Global number field 8.0.25857479250625.1

• Label: 8.0.25857479250625.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $5^{4}\cdot 11^{4}\cdot 41^{4}$
• Root discriminant: $47.49$
• Ramified primes: $5, 11, 41$
• Class number: $60$
• Class group: $[2, 30]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 4 x^{6} + 234 x^{4} + 1795 x^{2} + 13225 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(25857479250625=5^{4}\cdot 11^{4}\cdot 41^{4}\)
Root discriminant:		$47.49$
Ramified primes:		$5, 11, 41$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{79990} a^{6} + \frac{4523}{39995} a^{4} - \frac{1}{2} a^{3} - \frac{3231}{79990} a^{2} - \frac{241}{7999}$, $\frac{1}{1839770} a^{7} - \frac{115462}{919885} a^{5} + \frac{156749}{1839770} a^{3} - \frac{168461}{367954} a - \frac{1}{2}$

Class group and class number

$C_{2}\times C_{30}$, which has order $60$

Unit group

Rank:		$3$
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:		\( 47.563837179 \)

Galois group

$D_4$ (as 8T4):

A solvable group of order 8

The 5 conjugacy class representatives for $D_4$

Character table for $D_4$

Intermediate fields

\(\Q(\sqrt{-2255}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-451}) \), \(\Q(\sqrt{5}, \sqrt{-451})\), 4.2.11275.1 x2, 4.0.1017005.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.11275.1, 4.0.1017005.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/3.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.1.0.1}{1} }^{8}$

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
$5$	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$11$	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$41$	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	41.2.1.2	$x^{2} + 246$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.11_41.2t1.1c1	$1$	$ 11 \cdot 41 $	$x^{2} - x + 113$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.5_11_41.2t1.1c1	$1$	$ 5 \cdot 11 \cdot 41 $	$x^{2} - x + 564$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.5_11_41.4t3.5c1	$2$	$ 5 \cdot 11 \cdot 41 $	$x^{8} - 4 x^{6} + 234 x^{4} + 1795 x^{2} + 13225$	$D_4$ (as 8T4)	$1$	$0$

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 *.