Global number field 8.0.1686616064.2

• Label: 8.0.1686616064.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{11}\cdot 7^{7}$
• Root discriminant: $14.24$
• Ramified primes: $2, 7$
• Class number: $1$
• Class group: Trivial
• Galois group: $Z_8 : Z_8^\times$ (as 8T15)

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 7 x^{4} - 14 x^{3} + 12 x + 4 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1686616064=2^{11}\cdot 7^{7}\)
Root discriminant:		$14.24$
Ramified primes:		$2, 7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a^{6} - a^{5} - a^{3} + 6 a^{2} - 9 a - 3 \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + 3 a^{3} - \frac{1}{2} a^{2} - 5 a - 2 \),  \( a^{7} - a^{5} - a^{4} + 6 a^{3} - 5 a^{2} - 10 a - 3 \)
Regulator:		\( 169.332018111 \)

Galois group

$C_8:C_2^2$ (as 8T15):

A solvable group of order 32

The 11 conjugacy class representatives for $Z_8 : Z_8^\times$

Character table for $Z_8 : Z_8^\times$

Intermediate fields

\(\Q(\sqrt{-7}) \), 4.0.2744.1

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 16 siblings:	data not computed
Arithmetically equvalently sibling:	8.0.1686616064.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/17.8.0.1}{8} }$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.2.3.3	$x^{2} + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.3.3	$x^{2} + 2$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.2.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.2.3.1	$x^{2} + 14$	$2$	$1$	$3$	$C_2$	$[3]$
$7$	7.8.7.1	$x^{8} + 14$	$8$	$1$	$7$	$D_{8}$	$[\ ]_{8}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_7.2t1.2c1	$1$	$ 2^{3} \cdot 7 $	$x^{2} + 14$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_7.2t1.1c1	$1$	$ 2^{3} \cdot 7 $	$x^{2} - 14$	$C_2$ (as 2T1)	$1$	$1$
*	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_7.2t1.1c1	$1$	$ 2^{2} \cdot 7 $	$x^{2} - 7$	$C_2$ (as 2T1)	$1$	$1$
	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e3_7e2.4t3.2c1	$2$	$ 2^{3} \cdot 7^{2}$	$x^{4} - x^{3} + 3 x^{2} + 3 x + 2$	$D_{4}$ (as 4T3)	$1$	$0$
	2.2e5_7e2.4t3.2c1	$2$	$ 2^{5} \cdot 7^{2}$	$x^{4} + 7 x^{2} + 14$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.2e8_7e4.8t15.1c1	$4$	$ 2^{8} \cdot 7^{4}$	$x^{8} - 2 x^{7} + 7 x^{4} - 14 x^{3} + 12 x + 4$	$Z_8 : Z_8^\times$ (as 8T15)	$1$	$0$

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 *.