Global number field 8.0.1662029824.2

• Label: 8.0.1662029824.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{12}\cdot 7^{4}\cdot 13^{2}$
• Root discriminant: $14.21$
• Ramified primes: $2, 7, 13$
• Class number: $1$
• Class group: Trivial
• Galois group: $((C_2 \times D_4): C_2):C_3$ (as 8T32)

Normalized defining polynomial

\( x^{8} - 2 x^{7} + 4 x^{6} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x + 2 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1662029824=2^{12}\cdot 7^{4}\cdot 13^{2}\)
Root discriminant:		$14.21$
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 \(a\))

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

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{5}{4} a^{7} - \frac{5}{4} a^{6} + \frac{13}{4} a^{5} + \frac{17}{4} a^{4} + \frac{1}{2} a^{3} + 9 a^{2} + \frac{9}{2} a - \frac{5}{2} \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{3}{2} a^{5} + \frac{3}{2} a^{4} + a^{3} + 4 a^{2} + 3 a \),  \( \frac{1}{2} a^{7} - \frac{1}{2} a^{6} + \frac{3}{2} a^{5} + \frac{3}{2} a^{4} + 5 a^{2} + a - 2 \)
Regulator:		\( 50.8070750037 \)

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

4.0.3136.1

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	${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$	${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$	R	${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$	R	${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$	2.4.6.7	$x^{4} + 2 x^{3} + 2 x^{2} + 2$	$4$	$1$	$6$	$A_4$	$[2, 2]^{3}$
	2.4.6.7	$x^{4} + 2 x^{3} + 2 x^{2} + 2$	$4$	$1$	$6$	$A_4$	$[2, 2]^{3}$
$7$	7.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	7.6.4.3	$x^{6} + 56 x^{3} + 1323$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$13$	13.4.2.2	$x^{4} - 13 x^{2} + 338$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
	13.4.0.1	$x^{4} + x^{2} - x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$

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_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)	$1$	$-1$
*	3.2e6_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)	$1$	$-1$
	3.7e2_13e2.4t4.1c1	$3$	$ 7^{2} \cdot 13^{2}$	$x^{4} - x^{3} + 5 x^{2} - 4 x + 3$	$A_4$ (as 4T4)	$1$	$-1$
*	4.2e6_7e2_13e2.8t32.2c1	$4$	$ 2^{6} \cdot 7^{2} \cdot 13^{2}$	$x^{8} - 2 x^{7} + 4 x^{6} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x + 2$	$((C_2 \times D_4): C_2):C_3$ (as 8T32)	$1$	$0$
	4.2e6_7e3_13e2.24t97.2c1	$4$	$ 2^{6} \cdot 7^{3} \cdot 13^{2}$	$x^{8} - 2 x^{7} + 4 x^{6} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x + 2$	$((C_2 \times D_4): C_2):C_3$ (as 8T32)	$0$	$0$
	4.2e6_7e3_13e2.24t97.2c2	$4$	$ 2^{6} \cdot 7^{3} \cdot 13^{2}$	$x^{8} - 2 x^{7} + 4 x^{6} - x^{4} + 6 x^{3} - 2 x^{2} - 4 x + 2$	$((C_2 \times D_4): C_2):C_3$ (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 *.