Global number field 8.2.1871773696.1

• Label: 8.2.1871773696.1
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,2^{16}\cdot 13^{4}$
• Root discriminant: $14.42$
• Ramified primes: $2, 13$
• Class number: $1$
• Class group: Trivial
• Galois group: $Z_8 : Z_8^\times$ (as 8T15)

Normalized defining polynomial

\( x^{8} - 2 x^{6} - 8 x^{4} + 22 x^{2} - 9 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-1871773696=-\,2^{16}\cdot 13^{4}\)
Root discriminant:		$14.42$
Ramified primes:		$2, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{4}$, $\frac{1}{12} a^{7} + \frac{1}{12} a^{5} + \frac{1}{12} a^{3} + \frac{1}{12} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{7}{4} a^{2} + \frac{11}{4} \),  \( \frac{1}{12} a^{7} - \frac{5}{12} a^{5} - \frac{11}{12} a^{3} + \frac{43}{12} a \),  \( \frac{1}{2} a^{7} + \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{9}{2} a^{3} - \frac{9}{2} a^{2} + \frac{13}{2} a + \frac{11}{2} \),  \( \frac{1}{2} a^{7} - \frac{3}{4} a^{6} - a^{5} + \frac{3}{4} a^{4} - \frac{9}{2} a^{3} + \frac{25}{4} a^{2} + 11 a - \frac{37}{4} \)
Regulator:		\( 61.0598314827 \)

Galois group

$C_8:C_2^2$ (as 8T15):

A solvable group of order 32

The 11 conjugacy class representatives for $Z_8 : Z_8^\times$

Character table for $Z_8 : Z_8^\times$

Intermediate fields

\(\Q(\sqrt{13}) \), 4.2.2704.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure:	data not computed
Degree 16 siblings:	data not computed
Arithmetically equvalently sibling:	8.2.1871773696.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	R	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.8.0.1}{8} }$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.8.0.1}{8} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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)	$1$	$-1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
*	1.13.2t1.1c1	$1$	$ 13 $	$x^{2} - x - 3$	$C_2$ (as 2T1)	$1$	$1$
	1.2e3_13.2t1.2c1	$1$	$ 2^{3} \cdot 13 $	$x^{2} + 26$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_13.2t1.1c1	$1$	$ 2^{2} \cdot 13 $	$x^{2} + 13$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_13.2t1.1c1	$1$	$ 2^{3} \cdot 13 $	$x^{2} - 26$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e4_13.4t3.1c1	$2$	$ 2^{4} \cdot 13 $	$x^{4} - 3 x^{2} - 1$	$D_{4}$ (as 4T3)	$1$	$0$
	2.2e6_13.4t3.1c1	$2$	$ 2^{6} \cdot 13 $	$x^{4} - 6 x^{2} - 4$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.2e12_13e2.8t15.1c1	$4$	$ 2^{12} \cdot 13^{2}$	$x^{8} - 2 x^{6} - 8 x^{4} + 22 x^{2} - 9$	$Z_8 : Z_8^\times$ (as 8T15)	$1$	$0$

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 *.