Global number field 8.0.1809375813.2

• Label: 8.0.1809375813.2
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{6}\cdot 7^{2}\cdot 37^{3}$
• Root discriminant: $14.36$
• Ramified primes: $3, 7, 37$
• Class number: $2$
• Class group: $[2]$
• Galois group: $Z_8 : Z_8^\times$ (as 8T15)

Normalized defining polynomial

\( x^{8} - 3 x^{7} + 14 x^{6} - 27 x^{5} + 66 x^{4} - 57 x^{3} + 116 x^{2} + 6 x + 13 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1809375813=3^{6}\cdot 7^{2}\cdot 37^{3}\)
Root discriminant:		$14.36$
Ramified primes:		$3, 7, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{37} a^{6} + \frac{11}{37} a^{5} - \frac{1}{37} a^{4} - \frac{13}{37} a^{3} + \frac{16}{37} a^{2} - \frac{4}{37} a - \frac{17}{37}$, $\frac{1}{5587} a^{7} + \frac{51}{5587} a^{6} - \frac{856}{5587} a^{5} - \frac{2310}{5587} a^{4} + \frac{1864}{5587} a^{3} + \frac{2449}{5587} a^{2} + \frac{1747}{5587} a + \frac{2687}{5587}$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		\( -\frac{121}{5587} a^{7} + \frac{473}{5587} a^{6} - \frac{2124}{5587} a^{5} + \frac{4690}{5587} a^{4} - \frac{10218}{5587} a^{3} + \frac{11107}{5587} a^{2} - \frac{14483}{5587} a + \frac{3298}{5587} \) (order $6$)
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:		\( 43.5848861293 \)

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

\(\Q(\sqrt{-3}) \), 4.0.333.1

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:	8.0.1809375813.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.8.0.1}{8} }$	R	${\href{/LocalNumberField/5.8.0.1}{8} }$	R	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/17.8.0.1}{8} }$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{4}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.8.0.1}{8} }$

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
$3$	3.4.3.1	$x^{4} + 3$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
	3.4.3.1	$x^{4} + 3$	$4$	$1$	$3$	$D_{4}$	$[\ ]_{4}^{2}$
$7$	7.4.0.1	$x^{4} + x^{2} - 3 x + 5$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	7.4.2.2	$x^{4} - 7 x^{2} + 147$	$2$	$2$	$2$	$C_4$	$[\ ]_{2}^{2}$
$37$	$\Q_{37}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{37}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.7_37.2t1.1c1	$1$	$ 7 \cdot 37 $	$x^{2} - x + 65$	$C_2$ (as 2T1)	$1$	$-1$
	1.7.2t1.1c1	$1$	$ 7 $	$x^{2} - x + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_7_37.2t1.1c1	$1$	$ 3 \cdot 7 \cdot 37 $	$x^{2} - x - 194$	$C_2$ (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_37.2t1.1c1	$1$	$ 3 \cdot 37 $	$x^{2} - x + 28$	$C_2$ (as 2T1)	$1$	$-1$
*	2.3_37.4t3.2c1	$2$	$ 3 \cdot 37 $	$x^{4} - x^{3} - 2 x^{2} + 3$	$D_{4}$ (as 4T3)	$1$	$0$
	2.3_7e2_37.4t3.2c1	$2$	$ 3 \cdot 7^{2} \cdot 37 $	$x^{4} - x^{3} + 20 x^{2} - 2 x + 109$	$D_{4}$ (as 4T3)	$1$	$0$
*	4.3e4_7e2_37e2.8t15.1c1	$4$	$ 3^{4} \cdot 7^{2} \cdot 37^{2}$	$x^{8} - 3 x^{7} + 14 x^{6} - 27 x^{5} + 66 x^{4} - 57 x^{3} + 116 x^{2} + 6 x + 13$	$Z_8 : Z_8^\times$ (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 *.