Global number field 8.8.105226698752.1

• Label: 8.8.105226698752.1
• Degree: $8$
• Signature: $[8, 0]$
• Discriminant: $2^{31}\cdot 7^{2}$
• Root discriminant: $23.87$
• Ramified primes: $2, 7$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_8:C_2$ (as 8T7)

Normalized defining polynomial

\( x^{8} - 24 x^{6} + 180 x^{4} - 448 x^{2} + 98 \)

Invariants

Degree:		$8$
Signature:		$[8, 0]$
Discriminant:		\(105226698752=2^{31}\cdot 7^{2}\)
Root discriminant:		$23.87$
Ramified primes:		$2, 7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{217} a^{6} + \frac{67}{217} a^{4} - \frac{16}{217} a^{2} + \frac{7}{31}$, $\frac{1}{217} a^{7} + \frac{67}{217} a^{5} - \frac{16}{217} a^{3} + \frac{7}{31} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$7$
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:		\( 1043.69189352 \)

Galois group

$OD_{16}$ (as 8T7):

A solvable group of order 16

The 10 conjugacy class representatives for $C_8:C_2$

Character table for $C_8:C_2$

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\)

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

Sibling fields

Galois closure:

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	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.8.0.1}{8} }$	R	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.8.31.17	$x^{8} + 16 x^{5} + 12 x^{4} + 16 x^{3} + 2$	$8$	$1$	$31$	$C_8:C_2$	$[2, 3, 4, 5]$
$7$	7.4.2.2	$x^{4} - 7 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.0.1	$x^{4} + x^{2} - 3 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e3_7.2t1.1c1	$1$	$ 2^{3} \cdot 7 $	$x^{2} - 14$	$C_2$ (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)	$1$	$1$
	1.2e4_7.4t1.1c1	$1$	$ 2^{4} \cdot 7 $	$x^{4} - 28 x^{2} + 98$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e4.4t1.1c1	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e4.4t1.1c2	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
	1.2e4_7.4t1.1c2	$1$	$ 2^{4} \cdot 7 $	$x^{4} - 28 x^{2} + 98$	$C_4$ (as 4T1)	$0$	$1$
*	2.2e10_7.8t7.1c1	$2$	$ 2^{10} \cdot 7 $	$x^{8} - 24 x^{6} + 180 x^{4} - 448 x^{2} + 98$	$C_8:C_2$ (as 8T7)	$0$	$2$
*	2.2e10_7.8t7.1c2	$2$	$ 2^{10} \cdot 7 $	$x^{8} - 24 x^{6} + 180 x^{4} - 448 x^{2} + 98$	$C_8:C_2$ (as 8T7)	$0$	$2$

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 *.