Global number field 8.0.61334280470528.11

• Label: 8.0.61334280470528.11
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{31}\cdot 13^{4}$
• Root discriminant: $52.90$
• Ramified primes: $2, 13$
• Class number: $162$
• Class group: $[9, 18]$
• Galois group: $C_8$ (as 8T1)

Normalized defining polynomial

\( x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(61334280470528=2^{31}\cdot 13^{4}\)
Root discriminant:		$52.90$
Ramified primes:		$2, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(416=2^{5}\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(259,·)$, $\chi_{416}(105,·)$, $\chi_{416}(363,·)$, $\chi_{416}(209,·)$, $\chi_{416}(51,·)$, $\chi_{416}(313,·)$, $\chi_{416}(155,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{13} a^{2}$, $\frac{1}{13} a^{3}$, $\frac{1}{169} a^{4}$, $\frac{1}{169} a^{5}$, $\frac{1}{2197} a^{6}$, $\frac{1}{2197} a^{7}$

Class group and class number

$C_{9}\times C_{18}$, which has order $162$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{169} a^{4} + \frac{4}{13} a^{2} + 1 \),  \( \frac{1}{13} a^{2} + 1 \),  \( \frac{1}{2197} a^{6} + \frac{6}{169} a^{4} + \frac{9}{13} a^{2} + 3 \)
Regulator:		\( 19.534360053 \)

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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{16})^+\)

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	R	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.8.0.1}{8} }$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.8.31.2	$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$	$8$	$1$	$31$	$C_8$	$[3, 4, 5]$
$13$	13.8.4.2	$x^{8} + 169 x^{4} - 2197 x^{2} + 57122$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e5_13.8t1.1c1	$1$	$ 2^{5} \cdot 13 $	$x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$	$C_8$ (as 8T1)	$0$	$-1$
*	1.2e4.4t1.1c1	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e5_13.8t1.1c2	$1$	$ 2^{5} \cdot 13 $	$x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$	$C_8$ (as 8T1)	$0$	$-1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e5_13.8t1.1c3	$1$	$ 2^{5} \cdot 13 $	$x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$	$C_8$ (as 8T1)	$0$	$-1$
*	1.2e4.4t1.1c2	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e5_13.8t1.1c4	$1$	$ 2^{5} \cdot 13 $	$x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$	$C_8$ (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 *.