Global number field 8.0.3967389600768.2

• Label: 8.0.3967389600768.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{10}\cdot 3^{7}\cdot 11^{6}$
• Root discriminant: $37.57$
• Ramified primes: $2, 3, 11$
• Class number: $24$
• Class group: $[2, 2, 6]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} + 33 x^{6} + 330 x^{4} + 1089 x^{2} + 363 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(3967389600768=2^{10}\cdot 3^{7}\cdot 11^{6}\)
Root discriminant:		$37.57$
Ramified primes:		$2, 3, 11$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{11} a^{4}$, $\frac{1}{11} a^{5}$, $\frac{1}{22} a^{6} - \frac{1}{2}$, $\frac{1}{44} a^{7} - \frac{1}{44} a^{6} - \frac{1}{22} a^{5} - \frac{1}{22} a^{4} - \frac{1}{4} a + \frac{1}{4}$

Class group and class number

$C_{2}\times C_{2}\times C_{6}$, which has order $24$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{5}{22} a^{6} + \frac{64}{11} a^{4} + 32 a^{2} + \frac{23}{2} \),  \( \frac{4}{11} a^{6} + \frac{103}{11} a^{4} + 52 a^{2} + 16 \),  \( \frac{2}{11} a^{6} + \frac{50}{11} a^{4} + 23 a^{2} - 2 \)
Regulator:		\( 86.1661860126 \)

Galois group

$SD_{16}$ (as 8T8):

A solvable group of order 16

The 7 conjugacy class representatives for $QD_{16}$

Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{33}) \), 4.4.13068.1

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

Sibling fields

Galois closure:

data not computed

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	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.8.0.1}{8} }$	R	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.4.6.8	$x^{4} + 2 x^{3} + 2$	$4$	$1$	$6$	$D_{4}$	$[2, 2]^{2}$
	2.4.4.3	$x^{4} + 2 x^{2} + 4 x + 4$	$2$	$2$	$4$	$D_{4}$	$[2, 2]^{2}$
$3$	3.8.7.1	$x^{8} + 3$	$8$	$1$	$7$	$QD_{16}$	$[\ ]_{8}^{2}$
$11$	11.8.6.1	$x^{8} + 143 x^{4} + 5929$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_11.2t1.1c1	$1$	$ 3 \cdot 11 $	$x^{2} - x - 8$	$C_2$ (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_11.2t1.1c1	$1$	$ 2^{2} \cdot 11 $	$x^{2} - 11$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_3e2_11.4t3.2c1	$2$	$ 2^{2} \cdot 3^{2} \cdot 11 $	$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$	$D_{4}$ (as 4T3)	$1$	$2$
*	2.2e4_3e2_11e2.8t8.2c1	$2$	$ 2^{4} \cdot 3^{2} \cdot 11^{2}$	$x^{8} + 33 x^{6} + 330 x^{4} + 1089 x^{2} + 363$	$QD_{16}$ (as 8T8)	$0$	$-2$
*	2.2e4_3e2_11e2.8t8.2c2	$2$	$ 2^{4} \cdot 3^{2} \cdot 11^{2}$	$x^{8} + 33 x^{6} + 330 x^{4} + 1089 x^{2} + 363$	$QD_{16}$ (as 8T8)	$0$	$-2$

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 *.