Global number field 6.6.19766016272.1

• Label: 6.6.19766016272.1
• Degree: $6$
• Signature: $[6, 0]$
• Discriminant: $2^{4}\cdot 29^{3}\cdot 37^{3}$
• Root discriminant: $52.00$
• Ramified primes: $2, 29, 37$
• Class number: $2$
• Class group: $[2]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 107 x^{4} + 205 x^{3} + 2436 x^{2} - 3638 x - 10982 \)

Invariants

Degree:		$6$
Signature:		$[6, 0]$
Discriminant:		\(19766016272=2^{4}\cdot 29^{3}\cdot 37^{3}\)
Root discriminant:		$52.00$
Ramified primes:		$2, 29, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4765576} a^{5} + \frac{254655}{1191394} a^{4} - \frac{40023}{4765576} a^{3} + \frac{288639}{1191394} a^{2} - \frac{134701}{595697} a - \frac{62207}{140164}$

Class group and class number

$C_{2}$, which has order $2$

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:		\( 2216.29362216 \)

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

\(\Q(\sqrt{1073}) \), 3.3.148.1

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:	3.3.148.1 $\times$ \(\Q(\sqrt{29}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.6.534216656.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
$29$	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.4.2.1	$x^{4} + 145 x^{2} + 7569$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$37$	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	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
	1.29.2t1.1c1	$1$	$ 29 $	$x^{2} - x - 7$	$C_2$ (as 2T1)	$1$	$1$
*	1.29_37.2t1.1c1	$1$	$ 29 \cdot 37 $	$x^{2} - x - 268$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_37.3t2.1c1	$2$	$ 2^{2} \cdot 37 $	$x^{3} - x^{2} - 3 x + 1$	$S_3$ (as 3T2)	$1$	$2$
*	2.2e2_29e2_37.6t3.1c1	$2$	$ 2^{2} \cdot 29^{2} \cdot 37 $	$x^{6} - 3 x^{5} - 107 x^{4} + 205 x^{3} + 2436 x^{2} - 3638 x - 10982$	$D_{6}$ (as 6T3)	$1$	$2$

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 *.