Global number field 6.6.332334333.1

• Label: 6.6.332334333.1
• Degree: $6$
• Signature: $[6, 0]$
• Discriminant: $3^{8}\cdot 37^{3}$
• Root discriminant: $26.32$
• Ramified primes: $3, 37$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 30 x^{4} + 63 x^{3} + 219 x^{2} - 300 x - 271 \)

Invariants

Degree:		$6$
Signature:		$[6, 0]$
Discriminant:		\(332334333=3^{8}\cdot 37^{3}\)
Root discriminant:		$26.32$
Ramified primes:		$3, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(333=3^{2}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{333}(112,·)$, $\chi_{333}(1,·)$, $\chi_{333}(295,·)$, $\chi_{333}(184,·)$, $\chi_{333}(73,·)$, $\chi_{333}(223,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{28927} a^{5} - \frac{8085}{28927} a^{4} - \frac{3153}{28927} a^{3} - \frac{2078}{28927} a^{2} - \frac{11972}{28927} a - \frac{3411}{28927}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$5$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
Regulator:		\( 234.358129213 \)

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

\(\Q(\sqrt{37}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{37}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.6.0.1}{6} }$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.6.0.1}{6} }$	R	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
$37$	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_37.6t1.2c1	$1$	$ 3^{2} \cdot 37 $	$x^{6} - 3 x^{5} - 30 x^{4} + 63 x^{3} + 219 x^{2} - 300 x - 271$	$C_6$ (as 6T1)	$0$	$1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_37.6t1.2c2	$1$	$ 3^{2} \cdot 37 $	$x^{6} - 3 x^{5} - 30 x^{4} + 63 x^{3} + 219 x^{2} - 300 x - 271$	$C_6$ (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 *.