Global number field 6.2.1335255248.1

• Label: 6.2.1335255248.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 19^{3}\cdot 23^{3}$
• Root discriminant: $33.18$
• Ramified primes: $2, 19, 23$
• Class number: $2$
• Class group: $[2]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 3 x^{5} - 26 x^{4} + 45 x^{3} + 298 x^{2} - 795 x - 3257 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(1335255248=2^{4}\cdot 19^{3}\cdot 23^{3}\)
Root discriminant:		$33.18$
Ramified primes:		$2, 19, 23$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{1316551} a^{5} - \frac{348108}{1316551} a^{4} + \frac{148172}{1316551} a^{3} + \frac{421063}{1316551} a^{2} + \frac{120615}{1316551} a - \frac{557429}{1316551}$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
Regulator:		\( 82.9852576792 \)

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

\(\Q(\sqrt{437}) \), 3.1.76.1

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:	3.1.76.1 $\times$ \(\Q(\sqrt{-23}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.0.70276592.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	R	R	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$	2.6.4.1	$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$19$	19.2.1.1	$x^{2} - 19$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.4.2.1	$x^{4} + 57 x^{2} + 1444$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$23$	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.2.1.2	$x^{2} + 46$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.19.2t1.1c1	$1$	$ 19 $	$x^{2} - x + 5$	$C_2$ (as 2T1)	$1$	$-1$
	1.23.2t1.1c1	$1$	$ 23 $	$x^{2} - x + 6$	$C_2$ (as 2T1)	$1$	$-1$
*	1.19_23.2t1.1c1	$1$	$ 19 \cdot 23 $	$x^{2} - x - 109$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_19.3t2.1c1	$2$	$ 2^{2} \cdot 19 $	$x^{3} - 2 x - 2$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_19_23e2.6t3.1c1	$2$	$ 2^{2} \cdot 19 \cdot 23^{2}$	$x^{6} - 3 x^{5} - 26 x^{4} + 45 x^{3} + 298 x^{2} - 795 x - 3257$	$D_{6}$ (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 *.