Global number field 8.0.1743862431744.1

• Label: 8.0.1743862431744.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $2^{16}\cdot 3^{7}\cdot 23^{3}$
• Root discriminant: $33.90$
• Ramified primes: $2, 3, 23$
• Class number: $12$
• Class group: $[2, 6]$
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 4 x^{7} + 10 x^{6} + 8 x^{5} - 32 x^{4} - 28 x^{3} + 202 x^{2} + 284 x + 157 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(1743862431744=2^{16}\cdot 3^{7}\cdot 23^{3}\)
Root discriminant:		$33.90$
Ramified primes:		$2, 3, 23$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2896153} a^{7} + \frac{138885}{2896153} a^{6} + \frac{1219795}{2896153} a^{5} - \frac{154278}{2896153} a^{4} + \frac{1118873}{2896153} a^{3} + \frac{270548}{2896153} a^{2} - \frac{1443801}{2896153} a - \frac{1339238}{2896153}$

Class group and class number

$C_{2}\times C_{6}$, which has order $12$

Unit group

Rank:		$3$
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:		\( 147.466265671 \)

Galois group

$SD_{16}$ (as 8T8):

A solvable group of order 16

The 7 conjugacy class representatives for $QD_{16}$

Character table for $QD_{16}$

Intermediate fields

\(\Q(\sqrt{3}) \), 4.4.39744.1

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

Sibling fields

Galois closure:

data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/13.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.8.0.1}{8} }$	R	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.8.0.1}{8} }$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$	2.8.16.66	$x^{8} + 8 x^{4} + 336$	$8$	$1$	$16$	$QD_{16}$	$[2, 2, 5/2]^{2}$
$3$	3.8.7.2	$x^{8} - 3$	$8$	$1$	$7$	$QD_{16}$	$[\ ]_{8}^{2}$
$23$	$\Q_{23}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{23}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	23.2.1.1	$x^{2} - 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.1	$x^{2} - 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.1	$x^{2} - 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_3.2t1.1c1	$1$	$ 2^{2} \cdot 3 $	$x^{2} - 3$	$C_2$ (as 2T1)	$1$	$1$
	1.3_23.2t1.1c1	$1$	$ 3 \cdot 23 $	$x^{2} - x - 17$	$C_2$ (as 2T1)	$1$	$1$
	1.2e2_23.2t1.1c1	$1$	$ 2^{2} \cdot 23 $	$x^{2} - 23$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e4_3e2_23.4t3.1c1	$2$	$ 2^{4} \cdot 3^{2} \cdot 23 $	$x^{4} - 9 x^{2} + 3$	$D_{4}$ (as 4T3)	$1$	$2$
*	2.2e5_3e2_23.8t8.2c1	$2$	$ 2^{5} \cdot 3^{2} \cdot 23 $	$x^{8} - 4 x^{7} + 10 x^{6} + 8 x^{5} - 32 x^{4} - 28 x^{3} + 202 x^{2} + 284 x + 157$	$QD_{16}$ (as 8T8)	$0$	$-2$
*	2.2e5_3e2_23.8t8.2c2	$2$	$ 2^{5} \cdot 3^{2} \cdot 23 $	$x^{8} - 4 x^{7} + 10 x^{6} + 8 x^{5} - 32 x^{4} - 28 x^{3} + 202 x^{2} + 284 x + 157$	$QD_{16}$ (as 8T8)	$0$	$-2$

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 *.