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.

https://beta.lmfdb.org/NumberField/?hst=List&class_number=2-100&galois_group=C3&search_type=List — David Loeffler, Jul 22 '21 at 12:35

score 9 · Answer 1 · answered Jul 22 '21 at 21:20

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.

May be of interest: https://mathoverflow.net/questions/38341/class-numbers-and-163 — Tito Piezas III, Nov 15 '22 at 10:07

Gerry Myerson · Accepted Answer · 2021-07-22T12:51:15.947

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.

In the lmfdb you can also search for nontrivial class group: http://www.lmfdb.org/NumberField/?count=20&hst=List°ree=3&class_number=%7E1&galois_group=C3&search_type=List — Aurel, Jul 23 '21 at 10:30

score 1 · Answer 3 · answered Jul 22 '21 at 12:42

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)

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

3 Answers3