Global number field 8.0.694427904.6

• Label: 8.0.694427904.6
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 3^{6}\cdot 61^{2}$
• Root discriminant: $12.74$
• Ramified primes: $2, 3, 61$
• Class number: $1$
• Class group: Trivial
• Galois group: $Q_8:C_2$ (as 8T11)

Normalized defining polynomial

\( x^{8} - 4 x^{7} - x^{6} + 8 x^{5} + 31 x^{4} - 50 x^{3} - 43 x^{2} + 58 x + 97 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(694427904=2^{8}\cdot 3^{6}\cdot 61^{2}\)
Root discriminant:		$12.74$
Ramified primes:		$2, 3, 61$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{261997} a^{7} - \frac{50837}{261997} a^{6} + \frac{120809}{261997} a^{5} + \frac{125791}{261997} a^{4} - \frac{35090}{261997} a^{3} + \frac{54344}{261997} a^{2} + \frac{27773}{261997} a + \frac{1206}{2701}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{2149}{261997} a^{7} + \frac{4036}{261997} a^{6} - \frac{20486}{261997} a^{5} - \frac{56045}{261997} a^{4} + \frac{46726}{261997} a^{3} + \frac{196591}{261997} a^{2} - \frac{51139}{261997} a - \frac{3967}{2701} \) (order $12$)
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:		\( 40.7339437812 \)

Galois group

$D_4:C_2$ (as 8T11):

A solvable group of order 16

The 10 conjugacy class representatives for $Q_8:C_2$

Character table for $Q_8:C_2$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{12})\)

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

Sibling fields

Galois closure:	data not computed
Degree 8 siblings:	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	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.8.8.1	$x^{8} + 28 x^{4} + 144$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$3$	3.8.6.2	$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$	$4$	$2$	$6$	$D_4$	$[\ ]_{4}^{2}$
$61$	61.2.1.2	$x^{2} + 122$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	61.2.1.2	$x^{2} + 122$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	61.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	61.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.3_61.2t1.1c1	$1$	$ 3 \cdot 61 $	$x^{2} - x + 46$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_61.2t1.1c1	$1$	$ 2^{2} \cdot 61 $	$x^{2} + 61$	$C_2$ (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_3_61.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 61 $	$x^{2} - 183$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.61.2t1.1c1	$1$	$ 61 $	$x^{2} - x - 15$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_3e2_61.8t11.1c1	$2$	$ 2^{2} \cdot 3^{2} \cdot 61 $	$x^{8} - 4 x^{7} - x^{6} + 8 x^{5} + 31 x^{4} - 50 x^{3} - 43 x^{2} + 58 x + 97$	$Q_8:C_2$ (as 8T11)	$0$	$0$
*	2.2e2_3e2_61.8t11.1c2	$2$	$ 2^{2} \cdot 3^{2} \cdot 61 $	$x^{8} - 4 x^{7} - x^{6} + 8 x^{5} + 31 x^{4} - 50 x^{3} - 43 x^{2} + 58 x + 97$	$Q_8:C_2$ (as 8T11)	$0$	$0$

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 *.