Global number field 8.0.4335760898001.2

• Label: 8.0.4335760898001.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{4}\cdot 13^{4}\cdot 37^{4}$
• Root discriminant: $37.99$
• Ramified primes: $3, 13, 37$
• Class number: $8$
• Class group: $[2, 4]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} - 7 x^{6} + 157 x^{4} + 756 x^{2} + 11664 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(4335760898001=3^{4}\cdot 13^{4}\cdot 37^{4}\)
Root discriminant:		$37.99$
Ramified primes:		$3, 13, 37$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{5} - \frac{1}{6} a^{3} + \frac{1}{6} a$, $\frac{1}{33912} a^{6} - \frac{1}{12} a^{5} - \frac{31}{216} a^{4} + \frac{1}{12} a^{3} - \frac{107}{216} a^{2} - \frac{1}{12} a + \frac{7}{314}$, $\frac{1}{610416} a^{7} - \frac{247}{3888} a^{5} + \frac{541}{3888} a^{3} - \frac{1}{2} a^{2} - \frac{1331}{2826} a - \frac{1}{2}$

Class group and class number

$C_{2}\times C_{4}$, which has order $8$

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{7}{16956} a^{6} + \frac{1}{108} a^{4} - \frac{7}{108} a^{2} + \frac{108}{157} \) (order $6$)
Fundamental units:		\( \frac{67}{610416} a^{7} + \frac{7}{33912} a^{6} - \frac{25}{3888} a^{5} - \frac{1}{216} a^{4} + \frac{283}{3888} a^{3} + \frac{7}{216} a^{2} - \frac{1729}{5652} a - \frac{368}{157} \),  \( \frac{103}{305208} a^{7} - \frac{7}{33912} a^{6} - \frac{7}{1944} a^{5} + \frac{1}{216} a^{4} + \frac{157}{1944} a^{3} - \frac{7}{216} a^{2} - \frac{2483}{5652} a - \frac{677}{314} \),  \( \frac{4594891}{16956} a^{6} - \frac{656413}{108} a^{4} - \frac{4900685}{108} a^{2} - \frac{70892604}{157} \)
Regulator:		\( 1087.12975026 \)

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

\(\Q(\sqrt{-1443}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{481}) \), \(\Q(\sqrt{-3}, \sqrt{481})\), 4.0.4329.1 x2, 4.2.694083.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.694083.1, 4.0.4329.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{4}$	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	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$	R	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

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
$3$	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$13$	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	13.2.1.2	$x^{2} + 26$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$37$	37.4.2.1	$x^{4} + 333 x^{2} + 34225$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	37.4.2.1	$x^{4} + 333 x^{2} + 34225$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_37.2t1.1c1	$1$	$ 13 \cdot 37 $	$x^{2} - x - 120$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_13_37.2t1.1c1	$1$	$ 3 \cdot 13 \cdot 37 $	$x^{2} - x + 361$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.3_13_37.4t3.6c1	$2$	$ 3 \cdot 13 \cdot 37 $	$x^{8} - 7 x^{6} + 157 x^{4} + 756 x^{2} + 11664$	$D_4$ (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 *.