Global number field 8.0.58247424.1

• Label: 8.0.58247424.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 3^{4}\cdot 53^{2}$
• Root discriminant: $9.35$
• Ramified primes: $2, 3, 53$
• Class number: $1$
• Class group: Trivial
• Galois group: $V_4^2:(S_3\times C_2)$ (as 8T41)

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 10 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 1 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(58247424=2^{8}\cdot 3^{4}\cdot 53^{2}\)
Root discriminant:		$9.35$
Ramified primes:		$2, 3, 53$
This field is not Galois over $\Q$.
This is not a CM field.

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

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -2 a^{7} + 4 a^{6} - 12 a^{5} + 7 a^{4} - 18 a^{3} + 7 a^{2} - 8 a + 4 \) (order $6$)
Fundamental units:		\( a \),  \( a^{7} - 4 a^{6} + 9 a^{5} - 12 a^{4} + 8 a^{3} - 12 a^{2} + 4 a \),  \( a^{5} - 2 a^{4} + 5 a^{3} - 2 a^{2} + 4 a - 3 \)
Regulator:		\( 12.905424783 \)

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	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$	${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$	2.8.8.3	$x^{8} + 2 x^{7} + 2 x^{6} + 16$	$2$	$4$	$8$	$C_2^3: C_4$	$[2, 2, 2]^{4}$
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$53$	53.4.0.1	$x^{4} - x + 18$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	53.4.2.2	$x^{4} - 53 x^{2} + 14045$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2_3_53.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 53 $	$x^{2} - 159$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_53.2t1.1c1	$1$	$ 2^{2} \cdot 53 $	$x^{2} + 53$	$C_2$ (as 2T1)	$1$	$-1$
	2.2e2_3e2_53.6t3.2c1	$2$	$ 2^{2} \cdot 3^{2} \cdot 53 $	$x^{6} - 2 x^{5} - 13 x^{4} + 24 x^{3} + 39 x^{2} - 70 x - 134$	$D_{6}$ (as 6T3)	$1$	$0$
	2.2e2_53.3t2.1c1	$2$	$ 2^{2} \cdot 53 $	$x^{3} - x^{2} + 4 x - 2$	$S_3$ (as 3T2)	$1$	$0$
	3.2e6_53e2.6t8.3c1	$3$	$ 2^{6} \cdot 53^{2}$	$x^{4} - x^{2} - 2 x + 1$	$S_4$ (as 4T5)	$1$	$-1$
	3.2e4_3e3_53.6t11.1c1	$3$	$ 2^{4} \cdot 3^{3} \cdot 53 $	$x^{6} + 3 x^{4} - 117 x^{2} - 1431$	$S_4\times C_2$ (as 6T11)	$1$	$-1$
	3.2e4_53.4t5.1c1	$3$	$ 2^{4} \cdot 53 $	$x^{4} - x^{2} - 2 x + 1$	$S_4$ (as 4T5)	$1$	$1$
	3.2e6_3e3_53e2.6t11.1c1	$3$	$ 2^{6} \cdot 3^{3} \cdot 53^{2}$	$x^{6} + 3 x^{4} - 117 x^{2} - 1431$	$S_4\times C_2$ (as 6T11)	$1$	$1$
*	6.2e8_3e3_53e2.8t41.1c1	$6$	$ 2^{8} \cdot 3^{3} \cdot 53^{2}$	$x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 10 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$
	6.2e12_3e3_53e3.8t41.1c1	$6$	$ 2^{12} \cdot 3^{3} \cdot 53^{3}$	$x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 10 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$2$
	6.2e12_3e3_53e3.12t108.1c1	$6$	$ 2^{12} \cdot 3^{3} \cdot 53^{3}$	$x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 10 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$-2$
	6.2e12_3e3_53e4.12t108.1c1	$6$	$ 2^{12} \cdot 3^{3} \cdot 53^{4}$	$x^{8} - 2 x^{7} + 6 x^{6} - 4 x^{5} + 10 x^{4} - 6 x^{3} + 5 x^{2} - 4 x + 1$	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$

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 *.