Global number field 8.2.1517535243.1

• Label: 8.2.1517535243.1
• Degree: $8$
• Signature: $[2, 3]$
• Discriminant: $-\,3^{7}\cdot 7^{4}\cdot 17^{2}$
• Root discriminant: $14.05$
• Ramified primes: $3, 7, 17$
• Class number: $1$
• Class group: Trivial
• Galois group: $Z_8 : Z_8^\times$ (as 8T15)

Normalized defining polynomial

\( x^{8} - x^{7} + 4 x^{6} - x^{5} + 7 x^{4} - 10 x^{3} - 8 x^{2} + 14 x - 5 \)

Invariants

Degree:		$8$
Signature:		$[2, 3]$
Discriminant:		\(-1517535243=-\,3^{7}\cdot 7^{4}\cdot 17^{2}\)
Root discriminant:		$14.05$
Ramified primes:		$3, 7, 17$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} - \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{1}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{125} a^{7} + \frac{11}{125} a^{6} + \frac{11}{125} a^{5} + \frac{6}{125} a^{4} - \frac{46}{125} a^{3} - \frac{62}{125} a^{2} - \frac{2}{125} a - \frac{2}{25}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$4$
Torsion generator:		\( -1 \) (order $2$)
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:		\( 37.4830482781 \)

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

\(\Q(\sqrt{21}) \), 4.2.1323.1

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:	8.2.1517535243.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.8.0.1}{8} }$	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/19.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$3$	3.8.7.2	$x^{8} - 3$	$8$	$1$	$7$	$QD_{16}$	$[\ ]_{8}^{2}$
$7$	7.8.4.1	$x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$17$	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	17.2.1.1	$x^{2} - 17$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_17.2t1.1c1	$1$	$ 3 \cdot 17 $	$x^{2} - x + 13$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_7.2t1.1c1	$1$	$ 3 \cdot 7 $	$x^{2} - x - 5$	$C_2$ (as 2T1)	$1$	$1$
	1.3_7_17.2t1.1c1	$1$	$ 3 \cdot 7 \cdot 17 $	$x^{2} - x - 89$	$C_2$ (as 2T1)	$1$	$1$
	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.7_17.2t1.1c1	$1$	$ 7 \cdot 17 $	$x^{2} - x + 30$	$C_2$ (as 2T1)	$1$	$-1$
*	2.3e2_7.4t3.2c1	$2$	$ 3^{2} \cdot 7 $	$x^{4} - x^{3} + 2 x + 1$	$D_{4}$ (as 4T3)	$1$	$0$
	2.3e2_7_17e2.4t3.2c1	$2$	$ 3^{2} \cdot 7 \cdot 17^{2}$	$x^{4} - x^{3} + 39 x^{2} - 13 x + 373$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.3e4_7e2_17e2.8t15.2c1	$4$	$ 3^{4} \cdot 7^{2} \cdot 17^{2}$	$x^{8} - x^{7} + 4 x^{6} - x^{5} + 7 x^{4} - 10 x^{3} - 8 x^{2} + 14 x - 5$	$Z_8 : Z_8^\times$ (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 *.