Number field 8.0.2067245.1

• Label: 8.0.2067245.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2067245$
• Root discriminant: $6.16$
• Ramified primes: $5, 643$
• Class number: $1$
• Class group: trivial
• Galois group: $C_2 \wr S_4$ (as 8T44)

Normalized defining polynomial

\(x^{8} - x^{7} + x^{6} - x^{4} + x^{2} - x + 1\)  

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(2067245\)\(\medspace = 5\cdot 643^{2}\)
Root discriminant:		$6.16$
Ramified primes:		$5, 643$
$|\Aut(K/\Q)|$:		$2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$  

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( a^{7} - a^{6} + a^{5} - a^{3} + a - 1 \),  \( a^{6} - a^{5} - a^{2} + 1 \),  \( a^{4} + a - 1 \)
Regulator:		\( 0.453534943932 \)

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{4}\cdot 0.453534943932 \cdot 1}{2\sqrt{2067245}}\approx 0.2458126658969$

Galois group

$C_2^3:S_4.C_2$ (as 8T44):

A solvable group of order 384

The 20 conjugacy class representatives for $C_2 \wr S_4$

Character table for $C_2 \wr S_4$

Intermediate fields

4.2.643.1

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

Sibling fields

Degree 8 siblings:	data not computed
Degree 16 siblings:	data not computed
Degree 24 siblings:	data not computed
Degree 32 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	${\href{/LocalNumberField/2.8.0.1}{8} }$	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.8.0.1}{8} }$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$5$	5.2.1.1	$x^{2} - 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.4.0.1	$x^{4} + x^{2} - 2 x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
643	Data not computed

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	\(\Q\)	$C_1$	$1$	$1$
	1.643.2t1.a.a	$1$	$ 643 $	\(\Q(\sqrt{-643}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.3215.2t1.a.a	$1$	$ 5 \cdot 643 $	\(\Q(\sqrt{-3215}) \)	$C_2$ (as 2T1)	$1$	$-1$
	1.5.2t1.a.a	$1$	$ 5 $	\(\Q(\sqrt{5}) \)	$C_2$ (as 2T1)	$1$	$1$
	2.643.3t2.a.a	$2$	$ 643 $	3.1.643.1	$S_3$ (as 3T2)	$1$	$0$
	2.16075.6t3.b.a	$2$	$ 5^{2} \cdot 643 $	6.0.33230963375.1	$D_{6}$ (as 6T3)	$1$	$0$
*	3.643.4t5.a.a	$3$	$ 643 $	4.2.643.1	$S_4$ (as 4T5)	$1$	$1$
	3.413449.6t8.a.a	$3$	$ 643^{2}$	4.2.643.1	$S_4$ (as 4T5)	$1$	$-1$
	3.80375.6t11.a.a	$3$	$ 5^{3} \cdot 643 $	6.2.51681125.1	$S_4\times C_2$ (as 6T11)	$1$	$1$
	3.51681125.6t11.a.a	$3$	$ 5^{3} \cdot 643^{2}$	6.2.51681125.1	$S_4\times C_2$ (as 6T11)	$1$	$-1$
*	4.3215.8t44.b.a	$4$	$ 5 \cdot 643 $	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$-2$
	4.80375.8t44.b.a	$4$	$ 5^{3} \cdot 643 $	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$-2$
	4.1329238535.8t44.b.a	$4$	$ 5 \cdot 643^{3}$	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$2$
	4.33230963375.8t44.b.a	$4$	$ 5^{3} \cdot 643^{3}$	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$2$
	6.51681125.8t41.b.a	$6$	$ 5^{3} \cdot 643^{2}$	8.4.258405625.1	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$2$
	6.213...125.12t108.b.a	$6$	$ 5^{3} \cdot 643^{4}$	8.4.258405625.1	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$-2$
	6.33230963375.8t41.b.a	$6$	$ 5^{3} \cdot 643^{3}$	8.4.258405625.1	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$
	6.33230963375.12t108.b.a	$6$	$ 5^{3} \cdot 643^{3}$	8.4.258405625.1	$V_4^2:(S_3\times C_2)$ (as 8T41)	$1$	$0$
	8.267...625.24t708.b.a	$8$	$ 5^{6} \cdot 643^{4}$	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$0$
	8.427...025.24t708.b.a	$8$	$ 5^{2} \cdot 643^{4}$	8.0.2067245.1	$C_2 \wr S_4$ (as 8T44)	$1$	$0$

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 *.