Global number field 8.0.9437184.1

• Label: 8.0.9437184.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{20}\cdot 3^{2}$
• Root discriminant: $7.44$
• Ramified primes: $2, 3$
• Class number: $1$
• Class group: Trivial
• Galois group: $Q_8:C_2$ (as 8T11)

Normalized defining polynomial

\( x^{8} - 4 x^{6} - 4 x^{5} + 6 x^{4} + 16 x^{3} + 16 x^{2} + 8 x + 2 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(9437184=2^{20}\cdot 3^{2}\)
Root discriminant:		$7.44$
Ramified primes:		$2, 3$
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:		\( -a^{7} + 5 a^{5} + 3 a^{4} - 10 a^{3} - 15 a^{2} - 8 a - 1 \) (order $8$)
Fundamental units:		\( a^{7} + 3 a^{6} - 8 a^{5} - 14 a^{4} + 11 a^{3} + 39 a^{2} + 31 a + 11 \),  \( a^{7} - 2 a^{6} - 2 a^{5} + 3 a^{4} + 6 a^{3} + a^{2} - a - 1 \),  \( 2 a^{7} - 10 a^{5} - 5 a^{4} + 17 a^{3} + 30 a^{2} + 19 a + 5 \)
Regulator:		\( 7.01287439394 \)

Galois group

$D_4:C_2$ (as 8T11):

A solvable group of order 16

The 10 conjugacy class representatives for $Q_8:C_2$

Character table for $Q_8:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\zeta_{8})\)

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

Sibling fields

Galois closure:	data not computed
Degree 8 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	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.8.20.55	$x^{8} + 4 x^{6} + 4 x^{5} + 6 x^{4} + 2$	$8$	$1$	$20$	$Q_8:C_2$	$[2, 3, 3]^{2}$
$3$	3.4.0.1	$x^{4} - x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	3.4.2.2	$x^{4} - 3 x^{2} + 18$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$

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_3.2t1.1c1	$1$	$ 2^{2} \cdot 3 $	$x^{2} - 3$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_3.2t1.2c1	$1$	$ 2^{3} \cdot 3 $	$x^{2} + 6$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_3.2t1.1c1	$1$	$ 2^{3} \cdot 3 $	$x^{2} - 6$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e6_3.8t11.3c1	$2$	$ 2^{6} \cdot 3 $	$x^{8} - 4 x^{6} - 4 x^{5} + 6 x^{4} + 16 x^{3} + 16 x^{2} + 8 x + 2$	$Q_8:C_2$ (as 8T11)	$0$	$0$
*	2.2e6_3.8t11.3c2	$2$	$ 2^{6} \cdot 3 $	$x^{8} - 4 x^{6} - 4 x^{5} + 6 x^{4} + 16 x^{3} + 16 x^{2} + 8 x + 2$	$Q_8:C_2$ (as 8T11)	$0$	$0$

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 *.