Global number field 6.2.98064208.2

• Label: 6.2.98064208.2
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 11^{2}\cdot 37^{3}$
• Root discriminant: $21.47$
• Ramified primes: $2, 11, 37$
• Class number: $3$
• Class group: $[3]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - x^{5} - 26 x^{4} + 15 x^{3} + 254 x^{2} - 157 x - 889 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(98064208=2^{4}\cdot 11^{2}\cdot 37^{3}\)
Root discriminant:		$21.47$
Ramified primes:		$2, 11, 37$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{56321} a^{5} + \frac{5731}{56321} a^{4} + \frac{14923}{56321} a^{3} - \frac{12948}{56321} a^{2} + \frac{788}{3313} a + \frac{20192}{56321}$

Class group and class number

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

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{98}{56321} a^{5} - \frac{1572}{56321} a^{4} - \frac{1892}{56321} a^{3} + \frac{26479}{56321} a^{2} + \frac{1025}{3313} a - \frac{161382}{56321} \),  \( \frac{403}{56321} a^{5} + \frac{432}{56321} a^{4} - \frac{12378}{56321} a^{3} + \frac{19809}{56321} a^{2} + \frac{2829}{3313} a - \frac{85490}{56321} \),  \( \frac{452}{56321} a^{5} - \frac{354}{56321} a^{4} - \frac{13324}{56321} a^{3} + \frac{4888}{56321} a^{2} + \frac{11624}{3313} a + \frac{284387}{56321} \)
Regulator:		\( 20.7491860853 \)

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

\(\Q(\sqrt{37}) \), 3.1.44.1

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:	\(\Q\) $\times$ \(\Q(\sqrt{-407}) \) $\times$ 3.1.44.1
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	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	R	${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$	2.6.4.1	$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$11$	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{11}$	$x + 3$	$1$	$1$	$0$	Trivial	$[\ ]$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$37$	37.6.3.1	$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
*	1.37.2t1.1c1	$1$	$ 37 $	$x^{2} - x - 9$	$C_2$ (as 2T1)	$1$	$1$
	1.11_37.2t1.1c1	$1$	$ 11 \cdot 37 $	$x^{2} - x + 102$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e2_11.3t2.1c1	$2$	$ 2^{2} \cdot 11 $	$x^{3} - x^{2} + x + 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_11_37e2.6t3.1c1	$2$	$ 2^{2} \cdot 11 \cdot 37^{2}$	$x^{6} - x^{5} - 26 x^{4} + 15 x^{3} + 254 x^{2} - 157 x - 889$	$D_{6}$ (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 *.