Global number field 8.8.16098453125.1

• Label: 8.8.16098453125.1
• Degree: $8$
• Signature: $[8, 0]$
• Discriminant: $5^{6}\cdot 101^{3}$
• Root discriminant: $18.87$
• Ramified primes: $5, 101$
• Class number: $1$
• Class group: Trivial
• Galois group: $QD_{16}$ (as 8T8)

Normalized defining polynomial

\( x^{8} - 3 x^{7} - 8 x^{6} + 35 x^{5} - 19 x^{4} - 45 x^{3} + 53 x^{2} - 14 x + 1 \)

Invariants

Degree:		$8$
Signature:		$[8, 0]$
Discriminant:		\(16098453125=5^{6}\cdot 101^{3}\)
Root discriminant:		$18.87$
Ramified primes:		$5, 101$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$7$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( 2 a^{7} - 4 a^{6} - 19 a^{5} + 50 a^{4} + 2 a^{3} - 73 a^{2} + 45 a - 5 \),  \( a^{7} - 2 a^{6} - 10 a^{5} + 25 a^{4} + 6 a^{3} - 39 a^{2} + 14 a \),  \( a^{7} - 2 a^{6} - 10 a^{5} + 25 a^{4} + 6 a^{3} - 39 a^{2} + 14 a + 1 \),  \( a \),  \( a^{6} - 2 a^{5} - 11 a^{4} + 24 a^{3} + 14 a^{2} - 37 a + 7 \),  \( a^{7} - a^{6} - 11 a^{5} + 15 a^{4} + 21 a^{3} - 27 a^{2} - 7 a + 3 \),  \( 2 a^{7} - 5 a^{6} - 19 a^{5} + 60 a^{4} - 4 a^{3} - 90 a^{2} + 58 a - 6 \)
Regulator:		\( 297.817403022 \)

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

\(\Q(\sqrt{5}) \), 4.4.2525.1

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	${\href{/LocalNumberField/2.8.0.1}{8} }$	${\href{/LocalNumberField/3.8.0.1}{8} }$	R	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/17.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/53.8.0.1}{8} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$5$	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
$101$	101.2.1.2	$x^{2} + 202$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	101.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	101.4.2.1	$x^{4} + 505 x^{2} + 91809$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
	1.101.2t1.1c1	$1$	$ 101 $	$x^{2} - x - 25$	$C_2$ (as 2T1)	$1$	$1$
	1.5_101.2t1.1c1	$1$	$ 5 \cdot 101 $	$x^{2} - x - 126$	$C_2$ (as 2T1)	$1$	$1$
*	2.5_101.4t3.2c1	$2$	$ 5 \cdot 101 $	$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 5$	$D_{4}$ (as 4T3)	$1$	$2$
*	2.5e2_101.8t8.1c1	$2$	$ 5^{2} \cdot 101 $	$x^{8} - 3 x^{7} - 8 x^{6} + 35 x^{5} - 19 x^{4} - 45 x^{3} + 53 x^{2} - 14 x + 1$	$QD_{16}$ (as 8T8)	$0$	$2$
*	2.5e2_101.8t8.1c2	$2$	$ 5^{2} \cdot 101 $	$x^{8} - 3 x^{7} - 8 x^{6} + 35 x^{5} - 19 x^{4} - 45 x^{3} + 53 x^{2} - 14 x + 1$	$QD_{16}$ (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 *.