Global number field 6.2.296747776.1

• Label: 6.2.296747776.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{8}\cdot 13^{2}\cdot 19^{3}$
• Root discriminant: $25.83$
• Ramified primes: $2, 13, 19$
• Class number: $2$
• Class group: $[2]$
• Galois group: $S_4\times C_2$ (as 6T11)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 28 x^{4} + 2 x^{3} + 82 x^{2} + 420 x - 754 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(296747776=2^{8}\cdot 13^{2}\cdot 19^{3}\)
Root discriminant:		$25.83$
Ramified primes:		$2, 13, 19$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{150707} a^{5} - \frac{12751}{150707} a^{4} - \frac{50326}{150707} a^{3} + \frac{46477}{150707} a^{2} + \frac{44733}{150707} a - \frac{25309}{150707}$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
Regulator:		\( 56.0848461088 \)

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

3.1.76.1

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

Sibling algebras

Twin sextic algebra:	\(\Q(\sqrt{-1}) \) $\times$ 4.2.12844.1
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	${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.6.0.1}{6} }$	R	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$	${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$	2.6.8.3	$x^{6} + 2 x^{3} + 6$	$6$	$1$	$8$	$D_{6}$	$[2]_{3}^{2}$
$13$	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	13.4.2.2	$x^{4} - 13 x^{2} + 338$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
$19$	19.2.1.1	$x^{2} - 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.4.2.1	$x^{4} + 57 x^{2} + 1444$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_19.2t1.1c1	$1$	$ 2^{2} \cdot 19 $	$x^{2} - 19$	$C_2$ (as 2T1)	$1$	$1$
	1.19.2t1.1c1	$1$	$ 19 $	$x^{2} - x + 5$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	2.2e4_19.6t3.1c1	$2$	$ 2^{4} \cdot 19 $	$x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + 2$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.2e2_19.3t2.1c1	$2$	$ 2^{2} \cdot 19 $	$x^{3} - 2 x - 2$	$S_3$ (as 3T2)	$1$	$0$
	3.2e2_13e2_19.4t5.1c1	$3$	$ 2^{2} \cdot 13^{2} \cdot 19 $	$x^{4} - x^{3} + 3 x^{2} - 3 x - 4$	$S_4$ (as 4T5)	$1$	$1$
	3.2e6_13e2_19.6t11.1c1	$3$	$ 2^{6} \cdot 13^{2} \cdot 19 $	$x^{6} - 2 x^{5} + 28 x^{4} + 2 x^{3} + 82 x^{2} + 420 x - 754$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
*	3.2e6_13e2_19e2.6t11.1c1	$3$	$ 2^{6} \cdot 13^{2} \cdot 19^{2}$	$x^{6} - 2 x^{5} + 28 x^{4} + 2 x^{3} + 82 x^{2} + 420 x - 754$	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.2e2_13e2_19e2.6t8.1c1	$3$	$ 2^{2} \cdot 13^{2} \cdot 19^{2}$	$x^{4} - x^{3} + 3 x^{2} - 3 x - 4$	$S_4$ (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 *.