Global number field 8.4.26261675072.1

• Label: 8.4.26261675072.1
• Degree: $8$
• Signature: $[4, 2]$
• Discriminant: $2^{6}\cdot 17^{7}$
• Root discriminant: $20.06$
• Ramified primes: $2, 17$
• Class number: $2$
• Class group: $[2]$
• Galois group: $C_8:C_2$ (as 8T7)

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 5 x^{6} - 25 x^{5} + 42 x^{4} - 33 x^{3} + 3 x^{2} + 10 x - 1 \)

Invariants

Degree:		$8$
Signature:		$[4, 2]$
Discriminant:		\(26261675072=2^{6}\cdot 17^{7}\)
Root discriminant:		$20.06$
Ramified primes:		$2, 17$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3991} a^{7} - \frac{1509}{3991} a^{6} + \frac{1680}{3991} a^{5} + \frac{189}{3991} a^{4} - \frac{1231}{3991} a^{3} - \frac{1962}{3991} a^{2} + \frac{1435}{3991} a - \frac{1969}{3991}$

Class group and class number

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

Unit group

Rank:		$5$
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:		\( 83.4829009987 \)

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

\(\Q(\sqrt{17}) \), 4.4.4913.1

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	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.8.0.1}{8} }$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.4.6.4	$x^{4} - 2 x^{2} + 20$	$2$	$2$	$6$	$C_4$	$[3]^{2}$
	2.4.0.1	$x^{4} - x + 1$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$17$	17.8.7.1	$x^{8} - 1377$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_17.2t1.2c1	$1$	$ 2^{3} \cdot 17 $	$x^{2} + 34$	$C_2$ (as 2T1)	$1$	$-1$
*	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
*	1.17.4t1.1c1	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
	1.2e3_17.4t1.2c1	$1$	$ 2^{3} \cdot 17 $	$x^{4} + 34 x^{2} + 272$	$C_4$ (as 4T1)	$0$	$-1$
	1.2e3_17.4t1.2c2	$1$	$ 2^{3} \cdot 17 $	$x^{4} + 34 x^{2} + 272$	$C_4$ (as 4T1)	$0$	$-1$
*	1.17.4t1.1c2	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	2.2e3_17e2.8t7.1c1	$2$	$ 2^{3} \cdot 17^{2}$	$x^{8} - 3 x^{7} + 5 x^{6} - 25 x^{5} + 42 x^{4} - 33 x^{3} + 3 x^{2} + 10 x - 1$	$C_8:C_2$ (as 8T7)	$0$	$0$
*	2.2e3_17e2.8t7.1c2	$2$	$ 2^{3} \cdot 17^{2}$	$x^{8} - 3 x^{7} + 5 x^{6} - 25 x^{5} + 42 x^{4} - 33 x^{3} + 3 x^{2} + 10 x - 1$	$C_8:C_2$ (as 8T7)	$0$	$0$

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 *.