What I have done so far is first note this is the ring of integers over the seventh cyclotomic field. This has degree 6 and only complex embeddings. Then I computed minkowskis bound to be around 4.4.
Now we only have to examine (2) and (3) . I want to use dedekind here and I factored $X^6+...+1$ into two irreducible cubic over $\mathbf{F}_2 $ but I am unsure on showing these are principal and I am not sure how to factor over $\mathbf{F}_3 $.
Could someone give me a hint on how to proceed? Thanks