Global number field 6.2.29176128.1

• Label: 6.2.29176128.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{6}\cdot 3^{2}\cdot 37^{3}$
• Root discriminant: $17.55$
• Ramified primes: $2, 3, 37$
• Class number: $1$
• Class group: Trivial
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} + 10 x^{2} - 9 x - 3 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(29176128=2^{6}\cdot 3^{2}\cdot 37^{3}\)
Root discriminant:		$17.55$
Ramified primes:		$2, 3, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{2}{3} a^{5} - \frac{7}{6} a^{4} + \frac{4}{3} a^{3} - \frac{5}{6} a^{2} + 6 a + \frac{3}{2} \),  \( \frac{1}{2} a^{4} - 2 a^{3} + \frac{7}{2} a^{2} - 2 a + \frac{1}{2} \),  \( \frac{1}{3} a^{5} + \frac{7}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{6} a^{2} - 2 a - \frac{1}{2} \)
Regulator:		\( 102.5823062 \)

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

\(\Q(\sqrt{37}) \), 3.1.888.1

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:	3.1.888.1 $\times$ \(\Q(\sqrt{-6}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.0.18925056.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$	2.2.0.1	$x^{2} - x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	2.4.6.1	$x^{4} - 6 x^{2} + 4$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
$3$	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{3}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
37	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e3_3_37.2t1.2c1	$1$	$ 2^{3} \cdot 3 \cdot 37 $	$x^{2} + 222$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_3.2t1.2c1	$1$	$ 2^{3} \cdot 3 $	$x^{2} + 6$	$C_2$ (as 2T1)	$1$	$-1$
*	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e3_3_37.3t2.1c1	$2$	$ 2^{3} \cdot 3 \cdot 37 $	$x^{3} - x^{2} + 9 x + 3$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e3_3_37.6t3.3c1	$2$	$ 2^{3} \cdot 3 \cdot 37 $	$x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} + 10 x^{2} - 9 x - 3$	$D_{6}$ (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 *.