How do I find abelian cubic extension over $\mathbb{Q}$ with class number more than 1?

Question

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.

Answer 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 of a family of cyclic cubic fields.

Ennola and Turunen, On cyclic cubic fields, 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.

Answer 2

Because genus theory also applies to cyclic cubics, as soon as the conductor is divisible by at least two primes congruent to 1 modulo 3 (counting 9 as such a prime!), the class number will be divisible by 3, the smallest examples being the two cyclic cubic fields with conductor $63$ (discriminant $63^2$). If you want a genuine (i.e., outside genus theory) nontrivial class group, the smallest example is the cubic field with conductor $163$ (I believe only a coincidence with the well known $-163$ for imaginary quadratics), which has class group $C_2\times C_2$. If you want also odd class number, the smallest is conductor $313$ with class group $C_7$. Note that the class group is a $\mathbb Z[\zeta_3]$-module, so primes $p\equiv2\pmod3$ always appear squared, as in the example above.

Answer 3

The earliest result I know which leads to many examples is that of Uchida:

Uchida, K.: Class numbers of cubic cyclic fields. J. Math. Soc. Japan 26(3): 447-453 (July, 1974)

See also Washington, L.: Class Numbers of the Simplest Cubic Fields. Mathematics of Computation Vol. 48, No. 177 (Jan., 1987), pp. 371-384 (14 pages)