Global number field 8.8.33554432000000.2

• Label: 8.8.33554432000000.2
• Degree: $8$
• Signature: $[8, 0]$
• Discriminant: $2^{31}\cdot 5^{6}$
• Root discriminant: $49.06$
• Ramified primes: $2, 5$
• Class number: $2$
• Class group: $[2]$
• Galois group: $C_8$ (as 8T1)

Normalized defining polynomial

\( x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450 \)

Invariants

Degree:		$8$
Signature:		$[8, 0]$
Discriminant:		\(33554432000000=2^{31}\cdot 5^{6}\)
Root discriminant:		$49.06$
Ramified primes:		$2, 5$
This field is Galois and abelian over $\Q$.
Conductor:		\(160=2^{5}\cdot 5\)
Dirichlet character group:		$\lbrace$$\chi_{160}(1,·)$, $\chi_{160}(67,·)$, $\chi_{160}(9,·)$, $\chi_{160}(43,·)$, $\chi_{160}(81,·)$, $\chi_{160}(147,·)$, $\chi_{160}(89,·)$, $\chi_{160}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{35} a^{5} + \frac{1}{7} a^{3} - \frac{2}{7} a$, $\frac{1}{35} a^{6} - \frac{2}{35} a^{4} - \frac{2}{7} a^{2}$, $\frac{1}{35} a^{7} + \frac{3}{7} a$

Class group and class number

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

Unit group

Rank:		$7$
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:		\( 6237.2438059 \)

Galois group

$C_8$ (as 8T1):

A cyclic group of order 8

The 8 conjugacy class representatives for $C_8$

Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{2}) \), 4.4.51200.1

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	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$2$	2.8.31.3	$x^{8} + 8 x^{6} + 4 x^{4} + 16 x^{2} + 18$	$8$	$1$	$31$	$C_8$	$[3, 4, 5]$
$5$	5.8.6.3	$x^{8} + 25 x^{4} + 200$	$4$	$2$	$6$	$C_8$	$[\ ]_{4}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e5_5.8t1.5c1	$1$	$ 2^{5} \cdot 5 $	$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450$	$C_8$ (as 8T1)	$0$	$1$
*	1.2e4_5.4t1.3c1	$1$	$ 2^{4} \cdot 5 $	$x^{4} - 20 x^{2} + 50$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e5_5.8t1.5c2	$1$	$ 2^{5} \cdot 5 $	$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450$	$C_8$ (as 8T1)	$0$	$1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e5_5.8t1.5c3	$1$	$ 2^{5} \cdot 5 $	$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450$	$C_8$ (as 8T1)	$0$	$1$
*	1.2e4_5.4t1.3c2	$1$	$ 2^{4} \cdot 5 $	$x^{4} - 20 x^{2} + 50$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e5_5.8t1.5c4	$1$	$ 2^{5} \cdot 5 $	$x^{8} - 40 x^{6} + 500 x^{4} - 2000 x^{2} + 2450$	$C_8$ (as 8T1)	$0$	$1$

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 *.