Global number field 6.0.1851523947.2

• Label: 6.0.1851523947.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 7^{4}\cdot 13^{4}$
• Root discriminant: $35.04$
• Ramified primes: $3, 7, 13$
• Class number: $9$
• Class group: $[3, 3]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 31 x^{4} - 98 x^{3} + 964 x^{2} - 1920 x + 4096 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-1851523947=-\,3^{3}\cdot 7^{4}\cdot 13^{4}\)
Root discriminant:		$35.04$
Ramified primes:		$3, 7, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(273=3\cdot 7\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{273}(256,·)$, $\chi_{273}(1,·)$, $\chi_{273}(16,·)$, $\chi_{273}(74,·)$, $\chi_{273}(107,·)$, $\chi_{273}(92,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{894848} a^{5} + \frac{13951}{894848} a^{4} + \frac{14943}{894848} a^{3} + \frac{216239}{447424} a^{2} + \frac{105345}{223712} a - \frac{465}{6991}$

Class group and class number

$C_{3}\times C_{3}$, which has order $9$

Unit group

Rank:		$2$
Torsion generator:		\( \frac{465}{894848} a^{5} - \frac{433}{894848} a^{4} + \frac{13423}{894848} a^{3} - \frac{7409}{447424} a^{2} + \frac{104353}{223712} a + \frac{496}{6991} \) (order $6$)
Fundamental units:		\( \frac{27}{27964} a^{5} - \frac{837}{27964} a^{4} - \frac{2017}{27964} a^{3} - \frac{6507}{6991} a^{2} + \frac{12960}{6991} a - \frac{45219}{6991} \),  \( \frac{4121}{894848} a^{5} - \frac{1913}{894848} a^{4} + \frac{59303}{894848} a^{3} - \frac{35833}{447424} a^{2} + \frac{461033}{223712} a - \frac{28695}{6991} \)
Regulator:		\( 62.4891977899 \)

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

\(\Q(\sqrt{-3}) \), 3.3.8281.1

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

Sibling algebras

Twin sextic algebra:

3.3.8281.1 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	R	${\href{/LocalNumberField/11.6.0.1}{6} }$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$13$	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.7_13.3t1.2c1	$1$	$ 7 \cdot 13 $	$x^{3} - x^{2} - 30 x + 64$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_7_13.6t1.6c1	$1$	$ 3 \cdot 7 \cdot 13 $	$x^{6} - x^{5} + 31 x^{4} - 98 x^{3} + 964 x^{2} - 1920 x + 4096$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7_13.3t1.2c2	$1$	$ 7 \cdot 13 $	$x^{3} - x^{2} - 30 x + 64$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_7_13.6t1.6c2	$1$	$ 3 \cdot 7 \cdot 13 $	$x^{6} - x^{5} + 31 x^{4} - 98 x^{3} + 964 x^{2} - 1920 x + 4096$	$C_6$ (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 *.