Global number field 8.0.17555190016.3

• Label: 8.0.17555190016.3
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 7^{4}\cdot 13^{4}$
• Root discriminant: $19.08$
• Ramified primes: $2, 7, 13$
• Class number: $2$
• Class group: $[2]$
• Galois group: $((C_2 \times D_4): C_2):C_3$ (as 8T32)

Normalized defining polynomial

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

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(17555190016=2^{8}\cdot 7^{4}\cdot 13^{4}\)
Root discriminant:		$19.08$
Ramified primes:		$2, 7, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{4} - \frac{1}{7} a^{2} + \frac{1}{7}$, $\frac{1}{14} a^{7} + \frac{5}{14} a^{5} + \frac{3}{7} a^{3} + \frac{1}{14} a$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{3}{7} a^{6} - \frac{13}{7} a^{4} + \frac{25}{7} a^{2} + \frac{17}{7} \),  \( \frac{1}{14} a^{7} + \frac{1}{7} a^{6} - \frac{9}{14} a^{5} - \frac{2}{7} a^{4} + \frac{10}{7} a^{3} + \frac{6}{7} a^{2} - \frac{27}{14} a + \frac{8}{7} \),  \( \frac{2}{7} a^{7} + \frac{6}{7} a^{6} - \frac{4}{7} a^{5} - \frac{26}{7} a^{4} - \frac{9}{7} a^{3} + \frac{57}{7} a^{2} + \frac{65}{7} a + \frac{27}{7} \)
Regulator:		\( 130.67439539 \)

Galois group

$C_2^3:A_4$ (as 8T32):

A solvable group of order 96

The 11 conjugacy class representatives for $((C_2 \times D_4): C_2):C_3$

Character table for $((C_2 \times D_4): C_2):C_3$

Intermediate fields

4.0.8281.1

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

Sibling fields

Degree 8 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 32 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	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$	R	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$	2.2.2.2	$x^{2} + 2 x - 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.6.6.6	$x^{6} - 13 x^{4} + 7 x^{2} - 3$	$2$	$3$	$6$	$A_4\times C_2$	$[2, 2, 2]^{3}$
$7$	7.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	7.6.4.3	$x^{6} + 56 x^{3} + 1323$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$13$	13.8.4.1	$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
	3.2e6_7e2.4t4.1c1	$3$	$ 2^{6} \cdot 7^{2}$	$x^{4} - 2 x^{3} + 2 x^{2} + 2$	$A_4$ (as 4T4)	$1$	$-1$
	3.2e6_7e2_13e2.4t4.3c1	$3$	$ 2^{6} \cdot 7^{2} \cdot 13^{2}$	$x^{4} - 2 x^{3} + 6 x^{2} + 8 x + 42$	$A_4$ (as 4T4)	$1$	$-1$
	3.2e6_7e2_13e2.4t4.2c1	$3$	$ 2^{6} \cdot 7^{2} \cdot 13^{2}$	$x^{4} - 2 x^{3} - 14 x^{2} + 2 x + 27$	$A_4$ (as 4T4)	$1$	$3$
	3.2e6_7e2_13e2.4t4.1c1	$3$	$ 2^{6} \cdot 7^{2} \cdot 13^{2}$	$x^{4} - 2 x^{3} + 2 x^{2} + 12 x + 10$	$A_4$ (as 4T4)	$1$	$-1$
*	3.7e2_13e2.4t4.1c1	$3$	$ 7^{2} \cdot 13^{2}$	$x^{4} - x^{3} + 5 x^{2} - 4 x + 3$	$A_4$ (as 4T4)	$1$	$-1$
*	4.2e8_7e2_13e2.8t32.4c1	$4$	$ 2^{8} \cdot 7^{2} \cdot 13^{2}$	$x^{8} - 5 x^{6} + 12 x^{4} - 3 x^{2} + 4$	$((C_2 \times D_4): C_2):C_3$ (as 8T32)	$1$	$0$
	4.2e8_7e3_13e2.24t97.4c1	$4$	$ 2^{8} \cdot 7^{3} \cdot 13^{2}$	$x^{8} - 5 x^{6} + 12 x^{4} - 3 x^{2} + 4$	$((C_2 \times D_4): C_2):C_3$ (as 8T32)	$0$	$0$
	4.2e8_7e3_13e2.24t97.4c2	$4$	$ 2^{8} \cdot 7^{3} \cdot 13^{2}$	$x^{8} - 5 x^{6} + 12 x^{4} - 3 x^{2} + 4$	$((C_2 \times D_4): C_2):C_3$ (as 8T32)	$0$	$0$

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 *.