Global number field 8.0.1124864000.1

• Label: 8.0.1124864000.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{12}\cdot 5^{3}\cdot 13^{3}$
• Root discriminant: $13.53$
• Ramified primes: $2, 5, 13$
• Class number: $2$
• Class group: $[2]$
• Galois group: $D_{8}$ (as 8T6)

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 6 x^{6} + 14 x^{5} + 11 x^{4} - 46 x^{3} + 44 x^{2} - 12 x + 1 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1124864000=2^{12}\cdot 5^{3}\cdot 13^{3}\)
Root discriminant:		$13.53$
Ramified primes:		$2, 5, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} - \frac{2}{5} a^{4} + \frac{1}{5} a^{3} - \frac{2}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{35} a^{7} + \frac{1}{35} a^{6} - \frac{3}{35} a^{5} + \frac{1}{7} a^{4} - \frac{9}{35} a^{3} - \frac{3}{35} a^{2} - \frac{12}{35}$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{74}{35} a^{7} + \frac{136}{35} a^{6} + \frac{467}{35} a^{5} - \frac{193}{7} a^{4} - \frac{979}{35} a^{3} + \frac{3267}{35} a^{2} - 77 a + \frac{398}{35} \) (order $4$)
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:		\( 32.1335288983 \)

Galois group

$D_8$ (as 8T6):

A solvable group of order 16

The 7 conjugacy class representatives for $D_{8}$

Character table for $D_{8}$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.1040.1

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 8 sibling:	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	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/11.8.0.1}{8} }$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.8.0.1}{8} }$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.8.12.14	$x^{8} + 12 x^{4} + 144$	$4$	$2$	$12$	$D_4$	$[2, 2]^{2}$
$5$	$\Q_{5}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{5}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.2.1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$13$	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.5_13.2t1.1c1	$1$	$ 5 \cdot 13 $	$x^{2} - x - 16$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_5_13.2t1.1c1	$1$	$ 2^{2} \cdot 5 \cdot 13 $	$x^{2} + 65$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e2_5_13.4t3.4c1	$2$	$ 2^{2} \cdot 5 \cdot 13 $	$x^{4} - 3 x^{2} - 2 x + 5$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2e4_5_13.8t6.3c1	$2$	$ 2^{4} \cdot 5 \cdot 13 $	$x^{8} - 2 x^{7} - 6 x^{6} + 14 x^{5} + 11 x^{4} - 46 x^{3} + 44 x^{2} - 12 x + 1$	$D_{8}$ (as 8T6)	$1$	$0$
*	2.2e4_5_13.8t6.3c2	$2$	$ 2^{4} \cdot 5 \cdot 13 $	$x^{8} - 2 x^{7} - 6 x^{6} + 14 x^{5} + 11 x^{4} - 46 x^{3} + 44 x^{2} - 12 x + 1$	$D_{8}$ (as 8T6)	$1$	$0$

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 *.