Number field 6.0.19683.1: \(\Q(\zeta_{9})\)

• Label: 6.0.19683.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-19683$
• Root discriminant: \(5.20\)
• Ramified prime: $3$
• Class number: $1$
• Class group: trivial
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{3} + 1 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-19683\) \(\medspace = -\,3^{9}\)
Root discriminant:		\(5.20\)
Galois root discriminant:		$3^{3/2}\approx 5.196152422706632$
Ramified primes:		\(3\)
Discriminant root field:		\(\Q(\sqrt{-3}) \)
$\card{ \Gal(K/\Q) }$:		$6$
This field is Galois and abelian over $\Q$.
Conductor:		\(9=3^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{9}(1,·)$, $\chi_{9}(2,·)$, $\chi_{9}(4,·)$, $\chi_{9}(5,·)$, $\chi_{9}(7,·)$, $\chi_{9}(8,·)$$\rbrace$
This is a CM field.
Reflex fields:		\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})\)$^{3}$

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$

Monogenic:		Yes
Index:		$1$
Inessential primes:		None

Class group and class number

Trivial group, which has order $1$

Relative class number: $1$

Unit group

Rank:		$2$
Torsion generator:		\( a \)  (order $18$)
Fundamental units:		$a-1$, $a^{5}+a$
Regulator:		\( 3.39714980258 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 3.39714980258 \cdot 1}{18\cdot\sqrt{19683}}\cr\approx \mathstrut & 0.333684590840 \end{aligned}\]

Galois group

$C_6$ (as 6T1):

A cyclic group of order 6

The 6 conjugacy class representatives for $C_6$

Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra:	\(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\)
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	${\href{/padicField/2.6.0.1}{6} }$	R	${\href{/padicField/5.6.0.1}{6} }$	${\href{/padicField/7.3.0.1}{3} }^{2}$	${\href{/padicField/11.6.0.1}{6} }$	${\href{/padicField/13.3.0.1}{3} }^{2}$	${\href{/padicField/17.2.0.1}{2} }^{3}$	${\href{/padicField/19.1.0.1}{1} }^{6}$	${\href{/padicField/23.6.0.1}{6} }$	${\href{/padicField/29.6.0.1}{6} }$	${\href{/padicField/31.3.0.1}{3} }^{2}$	${\href{/padicField/37.1.0.1}{1} }^{6}$	${\href{/padicField/41.6.0.1}{6} }$	${\href{/padicField/43.3.0.1}{3} }^{2}$	${\href{/padicField/47.6.0.1}{6} }$	${\href{/padicField/53.2.0.1}{2} }^{3}$	${\href{/padicField/59.6.0.1}{6} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
\(3\)	3.6.9.10	$x^{6} + 6 x^{4} + 3$	$6$	$1$	$9$	$C_6$	$[2]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
*	1.3.2t1.a.a	$1$	$ 3 $	\(\Q(\sqrt{-3}) \)	$C_2$ (as 2T1)	$1$	$-1$
*	1.9.3t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*	1.9.6t1.a.a	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$
*	1.9.3t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})^+\)	$C_3$ (as 3T1)	$0$	$1$
*	1.9.6t1.a.b	$1$	$ 3^{2}$	\(\Q(\zeta_{9})\)	$C_6$ (as 6T1)	$0$	$-1$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.