Global number field 8.2.8680921875.1

• Label: 8.2.8680921875.1
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,3^{4}\cdot 5^{6}\cdot 19^{3}$
• Root discriminant: $17.47$
• Ramified primes: $3, 5, 19$
• Class number: $2$
• Class group: $[2]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 3 x^{7} - x^{6} - 7 x^{5} + 21 x^{4} + 80 x^{3} - 45 x^{2} - 190 x - 95 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-8680921875=-\,3^{4}\cdot 5^{6}\cdot 19^{3}\)
Root discriminant:		$17.47$
Ramified primes:		$3, 5, 19$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{19969} a^{7} + \frac{144}{1051} a^{6} + \frac{5528}{19969} a^{5} + \frac{4683}{19969} a^{4} + \frac{6660}{19969} a^{3} - \frac{9846}{19969} a^{2} + \frac{520}{1051} a + \frac{165}{1051}$

Class group and class number

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

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:		\( 32.4987605988 \)

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

\(\Q(\sqrt{5}) \), 4.2.475.1

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	${\href{/LocalNumberField/2.8.0.1}{8} }$	R	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
$5$	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
$19$	$\Q_{19}$	$x + 4$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19}$	$x + 4$	$1$	$1$	$0$	Trivial	$[\ ]$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

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)	$1$	$1$
	1.19.2t1.1c1	$1$	$ 19 $	$x^{2} - x + 5$	$C_2$ (as 2T1)	$1$	$-1$
	1.5_19.2t1.1c1	$1$	$ 5 \cdot 19 $	$x^{2} - x + 24$	$C_2$ (as 2T1)	$1$	$-1$
*	2.5_19.4t3.2c1	$2$	$ 5 \cdot 19 $	$x^{4} - x^{3} + 3 x^{2} + x + 1$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.3e2_5e2_19.8t8.1c1	$2$	$ 3^{2} \cdot 5^{2} \cdot 19 $	$x^{8} - 3 x^{7} - x^{6} - 7 x^{5} + 21 x^{4} + 80 x^{3} - 45 x^{2} - 190 x - 95$	$QD_{16}$ (as 8T8)	$0$	$0$
*	2.3e2_5e2_19.8t8.1c2	$2$	$ 3^{2} \cdot 5^{2} \cdot 19 $	$x^{8} - 3 x^{7} - x^{6} - 7 x^{5} + 21 x^{4} + 80 x^{3} - 45 x^{2} - 190 x - 95$	$QD_{16}$ (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 *.