Global number field 6.0.3116883375.2

• Label: 6.0.3116883375.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 5^{3}\cdot 31^{4}$
• Root discriminant: $38.22$
• Ramified primes: $3, 5, 31$
• Class number: $126$
• Class group: $[3, 42]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} - 9 x^{4} - 3 x^{3} + 152 x^{2} + 148 x + 736 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-3116883375=-\,3^{3}\cdot 5^{3}\cdot 31^{4}\)
Root discriminant:		$38.22$
Ramified primes:		$3, 5, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(465=3\cdot 5\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{465}(1,·)$, $\chi_{465}(211,·)$, $\chi_{465}(404,·)$, $\chi_{465}(149,·)$, $\chi_{465}(346,·)$, $\chi_{465}(284,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{4448} a^{5} + \frac{233}{4448} a^{4} + \frac{581}{4448} a^{3} - \frac{825}{4448} a^{2} + \frac{573}{2224} a - \frac{119}{278}$

Class group and class number

$C_{3}\times C_{42}$, which has order $126$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{17}{2224} a^{5} + \frac{69}{2224} a^{4} - \frac{131}{2224} a^{3} - \frac{1237}{2224} a^{2} - \frac{823}{1112} a + \frac{62}{139} \),  \( \frac{23}{2224} a^{5} - \frac{201}{2224} a^{4} + \frac{19}{2224} a^{3} + \frac{2153}{2224} a^{2} + \frac{947}{1112} a - \frac{1764}{139} \)
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

\(\Q(\sqrt{-15}) \), 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:

3.3.961.1 $\times$ \(\Q(\sqrt{-15}) \) $\times$ \(\Q\)

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	R	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$5$	5.6.3.2	$x^{6} - 25 x^{2} + 250$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$31$	31.3.2.1	$x^{3} - 31$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	31.3.2.1	$x^{3} - 31$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

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_5.2t1.1c1	$1$	$ 3 \cdot 5 $	$x^{2} - x + 4$	$C_2$ (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_5_31.6t1.2c1	$1$	$ 3 \cdot 5 \cdot 31 $	$x^{6} - x^{5} - 9 x^{4} - 3 x^{3} + 152 x^{2} + 148 x + 736$	$C_6$ (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_5_31.6t1.2c2	$1$	$ 3 \cdot 5 \cdot 31 $	$x^{6} - x^{5} - 9 x^{4} - 3 x^{3} + 152 x^{2} + 148 x + 736$	$C_6$ (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 *.