Global number field 6.0.83320384.2

• Label: 6.0.83320384.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 7^{2}\cdot 163^{2}$
• Root discriminant: $20.90$
• Ramified primes: $2, 7, 163$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 9 x^{4} - 10 x^{3} + 91 x^{2} + 290 x + 314 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-83320384=-\,2^{6}\cdot 7^{2}\cdot 163^{2}\)
Root discriminant:		$20.90$
Ramified primes:		$2, 7, 163$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{7} a^{4} + \frac{3}{7} a^{3} + \frac{3}{7} a^{2} + \frac{1}{7} a - \frac{2}{7}$, $\frac{1}{1911} a^{5} - \frac{121}{1911} a^{4} + \frac{709}{1911} a^{3} + \frac{67}{273} a^{2} + \frac{45}{91} a + \frac{59}{1911}$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{16}{1911} a^{5} + \frac{25}{1911} a^{4} + \frac{122}{1911} a^{3} + \frac{20}{273} a^{2} - \frac{83}{91} a - \frac{2855}{1911} \) (order $4$)
Fundamental units:		\( \frac{128}{1911} a^{5} - \frac{200}{1911} a^{4} - \frac{976}{1911} a^{3} + \frac{113}{273} a^{2} + \frac{482}{91} a + \frac{20929}{1911} \),  \( \frac{536}{1911} a^{5} - \frac{428}{1911} a^{4} - \frac{5725}{1911} a^{3} + \frac{461}{273} a^{2} + \frac{322}{13} a + \frac{118711}{1911} \)
Regulator:		\( 37.5212500781 \)

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

\(\Q(\sqrt{-1}) \), 3.1.4564.1

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:	3.1.4564.1 $\times$ \(\Q(\sqrt{1141}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.2.23767139536.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$	2.2.2.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.4.4.1	$x^{4} + 8 x^{2} + 4$	$2$	$2$	$4$	$C_2^2$	$[2]^{2}$
$7$	7.2.0.1	$x^{2} - x + 3$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	7.4.2.1	$x^{4} + 35 x^{2} + 441$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$163$	163.2.0.1	$x^{2} - x + 11$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	163.4.2.1	$x^{4} + 3423 x^{2} + 3214849$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.2e2_7_163.2t1.1c1	$1$	$ 2^{2} \cdot 7 \cdot 163 $	$x^{2} + 1141$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.7_163.2t1.1c1	$1$	$ 7 \cdot 163 $	$x^{2} - x - 285$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_7_163.3t2.1c1	$2$	$ 2^{2} \cdot 7 \cdot 163 $	$x^{3} + x - 26$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_7_163.6t3.3c1	$2$	$ 2^{2} \cdot 7 \cdot 163 $	$x^{6} - 9 x^{4} - 10 x^{3} + 91 x^{2} + 290 x + 314$	$D_{6}$ (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 *.