Global number field 8.0.7220000000.1

• Label: 8.0.7220000000.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 5^{7}\cdot 19^{2}$
• Root discriminant: $17.07$
• Ramified primes: $2, 5, 19$
• Class number: $4$
• Class group: $[4]$
• Galois group: $(C_8:C_2):C_2$ (as 8T16)

Normalized defining polynomial

\( x^{8} - 20 x^{6} + 220 x^{4} - 1045 x^{2} + 1805 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(7220000000=2^{8}\cdot 5^{7}\cdot 19^{2}\)
Root discriminant:		$17.07$
Ramified primes:		$2, 5, 19$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{21109} a^{6} - \frac{7221}{21109} a^{4} + \frac{7174}{21109} a^{2} - \frac{384}{1111}$, $\frac{1}{21109} a^{7} - \frac{7221}{21109} a^{5} + \frac{7174}{21109} a^{3} - \frac{384}{1111} a$

Class group and class number

$C_{4}$, which has order $4$

Unit group

Rank:		$3$
Torsion generator:		\( \frac{123}{21109} a^{6} - \frac{1605}{21109} a^{4} + \frac{16933}{21109} a^{2} - \frac{1681}{1111} \) (order $10$)
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:		\( 52.3346850559 \)

Galois group

$C_2^3.C_4$ (as 8T16):

A solvable group of order 32

The 11 conjugacy class representatives for $(C_8:C_2):C_2$

Character table for $(C_8:C_2):C_2$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\)

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

Sibling fields

Galois closure:	data not computed
Degree 8 sibling:	data not computed
Degree 16 siblings:	data not computed

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	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/13.8.0.1}{8} }$	${\href{/LocalNumberField/17.8.0.1}{8} }$	R	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.8.0.1}{8} }$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_5.4t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.1c1	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	1.2e2_5.4t1.1c2	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.1c2	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	2.2e4_5e2_19e2.4t3.3c1	$2$	$ 2^{4} \cdot 5^{2} \cdot 19^{2}$	$x^{4} - 2 x^{3} + 4 x^{2} + 92 x + 406$	$D_{4}$ (as 4T3)	$1$	$0$
	2.2e4_5_19e2.4t3.5c1	$2$	$ 2^{4} \cdot 5 \cdot 19^{2}$	$x^{4} - 2 x^{3} - 36 x^{2} + 18 x + 461$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.2e8_5e4_19e2.8t16.11c1	$4$	$ 2^{8} \cdot 5^{4} \cdot 19^{2}$	$x^{8} - 20 x^{6} + 220 x^{4} - 1045 x^{2} + 1805$	$(C_8:C_2):C_2$ (as 8T16)	$1$	$0$

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 *.