Global number field 6.2.2008483689113.2

• Label: 6.2.2008483689113.2
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $11^{3}\cdot 31^{3}\cdot 37^{3}$
• Root discriminant: $112.33$
• Ramified primes: $11, 31, 37$
• Class number: $40$
• Class group: $[2, 2, 10]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 205 x^{4} + 712 x^{3} + 9488 x^{2} + 93432 x - 32607053 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(2008483689113=11^{3}\cdot 31^{3}\cdot 37^{3}\)
Root discriminant:		$112.33$
Ramified primes:		$11, 31, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{102} a^{3} - \frac{1}{102} a^{2} - \frac{1}{102}$, $\frac{1}{102} a^{4} - \frac{1}{102} a^{2} - \frac{1}{102} a - \frac{1}{102}$, $\frac{1}{90146151906} a^{5} - \frac{151477591}{90146151906} a^{4} - \frac{46312489}{45073075953} a^{3} - \frac{40104989909}{90146151906} a^{2} - \frac{401119634}{15024358651} a - \frac{4632379499}{45073075953}$

Class group and class number

$C_{2}\times C_{2}\times C_{10}$, which has order $40$

Unit group

Rank:		$3$
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:		\( 181.787689042 \)

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

\(\Q(\sqrt{12617}) \), 3.1.31.1

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

Sibling algebras

Galois closure:	data not computed
Twin sextic algebra:	\(\Q\) $\times$ \(\Q(\sqrt{-407}) \) $\times$ 3.1.31.1
Degree 6 sibling:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.6.0.1}{6} }$	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	R	R	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$11$	11.2.1.1	$x^{2} - 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
31	Data not computed
$37$	37.2.1.2	$x^{2} + 74$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.4.2.1	$x^{4} + 333 x^{2} + 34225$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.31.2t1.1c1	$1$	$ 31 $	$x^{2} - x + 8$	$C_2$ (as 2T1)	$1$	$-1$
	1.11_37.2t1.1c1	$1$	$ 11 \cdot 37 $	$x^{2} - x + 102$	$C_2$ (as 2T1)	$1$	$-1$
*	1.11_31_37.2t1.1c1	$1$	$ 11 \cdot 31 \cdot 37 $	$x^{2} - x - 3154$	$C_2$ (as 2T1)	$1$	$1$
*	2.31.3t2.1c1	$2$	$ 31 $	$x^{3} + x - 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.11e2_31_37e2.6t3.1c1	$2$	$ 11^{2} \cdot 31 \cdot 37^{2}$	$x^{6} - 2 x^{5} + 205 x^{4} + 712 x^{3} + 9488 x^{2} + 93432 x - 32607053$	$D_{6}$ (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 *.