Global number field 6.0.5522223168.10

• Label: 6.0.5522223168.10
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 3^{3}\cdot 7^{4}\cdot 11^{3}$
• Root discriminant: $42.04$
• Ramified primes: $2, 3, 7, 11$
• Class number: $108$
• Class group: $[6, 18]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 96 x^{4} - 126 x^{3} + 3335 x^{2} - 2248 x + 41581 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-5522223168=-\,2^{6}\cdot 3^{3}\cdot 7^{4}\cdot 11^{3}\)
Root discriminant:		$42.04$
Ramified primes:		$2, 3, 7, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(924=2^{2}\cdot 3\cdot 7\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{924}(1,·)$, $\chi_{924}(659,·)$, $\chi_{924}(529,·)$, $\chi_{924}(263,·)$, $\chi_{924}(793,·)$, $\chi_{924}(527,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2550421} a^{5} - \frac{724625}{2550421} a^{4} - \frac{734009}{2550421} a^{3} - \frac{294385}{2550421} a^{2} + \frac{932750}{2550421} a + \frac{64554}{2550421}$

Class group and class number

$C_{6}\times C_{18}$, which has order $108$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{542}{2550421} a^{5} + \frac{18084}{2550421} a^{4} + \frac{32798}{2550421} a^{3} + \frac{1119853}{2550421} a^{2} + \frac{567142}{2550421} a + \frac{22236163}{2550421} \),  \( \frac{264}{2550421} a^{5} - \frac{19425}{2550421} a^{4} + \frac{53620}{2550421} a^{3} - \frac{1205010}{2550421} a^{2} + \frac{1405584}{2550421} a - \frac{23764480}{2550421} \)
Regulator:		\( 2.10181872849 \)

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

\(\Q(\sqrt{-33}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-33}) \) $\times$ \(\Q(\zeta_{7})^+\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	R	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.6.3	$x^{6} + 2 x^{4} + x^{2} - 7$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$3$	3.6.3.1	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$	7.3.2.2	$x^{3} - 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.2	$x^{3} - 7$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$11$	11.6.3.2	$x^{6} - 121 x^{2} + 3993$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_3_11.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 11 $	$x^{2} + 33$	$C_2$ (as 2T1)	$1$	$-1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3_7_11.6t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 7 \cdot 11 $	$x^{6} - 2 x^{5} + 96 x^{4} - 126 x^{3} + 3335 x^{2} - 2248 x + 41581$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3_7_11.6t1.1c2	$1$	$ 2^{2} \cdot 3 \cdot 7 \cdot 11 $	$x^{6} - 2 x^{5} + 96 x^{4} - 126 x^{3} + 3335 x^{2} - 2248 x + 41581$	$C_6$ (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 *.