Global number field 6.0.63078912.1

• Label: 6.0.63078912.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{9}\cdot 3^{6}\cdot 13^{2}$
• Root discriminant: $19.95$
• Ramified primes: $2, 3, 13$
• Class number: $3$
• Class group: $[3]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} + 12 x^{4} - 6 x^{3} + 21 x^{2} + 18 x + 11 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-63078912=-\,2^{9}\cdot 3^{6}\cdot 13^{2}\)
Root discriminant:		$19.95$
Ramified primes:		$2, 3, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{343} a^{5} + \frac{102}{343} a^{4} + \frac{18}{49} a^{3} + \frac{155}{343} a^{2} + \frac{53}{343} a - \frac{64}{343}$

Class group and class number

$C_{3}$, which has order $3$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{30}{343} a^{5} - \frac{27}{343} a^{4} + \frac{50}{49} a^{3} - \frac{495}{343} a^{2} + \frac{904}{343} a - \frac{205}{343} \),  \( \frac{17}{343} a^{5} + \frac{19}{343} a^{4} + \frac{12}{49} a^{3} - \frac{109}{343} a^{2} - \frac{128}{343} a - \frac{59}{343} \)
Regulator:		\( 18.9816670161 \)

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

\(\Q(\sqrt{-2}) \), 3.1.351.1

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

Sibling algebras

Galois closure:	Deg 12
Twin sextic algebra:	3.1.351.1 $\times$ \(\Q(\sqrt{78}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.2.2460077568.1

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	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.1.0.1}{1} }^{6}$	R	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.9.5	$x^{6} - 4 x^{4} + 4 x^{2} + 8$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
$3$	3.3.3.2	$x^{3} + 3 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
	3.3.3.2	$x^{3} + 3 x + 3$	$3$	$1$	$3$	$S_3$	$[3/2]_{2}$
$13$	13.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	13.4.2.1	$x^{4} + 39 x^{2} + 676$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_13.2t1.1c1	$1$	$ 3 \cdot 13 $	$x^{2} - x + 10$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_3_13.2t1.1c1	$1$	$ 2^{3} \cdot 3 \cdot 13 $	$x^{2} - 78$	$C_2$ (as 2T1)	$1$	$1$
*	2.3e3_13.3t2.1c1	$2$	$ 3^{3} \cdot 13 $	$x^{3} + 3 x - 3$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e6_3e3_13.6t3.6c1	$2$	$ 2^{6} \cdot 3^{3} \cdot 13 $	$x^{6} + 12 x^{4} - 6 x^{3} + 21 x^{2} + 18 x + 11$	$D_{6}$ (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 *.