Global number field 8.0.264777984.1

• Label: 8.0.264777984.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 3^{4}\cdot 113^{2}$
• Root discriminant: $11.29$
• Ramified primes: $2, 3, 113$
• Class number: $1$
• Class group: Trivial
• Galois group: $V_4^2:(S_3\times C_2)$ (as 8T41)

Normalized defining polynomial

\( x^{8} - 4 x^{6} - 2 x^{5} + 6 x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(264777984=2^{8}\cdot 3^{4}\cdot 113^{2}\)
Root discriminant:		$11.29$
Ramified primes:		$2, 3, 113$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{13} a^{7} - \frac{4}{13} a^{6} - \frac{1}{13} a^{5} + \frac{2}{13} a^{4} - \frac{2}{13} a^{3} + \frac{6}{13} a^{2} + \frac{5}{13} a - \frac{3}{13}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{8}{13} a^{7} - \frac{6}{13} a^{6} - \frac{34}{13} a^{5} + \frac{3}{13} a^{4} + \frac{62}{13} a^{3} - \frac{43}{13} a^{2} + \frac{40}{13} a + \frac{28}{13} \) (order $6$)
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:		\( 19.7523703725 \)

Galois group

$C_2^2:S_4:C_2$ (as 8T41):

A solvable group of order 192

The 14 conjugacy class representatives for $V_4^2:(S_3\times C_2)$

Character table for $V_4^2:(S_3\times C_2)$

Intermediate fields

\(\Q(\sqrt{-3}) \)

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

Sibling fields

Degree 8 sibling:	data not computed
Degree 12 siblings:	data not computed
Degree 16 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 32 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	${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$	2.8.8.2	$x^{8} + 2 x^{7} + 8 x^{2} + 48$	$2$	$4$	$8$	$C_2^2:C_4$	$[2, 2]^{4}$
$3$	3.8.4.1	$x^{8} + 36 x^{4} - 27 x^{2} + 324$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$113$	113.4.0.1	$x^{4} - x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	113.4.2.2	$x^{4} - 113 x^{2} + 127690$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$

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)	$1$	$-1$
	1.113.2t1.1c1	$1$	$ 113 $	$x^{2} - x - 28$	$C_2$ (as 2T1)	$1$	$1$
	1.3_113.2t1.1c1	$1$	$ 3 \cdot 113 $	$x^{2} - x + 85$	$C_2$ (as 2T1)	$1$	$-1$
	2.3_113.3t2.1c1	$2$	$ 3 \cdot 113 $	$x^{3} - x^{2} - x + 4$	$S_3$ (as 3T2)	$1$	$0$
	2.3_113.6t3.1c1	$2$	$ 3 \cdot 113 $	$x^{6} - 4 x^{4} + 4 x^{2} + 3$	$D_{6}$ (as 6T3)	$1$	$0$
	3.2e6_3e2_113e2.6t8.3c1	$3$	$ 2^{6} \cdot 3^{2} \cdot 113^{2}$	$x^{4} - 2 x^{3} - 4 x^{2} + 4 x - 2$	$S_4$ (as 4T5)	$1$	$-1$
	3.2e6_3_113e2.6t11.1c1	$3$	$ 2^{6} \cdot 3 \cdot 113^{2}$	$x^{6} - 4 x^{4} - 109 x^{2} - 452$	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.2e6_3_113.4t5.1c1	$3$	$ 2^{6} \cdot 3 \cdot 113 $	$x^{4} - 2 x^{3} - 4 x^{2} + 4 x - 2$	$S_4$ (as 4T5)	$1$	$1$
	3.2e6_3e2_113.6t11.1c1	$3$	$ 2^{6} \cdot 3^{2} \cdot 113 $	$x^{6} - 4 x^{4} - 109 x^{2} - 452$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
	6.2e10_3e2_113e3.8t41.1c1	$6$	$ 2^{10} \cdot 3^{2} \cdot 113^{3}$	$x^{8} - 4 x^{6} - 2 x^{5} + 6 x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$2$
*	6.2e8_3e3_113e2.8t41.1c1	$6$	$ 2^{8} \cdot 3^{3} \cdot 113^{2}$	$x^{8} - 4 x^{6} - 2 x^{5} + 6 x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$
	6.2e8_3e3_113e4.12t111.1c1	$6$	$ 2^{8} \cdot 3^{3} \cdot 113^{4}$	$x^{8} - 4 x^{6} - 2 x^{5} + 6 x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$
	6.2e10_3e4_113e3.12t108.1c1	$6$	$ 2^{10} \cdot 3^{4} \cdot 113^{3}$	$x^{8} - 4 x^{6} - 2 x^{5} + 6 x^{4} - 2 x^{3} + 3 x^{2} + 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$-2$

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 *.