Global number field 6.2.27931544.1

• Label: 6.2.27931544.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{3}\cdot 17\cdot 59^{3}$
• Root discriminant: $17.42$
• Ramified primes: $2, 17, 59$
• Class number: $2$
• Class group: $[2]$
• Galois group: $S_4\times C_2$ (as 6T11)

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 12 x^{4} + 26 x^{3} + 44 x^{2} - 114 x - 61 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(27931544=2^{3}\cdot 17\cdot 59^{3}\)
Root discriminant:		$17.42$
Ramified primes:		$2, 17, 59$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{1466} a^{5} - \frac{47}{1466} a^{4} - \frac{48}{733} a^{3} + \frac{681}{1466} a^{2} + \frac{185}{1466} a - \frac{188}{733}$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{33}{1466} a^{5} - \frac{85}{1466} a^{4} - \frac{118}{733} a^{3} + \frac{483}{1466} a^{2} + \frac{241}{1466} a - \frac{340}{733} \),  \( \frac{23}{733} a^{5} + \frac{37}{1466} a^{4} - \frac{751}{1466} a^{3} - \frac{463}{733} a^{2} + \frac{3379}{1466} a + \frac{5427}{1466} \),  \( \frac{76}{733} a^{5} + \frac{93}{733} a^{4} - \frac{699}{733} a^{3} - \frac{287}{733} a^{2} + \frac{3065}{733} a + \frac{1477}{733} \)
Regulator:		\( 24.792940531 \)

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

3.1.59.1

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

Sibling algebras

Twin sextic algebra:	4.2.1091264.5 $\times$ \(\Q(\sqrt{-34}) \)
Degree 6 sibling:	6.0.473416.1
Degree 8 siblings:	8.4.4145373626699776.4, 8.0.1190857117696.18
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	R	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$	${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$	R	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	R

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
$2$	2.2.3.4	$x^{2} + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.0.1	$x^{2} - x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	2.2.0.1	$x^{2} - x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
$17$	17.2.1.1	$x^{2} - 17$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
$59$	59.2.1.2	$x^{2} + 177$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	59.2.1.2	$x^{2} + 177$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	59.2.1.2	$x^{2} + 177$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e3_17.2t1.2c1	$1$	$ 2^{3} \cdot 17 $	$x^{2} + 34$	$C_2$ (as 2T1)	$1$	$-1$
	1.59.2t1.1c1	$1$	$ 59 $	$x^{2} - x + 15$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_17_59.2t1.1c1	$1$	$ 2^{3} \cdot 17 \cdot 59 $	$x^{2} - 2006$	$C_2$ (as 2T1)	$1$	$1$
	2.2e6_17e2_59.6t3.3c1	$2$	$ 2^{6} \cdot 17^{2} \cdot 59 $	$x^{6} + 106 x^{4} - 2 x^{3} + 3472 x^{2} + 200 x + 34817$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.59.3t2.1c1	$2$	$ 59 $	$x^{3} + 2 x - 1$	$S_3$ (as 3T2)	$1$	$0$
	3.2e6_17e2_59.4t5.3c1	$3$	$ 2^{6} \cdot 17^{2} \cdot 59 $	$x^{4} - 2 x^{3} + 11 x^{2} + 24 x + 8$	$S_4$ (as 4T5)	$1$	$1$
*	3.2e3_17_59e2.6t11.1c1	$3$	$ 2^{3} \cdot 17 \cdot 59^{2}$	$x^{6} - 2 x^{5} - 12 x^{4} + 26 x^{3} + 44 x^{2} - 114 x - 61$	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.2e3_17_59.6t11.1c1	$3$	$ 2^{3} \cdot 17 \cdot 59 $	$x^{6} - 2 x^{5} - 12 x^{4} + 26 x^{3} + 44 x^{2} - 114 x - 61$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
	3.2e6_17e2_59e2.6t8.3c1	$3$	$ 2^{6} \cdot 17^{2} \cdot 59^{2}$	$x^{4} - 2 x^{3} + 11 x^{2} + 24 x + 8$	$S_4$ (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 *.