Global number field 6.2.2824935696.3

• Label: 6.2.2824935696.3
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 3^{3}\cdot 11^{3}\cdot 17^{3}$
• Root discriminant: $37.60$
• Ramified primes: $2, 3, 11, 17$
• Class number: $6$
• Class group: $[6]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 9 x^{4} + 82 x^{3} - 47 x^{2} - 360 x - 3753 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(2824935696=2^{4}\cdot 3^{3}\cdot 11^{3}\cdot 17^{3}\)
Root discriminant:		$37.60$
Ramified primes:		$2, 3, 11, 17$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{6} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{4} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{155484} a^{5} + \frac{457}{17276} a^{4} + \frac{262}{12957} a^{3} - \frac{35593}{77742} a^{2} - \frac{5601}{17276} a - \frac{1971}{17276}$

Class group and class number

$C_{6}$, which has order $6$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{107}{155484} a^{5} - \frac{149}{51828} a^{4} - \frac{79}{25914} a^{3} + \frac{6932}{38871} a^{2} - \frac{3285}{17276} a - \frac{38137}{17276} \),  \( \frac{499}{25914} a^{5} + \frac{431}{12957} a^{4} - \frac{1630}{12957} a^{3} + \frac{3926}{4319} a^{2} + \frac{103421}{25914} a + \frac{64430}{4319} \),  \( \frac{67}{12957} a^{5} - \frac{845}{12957} a^{4} - \frac{3207}{4319} a^{3} + \frac{11671}{12957} a^{2} + \frac{77797}{12957} a + \frac{117790}{4319} \)
Regulator:		\( 357.304483008 \)

Galois group

$D_6$ (as 6T3):

A solvable group of order 12

The 6 conjugacy class representatives for $D_{6}$

Character table for $D_{6}$

Intermediate fields

\(\Q(\sqrt{561}) \), 3.1.204.1

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

Sibling algebras

Galois closure:	data not computed
Twin sextic algebra:	\(\Q\) $\times$ \(\Q(\sqrt{-11}) \) $\times$ 3.1.204.1
Degree 6 sibling:	data not computed

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	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/13.6.0.1}{6} }$	R	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	2.3.2.1	$x^{3} - 2$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
$3$	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.1	$x^{2} - 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$11$	11.6.3.2	$x^{6} - 121 x^{2} + 3993$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$17$	17.2.1.1	$x^{2} - 17$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.4.2.1	$x^{4} + 85 x^{2} + 2601$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_17.2t1.1c1	$1$	$ 3 \cdot 17 $	$x^{2} - x + 13$	$C_2$ (as 2T1)	$1$	$-1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_11_17.2t1.1c1	$1$	$ 3 \cdot 11 \cdot 17 $	$x^{2} - x - 140$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_3_17.3t2.1c1	$2$	$ 2^{2} \cdot 3 \cdot 17 $	$x^{3} - x^{2} + x - 3$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3_11e2_17.6t3.1c1	$2$	$ 2^{2} \cdot 3 \cdot 11^{2} \cdot 17 $	$x^{6} - 2 x^{5} - 9 x^{4} + 82 x^{3} - 47 x^{2} - 360 x - 3753$	$D_{6}$ (as 6T3)	$1$	$0$

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 *.