Global number field 8.4.68425984.1

• Label: 8.4.68425984.1
• Degree: $8$
• Signature: $[4, 2]$
• Discriminant: $2^{8}\cdot 11^{2}\cdot 47^{2}$
• Root discriminant: $9.54$
• Ramified primes: $2, 11, 47$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_2^3:S_4$ (as 8T39)

Normalized defining polynomial

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

Invariants

Degree:		$8$
Signature:		$[4, 2]$
Discriminant:		\(68425984=2^{8}\cdot 11^{2}\cdot 47^{2}\)
Root discriminant:		$9.54$
Ramified primes:		$2, 11, 47$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$5$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a \),  \( \frac{1}{2} a^{7} + a^{5} + \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - 2 a - \frac{1}{2} \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{2} a^{5} - \frac{3}{2} a^{3} + \frac{1}{2} a^{2} - \frac{3}{2} a \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{2} a^{2} - \frac{3}{2} a + 1 \),  \( \frac{1}{2} a^{7} + \frac{1}{2} a^{6} + \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{3}{2} a - 1 \)
Regulator:		\( 9.28949206428 \)

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

4.2.2068.1

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	${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$	R	${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$	2.4.8.6	$x^{4} + 6 x^{2} + 4 x + 2$	$4$	$1$	$8$	$D_{4}$	$[2, 3]^{2}$
	2.4.0.1	$x^{4} - x + 1$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$11$	11.2.0.1	$x^{2} - x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.2.0.1	$x^{2} - x + 7$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$47$	47.2.0.1	$x^{2} - x + 13$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	47.2.0.1	$x^{2} - x + 13$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	47.4.2.1	$x^{4} + 1175 x^{2} + 373321$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_11_47.2t1.1c1	$1$	$ 2^{2} \cdot 11 \cdot 47 $	$x^{2} + 517$	$C_2$ (as 2T1)	$1$	$-1$
	2.2e2_11_47.3t2.1c1	$2$	$ 2^{2} \cdot 11 \cdot 47 $	$x^{3} - x^{2} - 8 x - 10$	$S_3$ (as 3T2)	$1$	$0$
	3.2e6_11_47.4t5.2c1	$3$	$ 2^{6} \cdot 11 \cdot 47 $	$x^{4} - 2 x^{3} - 3 x^{2} - 2 x - 3$	$S_4$ (as 4T5)	$1$	$1$
	3.2e8_11e2_47e2.6t8.1c1	$3$	$ 2^{8} \cdot 11^{2} \cdot 47^{2}$	$x^{4} - 2 x^{3} + 3 x^{2} + 2 x - 5$	$S_4$ (as 4T5)	$1$	$-1$
*	3.2e2_11_47.4t5.1c1	$3$	$ 2^{2} \cdot 11 \cdot 47 $	$x^{4} - x^{3} - 2 x^{2} + 3 x + 1$	$S_4$ (as 4T5)	$1$	$1$
	3.2e4_11e2_47e2.6t8.3c1	$3$	$ 2^{4} \cdot 11^{2} \cdot 47^{2}$	$x^{4} - x^{3} - 2 x^{2} + 3 x + 1$	$S_4$ (as 4T5)	$1$	$-1$
	3.2e6_11_47.4t5.1c1	$3$	$ 2^{6} \cdot 11 \cdot 47 $	$x^{4} - 2 x^{3} + 3 x^{2} + 2 x - 5$	$S_4$ (as 4T5)	$1$	$1$
	3.2e8_11e2_47e2.6t8.2c1	$3$	$ 2^{8} \cdot 11^{2} \cdot 47^{2}$	$x^{4} - 2 x^{3} - 3 x^{2} - 2 x - 3$	$S_4$ (as 4T5)	$1$	$-1$
*	4.2e6_11_47.8t39.2c1	$4$	$ 2^{6} \cdot 11 \cdot 47 $	$x^{8} + x^{6} - 2 x^{4} - 3 x^{2} + 1$	$C_2^3:S_4$ (as 8T39)	$1$	$2$
	4.2e10_11e3_47e3.8t39.2c1	$4$	$ 2^{10} \cdot 11^{3} \cdot 47^{3}$	$x^{8} + x^{6} - 2 x^{4} - 3 x^{2} + 1$	$C_2^3:S_4$ (as 8T39)	$1$	$-2$
	6.2e14_11e3_47e3.8t34.1c1	$6$	$ 2^{14} \cdot 11^{3} \cdot 47^{3}$	$x^{8} - 4 x^{7} + 26 x^{6} - 48 x^{5} + 187 x^{4} - 160 x^{3} + 642 x^{2} - 116 x + 1358$	$V_4^2:S_3$ (as 8T34)	$1$	$0$
	8.2e16_11e4_47e4.24t333.3c1	$8$	$ 2^{16} \cdot 11^{4} \cdot 47^{4}$	$x^{8} + x^{6} - 2 x^{4} - 3 x^{2} + 1$	$C_2^3:S_4$ (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 *.