Global number field 8.0.31091776.1

• Label: 8.0.31091776.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{6}\cdot 17^{2}\cdot 41^{2}$
• Root discriminant: $8.64$
• Ramified primes: $2, 17, 41$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_2^3:S_4$ (as 8T39)

Normalized defining polynomial

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

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(31091776=2^{6}\cdot 17^{2}\cdot 41^{2}\)
Root discriminant:		$8.64$
Ramified primes:		$2, 17, 41$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a \),  \( \frac{1}{2} a^{7} + \frac{1}{2} a^{6} - \frac{1}{2} a^{5} + \frac{1}{2} a^{4} + a^{3} + a^{2} - \frac{3}{2} a + \frac{1}{2} \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{2} a^{5} + \frac{1}{2} a^{4} + 2 a^{3} - a^{2} + \frac{1}{2} a + \frac{1}{2} \)
Regulator:		\( 2.81041870785 \)

Galois group

$C_2^3:S_4$ (as 8T39):

A solvable group of order 192

The 13 conjugacy class representatives for $C_2^3:S_4$

Character table for $C_2^3:S_4$

Intermediate fields

4.0.697.1

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

Sibling fields

Degree 8 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	${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
$2$	2.2.0.1	$x^{2} - x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	2.6.6.2	$x^{6} - x^{4} - 5$	$2$	$3$	$6$	$A_4\times C_2$	$[2, 2]^{6}$
$17$	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.4.2.1	$x^{4} + 85 x^{2} + 2601$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$41$	41.2.0.1	$x^{2} - x + 12$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.0.1	$x^{2} - x + 12$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.17_41.2t1.1c1	$1$	$ 17 \cdot 41 $	$x^{2} - x - 174$	$C_2$ (as 2T1)	$1$	$1$
	2.17_41.3t2.1c1	$2$	$ 17 \cdot 41 $	$x^{3} - 7 x - 5$	$S_3$ (as 3T2)	$1$	$2$
	3.2e6_17_41.4t5.1c1	$3$	$ 2^{6} \cdot 17 \cdot 41 $	$x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 3$	$S_4$ (as 4T5)	$1$	$3$
	3.2e6_17e2_41e2.6t8.2c1	$3$	$ 2^{6} \cdot 17^{2} \cdot 41^{2}$	$x^{4} + 6 x^{2} - 8 x + 25$	$S_4$ (as 4T5)	$1$	$-1$
*	3.17_41.4t5.1c1	$3$	$ 17 \cdot 41 $	$x^{4} - x^{3} + 2 x^{2} - x + 2$	$S_4$ (as 4T5)	$1$	$-1$
	3.17e2_41e2.6t8.2c1	$3$	$ 17^{2} \cdot 41^{2}$	$x^{4} - x^{3} + 2 x^{2} - x + 2$	$S_4$ (as 4T5)	$1$	$-1$
	3.2e6_17_41.4t5.2c1	$3$	$ 2^{6} \cdot 17 \cdot 41 $	$x^{4} + 6 x^{2} - 8 x + 25$	$S_4$ (as 4T5)	$1$	$-1$
	3.2e6_17e2_41e2.6t8.1c1	$3$	$ 2^{6} \cdot 17^{2} \cdot 41^{2}$	$x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 3$	$S_4$ (as 4T5)	$1$	$3$
*	4.2e6_17_41.8t39.2c1	$4$	$ 2^{6} \cdot 17 \cdot 41 $	$x^{8} + 3 x^{4} - x^{2} + 1$	$C_2^3:S_4$ (as 8T39)	$1$	$0$
	4.2e6_17e3_41e3.8t39.2c1	$4$	$ 2^{6} \cdot 17^{3} \cdot 41^{3}$	$x^{8} + 3 x^{4} - x^{2} + 1$	$C_2^3:S_4$ (as 8T39)	$1$	$0$
	6.2e12_17e3_41e3.8t34.1c1	$6$	$ 2^{12} \cdot 17^{3} \cdot 41^{3}$	$x^{8} + 4 x^{6} - 16 x^{5} + 122 x^{4} - 32 x^{3} + 300 x^{2} - 944 x + 693$	$V_4^2:S_3$ (as 8T34)	$1$	$-2$
	8.2e12_17e4_41e4.24t333.4c1	$8$	$ 2^{12} \cdot 17^{4} \cdot 41^{4}$	$x^{8} + 3 x^{4} - x^{2} + 1$	$C_2^3:S_4$ (as 8T39)	$1$	$0$

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 *.