Global number field 6.2.74043072.3

• Label: 6.2.74043072.3
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{6}\cdot 3^{7}\cdot 23^{2}$
• Root discriminant: $20.49$
• Ramified primes: $2, 3, 23$
• Class number: $1$
• Class group: Trivial
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 9 x^{4} - 18 x^{3} + 42 x^{2} - 72 x + 24 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(74043072=2^{6}\cdot 3^{7}\cdot 23^{2}\)
Root discriminant:		$20.49$
Ramified primes:		$2, 3, 23$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{460} a^{5} + \frac{7}{115} a^{4} + \frac{17}{92} a^{3} - \frac{42}{115} a^{2} - \frac{31}{230} a + \frac{8}{115}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{115} a^{5} - \frac{3}{460} a^{4} - \frac{1}{92} a^{3} + \frac{9}{230} a^{2} - \frac{9}{230} a + \frac{32}{115} \),  \( \frac{169}{460} a^{5} + \frac{33}{115} a^{4} - \frac{347}{92} a^{3} - \frac{888}{115} a^{2} + \frac{2811}{230} a - \frac{2558}{115} \),  \( \frac{607}{230} a^{5} + \frac{2367}{460} a^{4} - \frac{2017}{92} a^{3} - \frac{22511}{230} a^{2} - \frac{2559}{230} a + \frac{2582}{115} \)
Regulator:		\( 291.573483376 \)

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

\(\Q(\sqrt{3}) \), 3.1.2484.1

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:	3.1.2484.1 $\times$ \(\Q(\sqrt{-23}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.0.141915888.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.6.0.1}{6} }$	R	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.2.2.2	$x^{2} + 2 x - 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.2.2.2	$x^{2} + 2 x - 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.2.2.2	$x^{2} + 2 x - 2$	$2$	$1$	$2$	$C_2$	$[2]$
$3$	3.6.7.1	$x^{6} + 6 x^{2} + 6$	$6$	$1$	$7$	$S_3$	$[3/2]_{2}$
$23$	$\Q_{23}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{23}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2_3_23.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 23 $	$x^{2} + 69$	$C_2$ (as 2T1)	$1$	$-1$
	1.23.2t1.1c1	$1$	$ 23 $	$x^{2} - x + 6$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e2_3.2t1.1c1	$1$	$ 2^{2} \cdot 3 $	$x^{2} - 3$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_3e3_23.3t2.1c1	$2$	$ 2^{2} \cdot 3^{3} \cdot 23 $	$x^{3} - 6 x - 20$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3e3_23.6t3.1c1	$2$	$ 2^{2} \cdot 3^{3} \cdot 23 $	$x^{6} - 9 x^{4} - 18 x^{3} + 42 x^{2} - 72 x + 24$	$D_{6}$ (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 *.