the original category, just twisted by some roots of unity factors typically. While this is not a proof, I think one could make it rigorous. The other fusion rules could potentially be obtained as a Z2-equivariantization of the near-group categories of type Z/N +(N-1) in the Evans-Gannon notation. Just a guess, but the numerology seems to work out.
When $j=1$ the product in the numerator is $1$ (the empty product). This can be used to verify the unitarity of a certain parameter-dependent solution to the Yang-Baxter equation.
I am aware of the result of Evertse and others showing that there are finitely many solutions, even in the case of number fields. I want to find all solutions if possible.
