Global number field 8.0.10070523904.15

• Label: 8.0.10070523904.15
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{22}\cdot 7^{4}$
• Root discriminant: $17.80$
• Ramified primes: $2, 7$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 4 x^{7} - 4 x^{6} + 12 x^{5} + 88 x^{4} - 84 x^{3} - 420 x^{2} + 124 x + 961 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(10070523904=2^{22}\cdot 7^{4}\)
Root discriminant:		$17.80$
Ramified primes:		$2, 7$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{85} a^{6} - \frac{6}{17} a^{5} - \frac{31}{85} a^{4} + \frac{38}{85} a^{3} - \frac{23}{85} a^{2} + \frac{23}{85} a + \frac{33}{85}$, $\frac{1}{297755} a^{7} + \frac{554}{297755} a^{6} + \frac{105954}{297755} a^{5} + \frac{9219}{297755} a^{4} + \frac{127994}{297755} a^{3} - \frac{19614}{297755} a^{2} - \frac{3835}{59551} a + \frac{3572}{9605}$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{421}{59551} a^{7} + \frac{3832}{297755} a^{6} + \frac{4071}{59551} a^{5} + \frac{4128}{297755} a^{4} - \frac{162269}{297755} a^{3} - \frac{212571}{297755} a^{2} + \frac{576301}{297755} a + \frac{27306}{9605} \) (order $8$)
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:		\( 78.4698556172 \)

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

\(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{2}) \), \(\Q(\zeta_{8})\), 4.0.25088.2 x2, 4.2.50176.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.50176.1, 4.0.25088.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.8.22.87	$x^{8} + 4 x^{7} + 6 x^{4} + 12 x^{2} + 2$	$8$	$1$	$22$	$D_4$	$[2, 3, 7/2]$
$7$	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.2e7_7e2.4t3.9c1	$2$	$ 2^{7} \cdot 7^{2}$	$x^{8} - 4 x^{7} - 4 x^{6} + 12 x^{5} + 88 x^{4} - 84 x^{3} - 420 x^{2} + 124 x + 961$	$D_4$ (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 *.