Global number field 8.0.5184000000.15

• Label: 8.0.5184000000.15
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{12}\cdot 3^{4}\cdot 5^{6}$
• Root discriminant: $16.38$
• Ramified primes: $2, 3, 5$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 31 x^{4} - 50 x^{3} + 2 x^{2} + 22 x + 121 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(5184000000=2^{12}\cdot 3^{4}\cdot 5^{6}\)
Root discriminant:		$16.38$
Ramified primes:		$2, 3, 5$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{33} a^{6} - \frac{1}{11} a^{5} + \frac{5}{33} a^{4} - \frac{5}{33} a^{3} + \frac{4}{33} a^{2} - \frac{2}{33} a - \frac{1}{3}$, $\frac{1}{5973} a^{7} + \frac{29}{1991} a^{6} + \frac{142}{5973} a^{5} + \frac{141}{1991} a^{4} - \frac{391}{5973} a^{3} - \frac{2689}{5973} a^{2} - \frac{357}{1991} a + \frac{53}{181}$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{10}{1991} a^{7} - \frac{35}{1991} a^{6} + \frac{153}{1991} a^{5} - \frac{295}{1991} a^{4} + \frac{615}{1991} a^{3} - \frac{645}{1991} a^{2} + \frac{1055}{1991} a - \frac{39}{181} \) (order $4$)
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:		\( 26.2773808012 \)

Galois group

$D_4$ (as 8T4):

A solvable group of order 8

The 5 conjugacy class representatives for $D_4$

Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(i, \sqrt{5})\), 4.0.72000.1 x2, 4.2.18000.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.18000.1, 4.0.72000.1

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	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$2$	2.8.12.14	$x^{8} + 12 x^{4} + 144$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
$3$	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$5$	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.2e4_3e2_5e2.4t3.5c1	$2$	$ 2^{4} \cdot 3^{2} \cdot 5^{2}$	$x^{8} - 4 x^{7} + 8 x^{6} - 10 x^{5} + 31 x^{4} - 50 x^{3} + 2 x^{2} + 22 x + 121$	$D_4$ (as 8T4)	$1$	$0$

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 *.