Global number field 8.0.343064484000000.1

• Label: 8.0.343064484000000.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 3^{6}\cdot 5^{6}\cdot 7^{6}$
• Root discriminant: $65.60$
• Ramified primes: $2, 3, 5, 7$
• Class number: $288$
• Class group: $[2, 2, 2, 6, 6]$
• Galois group: $Q_8$ (as 8T5)

Normalized defining polynomial

\( x^{8} + 105 x^{6} + 3780 x^{4} + 55125 x^{2} + 275625 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(343064484000000=2^{8}\cdot 3^{6}\cdot 5^{6}\cdot 7^{6}\)
Root discriminant:		$65.60$
Ramified primes:		$2, 3, 5, 7$
This field is Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{105} a^{4}$, $\frac{1}{105} a^{5}$, $\frac{1}{2100} a^{6} - \frac{1}{5} a^{2} + \frac{1}{4}$, $\frac{1}{10500} a^{7} - \frac{1}{525} a^{5} + \frac{9}{25} a^{3} + \frac{1}{4} a$

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}$, which has order $288$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{2100} a^{6} + \frac{4}{105} a^{4} + \frac{4}{5} a^{2} + \frac{13}{4} \),  \( \frac{1}{420} a^{6} + \frac{1}{5} a^{4} + 5 a^{2} + \frac{137}{4} \),  \( \frac{1}{350} a^{6} + \frac{26}{105} a^{4} + \frac{34}{5} a^{2} + \frac{117}{2} \)
Regulator:		\( 53.1582098182 \)

Galois group

$Q_8$ (as 8T5):

A solvable group of order 8

The 5 conjugacy class representatives for $Q_8$

Character table for $Q_8$

Intermediate fields

\(\Q(\sqrt{21}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{5}, \sqrt{21})\)

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

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	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$2$	2.4.4.2	$x^{4} - x^{2} + 5$	$2$	$2$	$4$	$C_4$	$[2]^{2}$
	2.4.4.2	$x^{4} - x^{2} + 5$	$2$	$2$	$4$	$C_4$	$[2]^{2}$
$3$	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	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
$7$	7.8.6.1	$x^{8} + 35 x^{4} + 441$	$4$	$2$	$6$	$Q_8$	$[\ ]_{4}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.3_7.2t1.1c1	$1$	$ 3 \cdot 7 $	$x^{2} - x - 5$	$C_2$ (as 2T1)	$1$	$1$
*	1.3_5_7.2t1.1c1	$1$	$ 3 \cdot 5 \cdot 7 $	$x^{2} - x - 26$	$C_2$ (as 2T1)	$1$	$1$
*^2	2.2e4_3e2_5e2_7e2.8t5.2c1	$2$	$ 2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$	$x^{8} + 105 x^{6} + 3780 x^{4} + 55125 x^{2} + 275625$	$Q_8$ (as 8T5)	$-1$	$-2$

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 *.