Global number field 8.0.378956381627633.1

• Label: 8.0.378956381627633.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $17^{7}\cdot 31^{4}$
• Root discriminant: $66.42$
• Ramified primes: $17, 31$
• Class number: $306$
• Class group: $[3, 102]$
• Galois group: $C_8$ (as 8T1)

Normalized defining polynomial

\( x^{8} - x^{7} + 129 x^{6} - 130 x^{5} + 3823 x^{4} - 7082 x^{3} + 39566 x^{2} - 84180 x + 136001 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(378956381627633=17^{7}\cdot 31^{4}\)
Root discriminant:		$66.42$
Ramified primes:		$17, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(527=17\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{527}(1,·)$, $\chi_{527}(497,·)$, $\chi_{527}(495,·)$, $\chi_{527}(433,·)$, $\chi_{527}(404,·)$, $\chi_{527}(373,·)$, $\chi_{527}(247,·)$, $\chi_{527}(185,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6} - \frac{2}{13} a^{5} - \frac{5}{13} a^{4} + \frac{4}{13} a^{3} + \frac{1}{13} a^{2} + \frac{4}{13} a - \frac{3}{13}$, $\frac{1}{79218665073197} a^{7} + \frac{994098796316}{79218665073197} a^{6} + \frac{558040874133}{6093743467169} a^{5} + \frac{35957529759096}{79218665073197} a^{4} - \frac{15006542934639}{79218665073197} a^{3} - \frac{39312357447875}{79218665073197} a^{2} - \frac{1564589944643}{6093743467169} a - \frac{1529410003523}{79218665073197}$

Class group and class number

$C_{3}\times C_{102}$, which has order $306$

Unit group

Rank:		$3$
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:		\( 27.6959098582 \)

Galois group

$C_8$ (as 8T1):

A cyclic group of order 8

The 8 conjugacy class representatives for $C_8$

Character table for $C_8$

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.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	R	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$17$	17.8.7.1	$x^{8} - 1377$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$
$31$	31.8.4.2	$x^{8} - 59582 x^{2} + 15699857$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.17_31.8t1.1c1	$1$	$ 17 \cdot 31 $	$x^{8} - x^{7} + 129 x^{6} - 130 x^{5} + 3823 x^{4} - 7082 x^{3} + 39566 x^{2} - 84180 x + 136001$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.4t1.1c1	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.17_31.8t1.1c2	$1$	$ 17 \cdot 31 $	$x^{8} - x^{7} + 129 x^{6} - 130 x^{5} + 3823 x^{4} - 7082 x^{3} + 39566 x^{2} - 84180 x + 136001$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
*	1.17_31.8t1.1c3	$1$	$ 17 \cdot 31 $	$x^{8} - x^{7} + 129 x^{6} - 130 x^{5} + 3823 x^{4} - 7082 x^{3} + 39566 x^{2} - 84180 x + 136001$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.4t1.1c2	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.17_31.8t1.1c4	$1$	$ 17 \cdot 31 $	$x^{8} - x^{7} + 129 x^{6} - 130 x^{5} + 3823 x^{4} - 7082 x^{3} + 39566 x^{2} - 84180 x + 136001$	$C_8$ (as 8T1)	$0$	$-1$

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 *.