Global number field 6.0.534534400.1

• Label: 6.0.534534400.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{8}\cdot 5^{2}\cdot 17^{4}$
• Root discriminant: $28.49$
• Ramified primes: $2, 5, 17$
• Class number: $6$
• Class group: $[6]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} + 23 x^{4} + 131 x^{2} + 25 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-534534400=-\,2^{8}\cdot 5^{2}\cdot 17^{4}\)
Root discriminant:		$28.49$
Ramified primes:		$2, 5, 17$
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{4} - \frac{1}{6}$, $\frac{1}{30} a^{5} + \frac{1}{10} a^{3} - \frac{1}{2} a^{2} - \frac{2}{15} a - \frac{1}{2}$

Class group and class number

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

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{1}{30} a^{5} - \frac{3}{5} a^{3} - \frac{71}{30} a \) (order $4$)
Fundamental units:		\( \frac{1}{30} a^{5} + \frac{8}{5} a^{3} + \frac{461}{30} a \),  \( \frac{38}{15} a^{5} + \frac{288}{5} a^{3} + \frac{4783}{15} a \)
Regulator:		\( 88.2934808747 \)

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

\(\Q(\sqrt{-1}) \), 3.3.5780.1

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:	\(\Q\) $\times$ \(\Q(\sqrt{-5}) \) $\times$ 3.3.5780.1
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	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$	2.6.8.1	$x^{6} + 2 x^{3} + 2$	$6$	$1$	$8$	$D_{6}$	$[2]_{3}^{2}$
$5$	$\Q_{5}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{5}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	5.2.1.1	$x^{2} - 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.2.1.1	$x^{2} - 5$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$17$	17.3.2.1	$x^{3} - 17$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$
	17.3.2.1	$x^{3} - 17$	$3$	$1$	$2$	$S_3$	$[\ ]_{3}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e2_5_17e2.3t2.2c1	$2$	$ 2^{2} \cdot 5 \cdot 17^{2}$	$x^{3} - x^{2} - 11 x + 5$	$S_3$ (as 3T2)	$1$	$2$
*	2.2e4_5_17e2.6t3.1c1	$2$	$ 2^{4} \cdot 5 \cdot 17^{2}$	$x^{6} + 23 x^{4} + 131 x^{2} + 25$	$D_{6}$ (as 6T3)	$1$	$-2$

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 *.