Global number field 8.0.1721829632.1

• Label: 8.0.1721829632.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{8}\cdot 17^{3}\cdot 37^{2}$
• Root discriminant: $14.27$
• Ramified primes: $2, 17, 37$
• Class number: $1$
• Class group: Trivial
• Galois group: $Z_8 : Z_8^\times$ (as 8T15)

Normalized defining polynomial

\( x^{8} - 2 x^{7} - 5 x^{6} + 24 x^{5} + 16 x^{4} - 72 x^{3} - 36 x^{2} + 74 x + 53 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1721829632=2^{8}\cdot 17^{3}\cdot 37^{2}\)
Root discriminant:		$14.27$
Ramified primes:		$2, 17, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2533} a^{7} + \frac{1005}{2533} a^{6} - \frac{1170}{2533} a^{5} - \frac{321}{2533} a^{4} + \frac{993}{2533} a^{3} - \frac{656}{2533} a^{2} + \frac{485}{2533} a - \frac{400}{2533}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{87}{149} a^{7} + \frac{326}{149} a^{6} - \frac{126}{149} a^{5} - \frac{1873}{149} a^{4} + \frac{1817}{149} a^{3} + \frac{3283}{149} a^{2} - \frac{2263}{149} a - \frac{2897}{149} \) (order $4$)
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:		\( 42.3287213263 \)

Galois group

$C_8:C_2^2$ (as 8T15):

A solvable group of order 32

The 11 conjugacy class representatives for $Z_8 : Z_8^\times$

Character table for $Z_8 : Z_8^\times$

Intermediate fields

\(\Q(\sqrt{-1}) \), 4.0.272.1

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 16 siblings:	data not computed
Arithmetically equvalently sibling:	8.0.1721829632.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.8.0.1}{8} }$	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.8.0.1}{8} }$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$2$	2.8.8.1	$x^{8} + 28 x^{4} + 144$	$2$	$4$	$8$	$C_4\times C_2$	$[2]^{4}$
$17$	$\Q_{17}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{17}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	17.2.1.1	$x^{2} - 17$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.4.2.1	$x^{4} + 85 x^{2} + 2601$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
37	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_37.2t1.1c1	$1$	$ 2^{2} \cdot 37 $	$x^{2} + 37$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_17_37.2t1.1c1	$1$	$ 2^{2} \cdot 17 \cdot 37 $	$x^{2} + 629$	$C_2$ (as 2T1)	$1$	$-1$
	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
	1.17_37.2t1.1c1	$1$	$ 17 \cdot 37 $	$x^{2} - x - 157$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_17.2t1.1c1	$1$	$ 2^{2} \cdot 17 $	$x^{2} + 17$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e2_17.4t3.1c1	$2$	$ 2^{2} \cdot 17 $	$x^{4} + x^{2} - 2 x + 1$	$D_{4}$ (as 4T3)	$1$	$0$
	2.2e2_17_37e2.4t3.1c1	$2$	$ 2^{2} \cdot 17 \cdot 37^{2}$	$x^{4} - 2 x^{3} - 17 x^{2} + 18 x + 1450$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.2e4_17e2_37e2.8t15.1c1	$4$	$ 2^{4} \cdot 17^{2} \cdot 37^{2}$	$x^{8} - 2 x^{7} - 5 x^{6} + 24 x^{5} + 16 x^{4} - 72 x^{3} - 36 x^{2} + 74 x + 53$	$Z_8 : Z_8^\times$ (as 8T15)	$1$	$0$

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 *.