Global number field 6.0.337779963.1

• Label: 6.0.337779963.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{9}\cdot 131^{2}$
• Root discriminant: $26.39$
• Ramified primes: $3, 131$
• Class number: $6$
• Class group: $[6]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 6 x^{4} - 38 x^{3} + 36 x^{2} + 114 x + 361 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-337779963=-\,3^{9}\cdot 131^{2}\)
Root discriminant:		$26.39$
Ramified primes:		$3, 131$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{6}$, $\frac{1}{6} a^{4} - \frac{1}{6} a$, $\frac{1}{342} a^{5} + \frac{1}{18} a^{4} + \frac{13}{342} a^{3} + \frac{5}{18} a^{2} + \frac{131}{342} a - \frac{7}{18}$

Class group and class number

$C_{6}$, which has order $6$

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{1}{114} a^{5} + \frac{1}{19} a^{3} + \frac{1}{6} a^{2} - \frac{6}{19} a \) (order $6$)
Fundamental units:		\( \frac{1}{57} a^{5} + \frac{7}{114} a^{3} - \frac{1}{3} a^{2} - \frac{7}{19} a - \frac{7}{6} \),  \( \frac{701}{342} a^{5} - \frac{38}{9} a^{4} - \frac{5023}{342} a^{3} - \frac{23}{18} a^{2} + \frac{9863}{171} a + \frac{2893}{18} \)
Regulator:		\( 77.1236060585 \)

Galois group

$D_6$ (as 6T3):

A solvable group of order 12

The 6 conjugacy class representatives for $D_{6}$

Character table for $D_{6}$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.10611.1

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

Sibling algebras

Galois closure:	Deg 12
Twin sextic algebra:	3.1.10611.1 $\times$ \(\Q(\sqrt{393}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.2.44249175153.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.6.9.11	$x^{6} + 6 x^{4} + 12$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
$131$	131.2.0.1	$x^{2} - x + 14$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	131.4.2.1	$x^{4} + 3537 x^{2} + 3363556$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.131.2t1.1c1	$1$	$ 131 $	$x^{2} - x + 33$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_131.2t1.1c1	$1$	$ 3 \cdot 131 $	$x^{2} - x - 98$	$C_2$ (as 2T1)	$1$	$1$
*	2.3e4_131.3t2.1c1	$2$	$ 3^{4} \cdot 131 $	$x^{3} + 6 x - 19$	$S_3$ (as 3T2)	$1$	$0$
*	2.3e4_131.6t3.2c1	$2$	$ 3^{4} \cdot 131 $	$x^{6} - 6 x^{4} - 38 x^{3} + 36 x^{2} + 114 x + 361$	$D_{6}$ (as 6T3)	$1$	$0$

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 *.