Global number field 6.0.608141349063.3

• Label: 6.0.608141349063.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 29^{3}\cdot 31^{4}$
• Root discriminant: $92.05$
• Ramified primes: $3, 29, 31$
• Class number: $1512$
• Class group: $[2, 6, 126]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 45 x^{4} - 39 x^{3} + 1538 x^{2} + 184 x + 22336 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-608141349063=-\,3^{3}\cdot 29^{3}\cdot 31^{4}\)
Root discriminant:		$92.05$
Ramified primes:		$3, 29, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(2697=3\cdot 29\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{2697}(1,·)$, $\chi_{2697}(869,·)$, $\chi_{2697}(2609,·)$, $\chi_{2697}(521,·)$, $\chi_{2697}(1741,·)$, $\chi_{2697}(2350,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{151904} a^{5} - \frac{1301}{75952} a^{4} + \frac{27107}{151904} a^{3} + \frac{8043}{75952} a^{2} + \frac{4273}{9494} a - \frac{1854}{4747}$

Class group and class number

$C_{2}\times C_{6}\times C_{126}$, which has order $1512$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{59}{75952} a^{5} - \frac{807}{37976} a^{4} + \frac{4321}{75952} a^{3} - \frac{19151}{37976} a^{2} + \frac{5263}{4747} a - \frac{90603}{4747} \),  \( \frac{5}{4747} a^{5} + \frac{177}{18988} a^{4} + \frac{491}{9494} a^{3} + \frac{3671}{18988} a^{2} + \frac{4859}{9494} a + \frac{11915}{4747} \)
Regulator:		\( 48.7831276503 \)

Galois group

$C_6$ (as 6T1):

A cyclic group of order 6

The 6 conjugacy class representatives for $C_6$

Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-87}) \), 3.3.961.1

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-87}) \) $\times$ 3.3.961.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	R	R	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$	3.6.3.1	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$29$	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	29.2.1.2	$x^{2} + 58$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$31$	31.6.4.1	$x^{6} + 1085 x^{3} + 1660608$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_29.2t1.1c1	$1$	$ 3 \cdot 29 $	$x^{2} - x + 22$	$C_2$ (as 2T1)	$1$	$-1$
*	1.31.3t1.1c1	$1$	$ 31 $	$x^{3} - x^{2} - 10 x + 8$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_29_31.6t1.1c1	$1$	$ 3 \cdot 29 \cdot 31 $	$x^{6} - x^{5} + 45 x^{4} - 39 x^{3} + 1538 x^{2} + 184 x + 22336$	$C_6$ (as 6T1)	$0$	$-1$
*	1.31.3t1.1c2	$1$	$ 31 $	$x^{3} - x^{2} - 10 x + 8$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_29_31.6t1.1c2	$1$	$ 3 \cdot 29 \cdot 31 $	$x^{6} - x^{5} + 45 x^{4} - 39 x^{3} + 1538 x^{2} + 184 x + 22336$	$C_6$ (as 6T1)	$0$	$-1$

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 *.