Global number field 8.0.171714816.5

• Label: 8.0.171714816.5
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 3^{4}\cdot 7^{2}\cdot 13^{2}$
• Root discriminant: $10.70$
• Ramified primes: $2, 3, 7, 13$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_2^3 : D_4 $ (as 8T22)

Normalized defining polynomial

\( x^{8} - 4 x^{7} - x^{6} + 16 x^{5} + x^{4} - 24 x^{3} - 12 x^{2} + 36 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(171714816=2^{8}\cdot 3^{4}\cdot 7^{2}\cdot 13^{2}\)
Root discriminant:		$10.70$
Ramified primes:		$2, 3, 7, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{12} a^{6} + \frac{1}{6} a^{5} + \frac{5}{12} a^{4} - \frac{1}{6} a^{3} - \frac{5}{12} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{72} a^{7} + \frac{1}{36} a^{6} + \frac{17}{72} a^{5} - \frac{7}{36} a^{4} + \frac{19}{72} a^{3} + \frac{1}{12} a^{2} - \frac{1}{12} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{17}{72} a^{7} + \frac{11}{18} a^{6} + \frac{83}{72} a^{5} - \frac{41}{18} a^{4} - \frac{263}{72} a^{3} + \frac{7}{6} a^{2} + \frac{47}{12} a + \frac{11}{2} \) (order $12$)
Fundamental units:		\( \frac{17}{72} a^{7} - \frac{11}{18} a^{6} - \frac{83}{72} a^{5} + \frac{41}{18} a^{4} + \frac{263}{72} a^{3} - \frac{7}{6} a^{2} - \frac{47}{12} a - \frac{13}{2} \),  \( \frac{5}{36} a^{7} - \frac{2}{9} a^{6} - \frac{23}{36} a^{5} + \frac{5}{9} a^{4} + \frac{59}{36} a^{3} + \frac{1}{3} a^{2} - \frac{5}{6} a - 2 \),  \( \frac{1}{72} a^{7} - \frac{1}{18} a^{6} + \frac{5}{72} a^{5} - \frac{11}{18} a^{4} + \frac{103}{72} a^{3} - \frac{1}{2} a^{2} + \frac{5}{12} a - \frac{3}{2} \)
Regulator:		\( 48.7827733585 \)

Galois group

$Q_8:C_2^2$ (as 8T22):

A solvable group of order 32

The 17 conjugacy class representatives for $C_2^3 : D_4 $

Character table for $C_2^3 : D_4 $

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\)

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 siblings:	data not computed
Degree 16 siblings:	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	R	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.4.4.1	$x^{4} + 8 x^{2} + 4$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
	2.4.4.1	$x^{4} + 8 x^{2} + 4$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
$3$	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$7$	7.4.2.2	$x^{4} - 7 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	7.4.0.1	$x^{4} + x^{2} - 3 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
$13$	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	13.2.1.1	$x^{2} - 13$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

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_7_13.2t1.1c1	$1$	$ 2^{2} \cdot 7 \cdot 13 $	$x^{2} - 91$	$C_2$ (as 2T1)	$1$	$1$
	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_13.2t1.1c1	$1$	$ 2^{2} \cdot 13 $	$x^{2} + 13$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_7_13.2t1.1c1	$1$	$ 3 \cdot 7 \cdot 13 $	$x^{2} - x - 68$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2_3.2t1.1c1	$1$	$ 2^{2} \cdot 3 $	$x^{2} - 3$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_3_7.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 7 $	$x^{2} + 21$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_13.2t1.1c1	$1$	$ 3 \cdot 13 $	$x^{2} - x + 10$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.7_13.2t1.1c1	$1$	$ 7 \cdot 13 $	$x^{2} - x + 23$	$C_2$ (as 2T1)	$1$	$-1$
	1.13.2t1.1c1	$1$	$ 13 $	$x^{2} - x - 3$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_7.2t1.1c1	$1$	$ 2^{2} \cdot 7 $	$x^{2} - 7$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_3_7_13.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 7 \cdot 13 $	$x^{2} + 273$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_3_13.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 13 $	$x^{2} - 39$	$C_2$ (as 2T1)	$1$	$1$
	1.3_7.2t1.1c1	$1$	$ 3 \cdot 7 $	$x^{2} - x - 5$	$C_2$ (as 2T1)	$1$	$1$
*	4.2e4_3e2_7e2_13e2.8t22.9c1	$4$	$ 2^{4} \cdot 3^{2} \cdot 7^{2} \cdot 13^{2}$	$x^{8} - 4 x^{7} - x^{6} + 16 x^{5} + x^{4} - 24 x^{3} - 12 x^{2} + 36$	$C_2^3 : D_4 $ (as 8T22)	$1$	$0$

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 *.