I am trying to see them as subfield $\mathbb{Q}(\zeta\_n).$ I feel it is a tiring job by using SageMath. Moreover, I am ending up with the abelian cubic field with the class number $1.$
I appreciate any alternative methods.



How do I find abelian cubic extension over $\mathbb{Q}$ with class number more than 1?
Günter Lettl, A lower bound for the class number of certain cubic fields, Math. Comp. 46, #174 (April 1986) 659-666, has abstract,


Let $K$ be a cyclic number field with generating polynomial $$x^3-{a-3\over2}x^2-{a+3\over2}x-1$$ and conductor $m$. We will derive a lower bound for the class number of these fields and list all such fields with conductor $m=(a^2+27)/4$ or $m=(1+27b^2)/4$ and small class number.


Lawrence C. Washington, Class numbers of the simplest cubic fields, Math. Comp. 48, #177 (Jan. 1987) 371-384, has abstract,


Using the "simplest cubic fields" of D. Shanks, we give a modified proof and an extension of a result of Uchida, showing how to obtain cyclic cubic fields with class number divisible by $n$, for any $n$.


See also Jaclyn Lang, [Properties](https://www.math.univ-paris13.fr/%7Elang/mathesis.pdf) of a family of cyclic cubic fields.


Ennola and Turunen, On cyclic cubic [fields](https://www.ams.org/journals/mcom/1985-45-172/S0025-5718-1985-0804947-3/S0025-5718-1985-0804947-3.pdf), tabulate class numbers and units in cyclic cubic fields with conductor less than $16000$.


<http://www.lmfdb.org/NumberField/?start=0&degree=3&galois_group=C3&count=20> lists over 4,000 cyclic cubic number fields, with class number.


