Global number field 8.2.210093400576.2

• Label: 8.2.210093400576.2
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,2^{9}\cdot 17^{7}$
• Root discriminant: $26.02$
• Ramified primes: $2, 17$
• Class number: $4$
• Class group: $[4]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 17 x^{6} + 17 x^{4} + 17 x^{2} - 34 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-210093400576=-\,2^{9}\cdot 17^{7}\)
Root discriminant:		$26.02$
Ramified primes:		$2, 17$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{152} a^{6} - \frac{3}{76} a^{4} - \frac{1}{2} a^{3} + \frac{27}{152} a^{2} - \frac{33}{76}$, $\frac{1}{152} a^{7} - \frac{3}{76} a^{5} + \frac{27}{152} a^{3} - \frac{1}{2} a^{2} - \frac{33}{76} a$

Class group and class number

$C_{4}$, which has order $4$

Unit group

Rank:		$4$
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:		\( 217.459624065 \)

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

\(\Q(\sqrt{17}) \), 4.2.39304.1

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	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.2.3.4	$x^{2} + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.2.0.1	$x^{2} - x + 1$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	2.4.6.2	$x^{4} - 2 x^{2} + 4$	$2$	$2$	$6$	$C_2^2$	$[3]^{2}$
$17$	17.8.7.4	$x^{8} - 12393$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
	1.2e3_17.2t1.2c1	$1$	$ 2^{3} \cdot 17 $	$x^{2} + 34$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e3_17e2.4t3.3c1	$2$	$ 2^{3} \cdot 17^{2}$	$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 50$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2e3_17e2.8t8.1c1	$2$	$ 2^{3} \cdot 17^{2}$	$x^{8} - 17 x^{6} + 17 x^{4} + 17 x^{2} - 34$	$QD_{16}$ (as 8T8)	$0$	$0$
*	2.2e3_17e2.8t8.1c2	$2$	$ 2^{3} \cdot 17^{2}$	$x^{8} - 17 x^{6} + 17 x^{4} + 17 x^{2} - 34$	$QD_{16}$ (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 *.