Global number field 6.2.2024284625.1

• Label: 6.2.2024284625.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $5^{3}\cdot 11^{3}\cdot 23^{3}$
• Root discriminant: $35.57$
• Ramified primes: $5, 11, 23$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 29 x^{4} - 152 x^{3} + 320 x^{2} - 1736 x - 58141 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(2024284625=5^{3}\cdot 11^{3}\cdot 23^{3}\)
Root discriminant:		$35.57$
Ramified primes:		$5, 11, 23$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{14} a^{3} - \frac{1}{14} a^{2} + \frac{1}{14}$, $\frac{1}{14} a^{4} - \frac{1}{14} a^{2} + \frac{1}{14} a + \frac{1}{14}$, $\frac{1}{22343482} a^{5} - \frac{70846}{11171741} a^{4} + \frac{360466}{11171741} a^{3} + \frac{2772236}{11171741} a^{2} - \frac{10581981}{22343482} a - \frac{9645197}{22343482}$

Class group and class number

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

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:		\( 55.0234450386 \)

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

\(\Q(\sqrt{1265}) \), 3.1.23.1

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:	3.1.23.1 $\times$ \(\Q(\sqrt{-55}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.0.88012375.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/3.6.0.1}{6} }$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$	5.2.1.2	$x^{2} + 10$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$11$	11.2.1.1	$x^{2} - 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$23$	23.2.1.1	$x^{2} - 23$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	23.4.2.1	$x^{4} + 299 x^{2} + 25921$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.23.2t1.1c1	$1$	$ 23 $	$x^{2} - x + 6$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5_11_23.2t1.1c1	$1$	$ 5 \cdot 11 \cdot 23 $	$x^{2} - x - 316$	$C_2$ (as 2T1)	$1$	$1$
	1.5_11.2t1.1c1	$1$	$ 5 \cdot 11 $	$x^{2} - x + 14$	$C_2$ (as 2T1)	$1$	$-1$
*	2.23.3t2.1c1	$2$	$ 23 $	$x^{3} - x^{2} + 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.5e2_11e2_23.6t3.1c1	$2$	$ 5^{2} \cdot 11^{2} \cdot 23 $	$x^{6} - 2 x^{5} + 29 x^{4} - 152 x^{3} + 320 x^{2} - 1736 x - 58141$	$D_{6}$ (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 *.