Global number field 8.2.3881196000000.9

• Label: 8.2.3881196000000.9
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,2^{8}\cdot 3^{6}\cdot 5^{6}\cdot 11^{3}$
• Root discriminant: $37.46$
• Ramified primes: $2, 3, 5, 11$
• Class number: $12$
• Class group: $[2, 6]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 15 x^{6} - 90 x^{4} + 1800 x^{2} - 2475 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-3881196000000=-\,2^{8}\cdot 3^{6}\cdot 5^{6}\cdot 11^{3}\)
Root discriminant:		$37.46$
Ramified primes:		$2, 3, 5, 11$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{75} a^{4} + \frac{2}{5} a^{2} + \frac{2}{5}$, $\frac{1}{75} a^{5} + \frac{2}{5} a^{3} + \frac{2}{5} a$, $\frac{1}{825} a^{6} + \frac{1}{825} a^{4} + \frac{9}{55} a^{2} + \frac{2}{5}$, $\frac{1}{825} a^{7} + \frac{1}{825} a^{5} + \frac{9}{55} a^{3} + \frac{2}{5} a$

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 184.653245376 \)

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

\(\Q(\sqrt{5}) \), 4.2.2475.1

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	R	${\href{/LocalNumberField/7.8.0.1}{8} }$	R	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.8.0.1}{8} }$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.8.8.4	$x^{8} + 2 x^{7} + 2 x^{6} + 8 x^{3} + 48$	$2$	$4$	$8$	$C_8$	$[2]^{4}$
$3$	3.8.6.1	$x^{8} + 9 x^{4} + 36$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$
$5$	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
$11$	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
	1.5_11.2t1.1c1	$1$	$ 5 \cdot 11 $	$x^{2} - x + 14$	$C_2$ (as 2T1)	$1$	$-1$
*	2.3e2_5_11.4t3.2c1	$2$	$ 3^{2} \cdot 5 \cdot 11 $	$x^{4} - x^{3} + 4 x^{2} - 6 x + 3$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2e4_3e2_5e2_11.8t8.1c1	$2$	$ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11 $	$x^{8} - 15 x^{6} - 90 x^{4} + 1800 x^{2} - 2475$	$QD_{16}$ (as 8T8)	$0$	$0$
*	2.2e4_3e2_5e2_11.8t8.1c2	$2$	$ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 11 $	$x^{8} - 15 x^{6} - 90 x^{4} + 1800 x^{2} - 2475$	$QD_{16}$ (as 8T8)	$0$	$0$

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 *.