Global number field 8.0.5955980625.3

• Label: 8.0.5955980625.3
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{4}\cdot 5^{4}\cdot 7^{6}$
• Root discriminant: $16.67$
• Ramified primes: $3, 5, 7$
• Class number: $2$
• Class group: $[2]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 4 x^{6} - 2 x^{5} - 28 x^{4} + 71 x^{3} - 6 x^{2} - 130 x + 109 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(5955980625=3^{4}\cdot 5^{4}\cdot 7^{6}\)
Root discriminant:		$16.67$
Ramified primes:		$3, 5, 7$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{12} a^{6} + \frac{1}{12} a^{5} + \frac{1}{12} a^{4} - \frac{5}{12} a^{3} + \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{5}{12}$, $\frac{1}{408} a^{7} + \frac{1}{51} a^{6} + \frac{23}{102} a^{5} + \frac{97}{204} a^{4} + \frac{11}{68} a^{3} - \frac{19}{408} a^{2} + \frac{193}{408} a - \frac{47}{408}$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 19.7261051853 \)

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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-35}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}, \sqrt{21})\), 4.0.25725.2 x2, 4.2.15435.2 x2

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

Sibling fields

Degree 4 siblings:

4.2.15435.2, 4.0.25725.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	R	R	R	${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/17.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$5$	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$7$	7.8.6.2	$x^{8} - 49 x^{4} + 3969$	$4$	$2$	$6$	$D_4$	$[\ ]_{4}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_7.2t1.1c1	$1$	$ 3 \cdot 7 $	$x^{2} - x - 5$	$C_2$ (as 2T1)	$1$	$1$
*	1.5_7.2t1.1c1	$1$	$ 5 \cdot 7 $	$x^{2} - x + 9$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_5.2t1.1c1	$1$	$ 3 \cdot 5 $	$x^{2} - x + 4$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.3_5_7e2.4t3.6c1	$2$	$ 3 \cdot 5 \cdot 7^{2}$	$x^{8} - 3 x^{7} + 4 x^{6} - 2 x^{5} - 28 x^{4} + 71 x^{3} - 6 x^{2} - 130 x + 109$	$D_4$ (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 *.