Global number field 8.0.204807268365921.2

• Label: 8.0.204807268365921.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{4}\cdot 13^{4}\cdot 97^{4}$
• Root discriminant: $61.51$
• Ramified primes: $3, 13, 97$
• Class number: $96$
• Class group: $[2, 2, 24]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 17 x^{6} + 532 x^{4} + 4131 x^{2} + 59049 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(204807268365921=3^{4}\cdot 13^{4}\cdot 97^{4}\)
Root discriminant:		$61.51$
Ramified primes:		$3, 13, 97$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{18} a^{5} + \frac{1}{18} a^{3} - \frac{1}{2} a^{2} - \frac{4}{9} a - \frac{1}{2}$, $\frac{1}{129276} a^{6} - \frac{43}{486} a^{4} + \frac{1}{243} a^{2} - \frac{1}{2} a + \frac{17}{532}$, $\frac{1}{6980904} a^{7} - \frac{1}{258552} a^{6} - \frac{143}{13122} a^{5} - \frac{50}{243} a^{4} - \frac{850}{6561} a^{3} + \frac{121}{243} a^{2} + \frac{5869}{28728} a - \frac{17}{1064}$

Class group and class number

$C_{2}\times C_{2}\times C_{24}$, which has order $96$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{17}{129276} a^{6} - \frac{1}{243} a^{4} + \frac{17}{243} a^{2} + \frac{289}{532} \) (order $6$)
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:		\( 1111.0802019 \)

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

\(\Q(\sqrt{-3783}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{1261}) \), \(\Q(\sqrt{-3}, \sqrt{1261})\), 4.0.11349.1 x2, 4.2.4770363.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.4770363.1, 4.0.11349.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$13$	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$97$	97.4.2.1	$x^{4} + 873 x^{2} + 235225$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	97.4.2.1	$x^{4} + 873 x^{2} + 235225$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.13_97.2t1.1c1	$1$	$ 13 \cdot 97 $	$x^{2} - x - 315$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_13_97.2t1.1c1	$1$	$ 3 \cdot 13 \cdot 97 $	$x^{2} - x + 946$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.3_13_97.4t3.6c1	$2$	$ 3 \cdot 13 \cdot 97 $	$x^{8} - 17 x^{6} + 532 x^{4} + 4131 x^{2} + 59049$	$D_4$ (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 *.