Global number field 6.2.112698432.1

• Label: 6.2.112698432.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{6}\cdot 3^{3}\cdot 7^{2}\cdot 11^{3}$
• Root discriminant: $21.98$
• Ramified primes: $2, 3, 7, 11$
• Class number: $4$
• Class group: $[4]$
• Galois group: $S_4\times C_2$ (as 6T11)

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 28 x^{4} + 36 x^{3} + 294 x^{2} - 250 x - 854 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(112698432=2^{6}\cdot 3^{3}\cdot 7^{2}\cdot 11^{3}\)
Root discriminant:		$21.98$
Ramified primes:		$2, 3, 7, 11$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{221} a^{5} - \frac{2}{17} a^{4} - \frac{67}{221} a^{3} + \frac{97}{221} a^{2} - \frac{45}{221} a - \frac{54}{221}$

Class group and class number

$C_{4}$, which has order $4$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{24}{221} a^{5} + \frac{3}{17} a^{4} - \frac{503}{221} a^{3} - \frac{987}{221} a^{2} + \frac{2898}{221} a + \frac{4671}{221} \),  \( \frac{16}{221} a^{5} + \frac{2}{17} a^{4} - \frac{409}{221} a^{3} - \frac{658}{221} a^{2} + \frac{2595}{221} a + \frac{4219}{221} \),  \( \frac{14}{221} a^{5} + \frac{6}{17} a^{4} - \frac{54}{221} a^{3} - \frac{1073}{221} a^{2} - \frac{1956}{221} a - \frac{977}{221} \)
Regulator:		\( 25.5018610066 \)

Galois group

$C_2\times S_4$ (as 6T11):

A solvable group of order 48

The 10 conjugacy class representatives for $S_4\times C_2$

Character table for $S_4\times C_2$

Intermediate fields

3.1.44.1

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

Sibling algebras

Twin sextic algebra:	\(\Q(\sqrt{-3}) \) $\times$ 4.2.8624.1
Degree 6 sibling:	data not computed
Degree 8 siblings:	data not computed
Degree 12 siblings:	data not computed
Degree 16 sibling:	data not computed
Degree 24 siblings:	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	R	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	R	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$	2.6.6.7	$x^{6} + 2 x^{2} + 2 x + 2$	$6$	$1$	$6$	$S_4$	$[4/3, 4/3]_{3}^{2}$
$3$	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$	$\Q_{7}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{7}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	7.4.2.2	$x^{4} - 7 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
$11$	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.1	$x^{2} - 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.3_11.2t1.1c1	$1$	$ 3 \cdot 11 $	$x^{2} - x - 8$	$C_2$ (as 2T1)	$1$	$1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	2.2e2_3e2_11.6t3.1c1	$2$	$ 2^{2} \cdot 3^{2} \cdot 11 $	$x^{6} - x^{5} - 3 x^{3} + 2 x^{2} + x + 1$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.2e2_11.3t2.1c1	$2$	$ 2^{2} \cdot 11 $	$x^{3} - x^{2} + x + 1$	$S_3$ (as 3T2)	$1$	$0$
	3.2e4_7e2_11.4t5.1c1	$3$	$ 2^{4} \cdot 7^{2} \cdot 11 $	$x^{4} - 2 x^{3} + 2 x^{2} + 6 x + 2$	$S_4$ (as 4T5)	$1$	$1$
	3.2e4_3e3_7e2_11.6t11.1c1	$3$	$ 2^{4} \cdot 3^{3} \cdot 7^{2} \cdot 11 $	$x^{6} - 2 x^{5} - 28 x^{4} + 36 x^{3} + 294 x^{2} - 250 x - 854$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
*	3.2e4_3e3_7e2_11e2.6t11.1c1	$3$	$ 2^{4} \cdot 3^{3} \cdot 7^{2} \cdot 11^{2}$	$x^{6} - 2 x^{5} - 28 x^{4} + 36 x^{3} + 294 x^{2} - 250 x - 854$	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.2e4_7e2_11e2.6t8.1c1	$3$	$ 2^{4} \cdot 7^{2} \cdot 11^{2}$	$x^{4} - 2 x^{3} + 2 x^{2} + 6 x + 2$	$S_4$ (as 4T5)	$1$	$-1$

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 *.