Global number field 8.0.49991146569.7

• Label: 8.0.49991146569.7
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{6}\cdot 7^{4}\cdot 13^{4}$
• Root discriminant: $21.75$
• Ramified primes: $3, 7, 13$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $D_4$ (as 8T4)

Normalized defining polynomial

\( x^{8} + 19 x^{6} + 120 x^{4} + 931 x^{2} + 2401 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(49991146569=3^{6}\cdot 7^{4}\cdot 13^{4}\)
Root discriminant:		$21.75$
Ramified primes:		$3, 7, 13$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{14} a^{5} - \frac{1}{7} a^{3} - \frac{1}{2} a^{2} - \frac{3}{7} a$, $\frac{1}{4312} a^{6} - \frac{16}{539} a^{4} + \frac{211}{539} a^{2} - \frac{29}{88}$, $\frac{1}{60368} a^{7} - \frac{1}{8624} a^{6} - \frac{8}{3773} a^{5} + \frac{8}{539} a^{4} - \frac{703}{3773} a^{3} - \frac{211}{1078} a^{2} - \frac{117}{1232} a + \frac{29}{176}$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{15}{15092} a^{7} - \frac{59}{3773} a^{5} - \frac{263}{7546} a^{3} - \frac{137}{308} a + \frac{1}{2} \) (order $6$)
Fundamental units:		\( \frac{1}{2156} a^{7} + \frac{15}{4312} a^{6} + \frac{13}{1078} a^{5} + \frac{59}{1078} a^{4} + \frac{151}{1078} a^{3} + \frac{401}{1078} a^{2} + \frac{127}{308} a + \frac{49}{88} \),  \( \frac{37}{15092} a^{7} - \frac{15}{4312} a^{6} + \frac{327}{7546} a^{5} - \frac{59}{1078} a^{4} + \frac{1583}{7546} a^{3} - \frac{401}{1078} a^{2} + \frac{401}{308} a - \frac{401}{88} \),  \( \frac{223}{7546} a^{7} + \frac{1898}{3773} a^{5} + \frac{7647}{3773} a^{3} + \frac{503}{22} a \)
Regulator:		\( 240.569294846 \)

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

\(\Q(\sqrt{-91}) \), \(\Q(\sqrt{273}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-3}, \sqrt{-91})\), 4.2.223587.2 x2, 4.0.2457.1 x2

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

Sibling fields

Degree 4 siblings:

4.2.223587.2, 4.0.2457.1

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	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/11.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$	3.8.6.2	$x^{8} + 4 x^{7} + 14 x^{6} + 28 x^{5} + 43 x^{4} + 44 x^{3} + 110 x^{2} + 92 x + 22$	$4$	$2$	$6$	$D_4$	$[\ ]_{4}^{2}$
$7$	7.2.1.1	$x^{2} - 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.1	$x^{2} - 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.1	$x^{2} - 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	7.2.1.1	$x^{2} - 7$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$13$	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_7_13.2t1.1c1	$1$	$ 3 \cdot 7 \cdot 13 $	$x^{2} - x - 68$	$C_2$ (as 2T1)	$1$	$1$
*	1.7_13.2t1.1c1	$1$	$ 7 \cdot 13 $	$x^{2} - x + 23$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.3e2_7_13.4t3.6c1	$2$	$ 3^{2} \cdot 7 \cdot 13 $	$x^{8} + 19 x^{6} + 120 x^{4} + 931 x^{2} + 2401$	$D_4$ (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 *.