Global number field 8.0.33237432513.1

• Label: 8.0.33237432513.1
• Degree: $8$
• Signature: $[0, 4]$
• Discriminant: $3^{4}\cdot 17^{7}$
• Root discriminant: $20.66$
• Ramified primes: $3, 17$
• Class number: $2$
• Class group: $[2]$
• Galois group: $C_8$ (as 8T1)

Normalized defining polynomial

\( x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103 \)

Invariants

Degree:		$8$
Signature:		$[0, 4]$
Discriminant:		\(33237432513=3^{4}\cdot 17^{7}\)
Root discriminant:		$20.66$
Ramified primes:		$3, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(51=3\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{51}(32,·)$, $\chi_{51}(1,·)$, $\chi_{51}(2,·)$, $\chi_{51}(4,·)$, $\chi_{51}(8,·)$, $\chi_{51}(13,·)$, $\chi_{51}(16,·)$, $\chi_{51}(26,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{798953} a^{7} + \frac{331532}{798953} a^{6} + \frac{236450}{798953} a^{5} + \frac{106338}{798953} a^{4} - \frac{43909}{798953} a^{3} - \frac{358898}{798953} a^{2} - \frac{58192}{798953} a - \frac{250292}{798953}$

Class group and class number

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

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:		\( 27.6959098582 \)

Galois group

$C_8$ (as 8T1):

A cyclic group of order 8

The 8 conjugacy class representatives for $C_8$

Character table for $C_8$

Intermediate fields

\(\Q(\sqrt{17}) \), 4.4.4913.1

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

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.4.0.1}{4} }^{2}$	R	${\href{/LocalNumberField/5.8.0.1}{8} }$	${\href{/LocalNumberField/7.8.0.1}{8} }$	${\href{/LocalNumberField/11.8.0.1}{8} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{4}$	R	${\href{/LocalNumberField/19.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/23.8.0.1}{8} }$	${\href{/LocalNumberField/29.8.0.1}{8} }$	${\href{/LocalNumberField/31.8.0.1}{8} }$	${\href{/LocalNumberField/37.8.0.1}{8} }$	${\href{/LocalNumberField/41.8.0.1}{8} }$	${\href{/LocalNumberField/43.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{8}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{2}$	${\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
$3$	3.8.4.2	$x^{8} - 27 x^{2} + 162$	$2$	$4$	$4$	$C_8$	$[\ ]_{2}^{4}$
$17$	17.8.7.1	$x^{8} - 1377$	$8$	$1$	$7$	$C_8$	$[\ ]_{8}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_17.8t1.1c1	$1$	$ 3 \cdot 17 $	$x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.4t1.1c1	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.3_17.8t1.1c2	$1$	$ 3 \cdot 17 $	$x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.2t1.1c1	$1$	$ 17 $	$x^{2} - x - 4$	$C_2$ (as 2T1)	$1$	$1$
*	1.3_17.8t1.1c3	$1$	$ 3 \cdot 17 $	$x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103$	$C_8$ (as 8T1)	$0$	$-1$
*	1.17.4t1.1c2	$1$	$ 17 $	$x^{4} - x^{3} - 6 x^{2} + x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.3_17.8t1.1c4	$1$	$ 3 \cdot 17 $	$x^{8} - x^{7} + 10 x^{6} - 11 x^{5} + 15 x^{4} - 61 x^{3} + 58 x^{2} - 47 x + 103$	$C_8$ (as 8T1)	$0$	$-1$

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 *.