Global number field 6.2.1693181072.2

• Label: 6.2.1693181072.2
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 11^{3}\cdot 43^{3}$
• Root discriminant: $34.52$
• Ramified primes: $2, 11, 43$
• Class number: $3$
• Class group: $[3]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 23 x^{4} - 94 x^{3} + 193 x^{2} - 792 x - 2961 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(1693181072=2^{4}\cdot 11^{3}\cdot 43^{3}\)
Root discriminant:		$34.52$
Ramified primes:		$2, 11, 43$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{60} a^{4} + \frac{1}{30} a^{3} - \frac{4}{15} a^{2} - \frac{1}{5} a - \frac{1}{20}$, $\frac{1}{14160} a^{5} - \frac{37}{4720} a^{4} - \frac{311}{7080} a^{3} + \frac{557}{1770} a^{2} - \frac{53}{240} a + \frac{2273}{4720}$

Class group and class number

$C_{3}$, which has order $3$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{2360} a^{5} + \frac{7}{2360} a^{4} + \frac{11}{3540} a^{3} - \frac{139}{1770} a^{2} - \frac{11}{120} a + \frac{113}{472} \),  \( \frac{9}{590} a^{5} - \frac{271}{3540} a^{4} + \frac{512}{885} a^{3} - \frac{1609}{590} a^{2} + \frac{121}{15} a - \frac{28711}{1180} \),  \( \frac{4}{3} a^{3} - \frac{4}{3} a^{2} + \frac{44}{3} a - 135 \)
Regulator:		\( 285.610219656 \)

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

\(\Q(\sqrt{473}) \), 3.1.172.1

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:	3.1.172.1 $\times$ \(\Q(\sqrt{-11}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.0.39376304.4

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	R	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$2$	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
$11$	11.6.3.2	$x^{6} - 121 x^{2} + 3993$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
43	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.43.2t1.1c1	$1$	$ 43 $	$x^{2} - x + 11$	$C_2$ (as 2T1)	$1$	$-1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
*	1.11_43.2t1.1c1	$1$	$ 11 \cdot 43 $	$x^{2} - x - 118$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_43.3t2.1c1	$2$	$ 2^{2} \cdot 43 $	$x^{3} - x^{2} - x + 3$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_11e2_43.6t3.1c1	$2$	$ 2^{2} \cdot 11^{2} \cdot 43 $	$x^{6} - 2 x^{5} + 23 x^{4} - 94 x^{3} + 193 x^{2} - 792 x - 2961$	$D_{6}$ (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 *.