Global number field 8.2.3596615680000.6

• Label: 8.2.3596615680000.6
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,2^{24}\cdot 5^{4}\cdot 7^{3}$
• Root discriminant: $37.11$
• Ramified primes: $2, 5, 7$
• Class number: $6$
• Class group: $[6]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 20 x^{6} + 100 x^{4} + 500 x^{2} - 4375 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-3596615680000=-\,2^{24}\cdot 5^{4}\cdot 7^{3}\)
Root discriminant:		$37.11$
Ramified primes:		$2, 5, 7$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{5} a^{2}$, $\frac{1}{5} a^{3}$, $\frac{1}{25} a^{4}$, $\frac{1}{25} a^{5}$, $\frac{1}{125} a^{6}$, $\frac{1}{125} a^{7}$

Class group and class number

$C_{6}$, which has order $6$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{5} a^{2} - 2 \),  \( \frac{1}{125} a^{6} - \frac{2}{25} a^{4} - \frac{1}{5} a^{2} + 3 \),  \( \frac{9}{125} a^{7} - \frac{26}{125} a^{6} - \frac{21}{25} a^{5} + \frac{61}{25} a^{4} + \frac{1}{5} a^{3} - \frac{4}{5} a^{2} + 38 a - 109 \),  \( \frac{13}{125} a^{7} + \frac{9}{25} a^{6} - \frac{27}{25} a^{5} - \frac{21}{5} a^{4} - \frac{13}{5} a^{3} - 7 a^{2} + 35 a + 146 \)
Regulator:		\( 510.059190604 \)

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

\(\Q(\sqrt{2}) \), 4.2.1792.1

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	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.8.24.4	$x^{8} + 14 x^{4} + 8 x^{3} + 12 x^{2} + 8 x + 14$	$8$	$1$	$24$	$Q_8$	$[2, 3, 4]$
$5$	5.8.4.2	$x^{8} + 25 x^{4} - 250 x^{2} + 1250$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
$7$	$\Q_{7}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{7}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	7.2.1.2	$x^{2} + 14$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 14$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.2	$x^{2} + 14$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_7.2t1.2c1	$1$	$ 2^{3} \cdot 7 $	$x^{2} + 14$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e5_7.4t3.2c1	$2$	$ 2^{5} \cdot 7 $	$x^{4} - x^{2} + 2$	$D_{4}$ (as 4T3)	$1$	$0$
*	2.2e8_5e2_7.8t8.1c1	$2$	$ 2^{8} \cdot 5^{2} \cdot 7 $	$x^{8} - 20 x^{6} + 100 x^{4} + 500 x^{2} - 4375$	$QD_{16}$ (as 8T8)	$0$	$0$
*	2.2e8_5e2_7.8t8.1c2	$2$	$ 2^{8} \cdot 5^{2} \cdot 7 $	$x^{8} - 20 x^{6} + 100 x^{4} + 500 x^{2} - 4375$	$QD_{16}$ (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 *.