Global number field 6.2.8571729177.1

• Label: 6.2.8571729177.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $3^{4}\cdot 11^{3}\cdot 43^{3}$
• Root discriminant: $45.24$
• Ramified primes: $3, 11, 43$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $S_4\times C_2$ (as 6T11)

Normalized defining polynomial

\( x^{6} - x^{5} - 3 x^{4} + 125 x^{3} - 569 x^{2} + 1152 x - 2109 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(8571729177=3^{4}\cdot 11^{3}\cdot 43^{3}\)
Root discriminant:		$45.24$
Ramified primes:		$3, 11, 43$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{185994} a^{5} - \frac{582}{10333} a^{4} - \frac{121}{61998} a^{3} + \frac{41335}{92997} a^{2} + \frac{6415}{61998} a + \frac{9091}{61998}$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{343}{61998} a^{5} + \frac{436}{10333} a^{4} - \frac{171}{20666} a^{3} + \frac{11362}{30999} a^{2} + \frac{51081}{20666} a - \frac{250345}{20666} \),  \( \frac{769}{185994} a^{5} + \frac{616}{30999} a^{4} - \frac{10385}{61998} a^{3} + \frac{105637}{92997} a^{2} - \frac{212699}{61998} a + \frac{109201}{61998} \),  \( \frac{1}{185994} a^{5} - \frac{582}{10333} a^{4} - \frac{121}{61998} a^{3} + \frac{41335}{92997} a^{2} - \frac{427571}{61998} a + \frac{1373047}{61998} \)
Regulator:		\( 238.005350427 \)

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

3.1.1419.1

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

Sibling algebras

Twin sextic algebra:	4.2.12771.1 $\times$ \(\Q(\sqrt{-3}) \)
Degree 6 sibling:	6.0.6040683.2
Degree 8 siblings:	8.4.36489851106489.1, 8.0.163098441.1
Degree 12 siblings:	Deg 12, Deg 12, Deg 12, Deg 12, Deg 12, Deg 12
Degree 16 sibling:	Deg 16
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	${\href{/LocalNumberField/2.4.0.1}{4} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	R	${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.4.3.2	$x^{4} - 3$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
$11$	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$43$	43.2.1.1	$x^{2} - 43$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	43.2.1.1	$x^{2} - 43$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	43.2.1.1	$x^{2} - 43$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_11_43.2t1.1c1	$1$	$ 3 \cdot 11 \cdot 43 $	$x^{2} - x + 355$	$C_2$ (as 2T1)	$1$	$-1$
	1.11_43.2t1.1c1	$1$	$ 11 \cdot 43 $	$x^{2} - x - 118$	$C_2$ (as 2T1)	$1$	$1$
	2.3_11_43.6t3.1c1	$2$	$ 3 \cdot 11 \cdot 43 $	$x^{6} - 4 x^{4} + 16 x^{2} - 36 x + 27$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.3_11_43.3t2.1c1	$2$	$ 3 \cdot 11 \cdot 43 $	$x^{3} - x^{2} + 10 x - 12$	$S_3$ (as 3T2)	$1$	$0$
	3.3e3_11_43.4t5.1c1	$3$	$ 3^{3} \cdot 11 \cdot 43 $	$x^{4} - x^{3} - 3 x^{2} + 5 x + 1$	$S_4$ (as 4T5)	$1$	$1$
*	3.3e3_11e2_43e2.6t11.1c1	$3$	$ 3^{3} \cdot 11^{2} \cdot 43^{2}$	$x^{6} - x^{5} - 3 x^{4} + 125 x^{3} - 569 x^{2} + 1152 x - 2109$	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.3e2_11_43.6t11.1c1	$3$	$ 3^{2} \cdot 11 \cdot 43 $	$x^{6} - x^{5} - 3 x^{4} + 125 x^{3} - 569 x^{2} + 1152 x - 2109$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
	3.3e2_11e2_43e2.6t8.2c1	$3$	$ 3^{2} \cdot 11^{2} \cdot 43^{2}$	$x^{4} - x^{3} - 3 x^{2} + 5 x + 1$	$S_4$ (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 *.