Global number field 8.2.124952510464.2

• Label: 8.2.124952510464.2
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,2^{22}\cdot 31^{3}$
• Root discriminant: $24.38$
• Ramified primes: $2, 31$
• Class number: $1$
• Class group: Trivial
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} - 124 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-124952510464=-\,2^{22}\cdot 31^{3}\)
Root discriminant:		$24.38$
Ramified primes:		$2, 31$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{16} a^{4} + \frac{1}{4} a^{2} + \frac{1}{8}$, $\frac{1}{16} a^{5} + \frac{1}{4} a^{3} + \frac{1}{8} a$, $\frac{1}{32} a^{6} + \frac{1}{16} a^{2} - \frac{1}{4}$, $\frac{1}{32} a^{7} + \frac{1}{16} a^{3} - \frac{1}{4} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{8} a^{4} - \frac{1}{2} a^{2} + \frac{5}{4} \),  \( \frac{1}{16} a^{4} + \frac{1}{4} a^{2} + \frac{1}{8} \),  \( \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{3}{8} a^{5} - \frac{1}{4} a^{4} + \frac{5}{8} a^{3} + \frac{7}{8} a^{2} + \frac{11}{4} a + 1 \),  \( \frac{1}{4} a^{7} + \frac{1}{2} a^{6} - \frac{7}{8} a^{5} - \frac{5}{4} a^{4} + 3 a^{3} + 6 a^{2} + \frac{33}{4} a + \frac{33}{2} \)
Regulator:		\( 595.966172184 \)

Galois group

$SD_{16}$ (as 8T8):

A solvable group of order 16

The 7 conjugacy class representatives for $QD_{16}$

Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.2.1984.1

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

Sibling fields

Galois closure:

data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.8.0.1}{8} }$	R	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.4.11.3	$x^{4} + 4 x^{2} + 18$	$4$	$1$	$11$	$C_4$	$[3, 4]$
	2.4.11.3	$x^{4} + 4 x^{2} + 18$	$4$	$1$	$11$	$C_4$	$[3, 4]$
$31$	$\Q_{31}$	$x + 7$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{31}$	$x + 7$	$1$	$1$	$0$	Trivial	$[\ ]$
	31.2.1.2	$x^{2} + 217$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 217$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	31.2.1.2	$x^{2} + 217$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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)	$1$	$1$
	1.31.2t1.1c1	$1$	$ 31 $	$x^{2} - x + 8$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_31.2t1.2c1	$1$	$ 2^{3} \cdot 31 $	$x^{2} + 62$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e3_31.4t3.2c1	$2$	$ 2^{3} \cdot 31 $	$x^{4} - x^{3} + 4 x^{2} + 2 x + 4$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2e8_31.8t8.1c1	$2$	$ 2^{8} \cdot 31 $	$x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} - 124$	$QD_{16}$ (as 8T8)	$0$	$0$
*	2.2e8_31.8t8.1c2	$2$	$ 2^{8} \cdot 31 $	$x^{8} - 8 x^{6} + 20 x^{4} - 16 x^{2} - 124$	$QD_{16}$ (as 8T8)	$0$	$0$

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 *.