Global number field 8.8.11719682839553.1

• Label: 8.8.11719682839553.1
• Degree: $8$
• Signature: $[8, 0]$
• Discriminant: $13^{4}\cdot 17^{7}$
• Root discriminant: $43.01$
• Ramified primes: $13, 17$
• Class number: $2$
• Class group: $[2]$
• Galois group: $C_8$ (as 8T1)

Normalized defining polynomial

\( x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753 \)

Invariants

Degree:		$8$
Signature:		$[8, 0]$
Discriminant:		\(11719682839553=13^{4}\cdot 17^{7}\)
Root discriminant:		$43.01$
Ramified primes:		$13, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(221=13\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{221}(1,·)$, $\chi_{221}(168,·)$, $\chi_{221}(77,·)$, $\chi_{221}(118,·)$, $\chi_{221}(183,·)$, $\chi_{221}(25,·)$, $\chi_{221}(155,·)$, $\chi_{221}(157,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{110344621} a^{7} + \frac{1225022}{110344621} a^{6} - \frac{6720152}{110344621} a^{5} + \frac{30030887}{110344621} a^{4} - \frac{38321305}{110344621} a^{3} - \frac{16181165}{110344621} a^{2} + \frac{8421575}{110344621} a - \frac{47262281}{110344621}$

Class group and class number

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

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:		\( 3630.8787207 \)

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	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	R	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.8.0.1}{8} }$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$13$	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$17$	17.8.7.3	$x^{8} - 17$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.13_17.8t1.1c1	$1$	$ 13 \cdot 17 $	$x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753$	$C_8$ (as 8T1)	$0$	$1$
*	1.17.4t1.1c1	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.13_17.8t1.1c2	$1$	$ 13 \cdot 17 $	$x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753$	$C_8$ (as 8T1)	$0$	$1$
*	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
*	1.13_17.8t1.1c3	$1$	$ 13 \cdot 17 $	$x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753$	$C_8$ (as 8T1)	$0$	$1$
*	1.17.4t1.1c2	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.13_17.8t1.1c4	$1$	$ 13 \cdot 17 $	$x^{8} - x^{7} - 58 x^{6} + 57 x^{5} + 831 x^{4} - 1285 x^{3} - 3070 x^{2} + 6889 x - 2753$	$C_8$ (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 *.