Global number field 6.0.54442800.1

• Label: 6.0.54442800.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{4}\cdot 3^{3}\cdot 5^{2}\cdot 71^{2}$
• Root discriminant: $19.47$
• Ramified primes: $2, 3, 5, 71$
• Class number: $2$
• Class group: $[2]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - x^{5} + 6 x^{4} - 29 x^{3} + 42 x^{2} - 85 x + 289 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-54442800=-\,2^{4}\cdot 3^{3}\cdot 5^{2}\cdot 71^{2}\)
Root discriminant:		$19.47$
Ramified primes:		$2, 3, 5, 71$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{5678} a^{5} + \frac{331}{2839} a^{4} - \frac{1133}{5678} a^{3} - \frac{856}{2839} a^{2} - \frac{2253}{5678} a - \frac{15}{167}$

Class group and class number

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

Unit group

Rank:		$2$
Torsion generator:		\( \frac{30}{2839} a^{5} - \frac{13}{2839} a^{4} + \frac{78}{2839} a^{3} - \frac{258}{2839} a^{2} + \frac{546}{2839} a + \frac{102}{167} \) (order $6$)
Fundamental units:		\( \frac{309}{5678} a^{5} + \frac{75}{2839} a^{4} + \frac{1939}{5678} a^{3} - \frac{3316}{2839} a^{2} - \frac{3461}{5678} a - \frac{1128}{167} \),  \( \frac{507}{5678} a^{5} + \frac{316}{2839} a^{4} + \frac{4725}{5678} a^{3} - \frac{2464}{2839} a^{2} + \frac{4685}{5678} a - \frac{1593}{167} \)
Regulator:		\( 36.7418037876 \)

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

\(\Q(\sqrt{-3}) \), 3.1.1420.1

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:	3.1.1420.1 $\times$ \(\Q(\sqrt{1065}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.2.19327194000.1

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	R	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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	$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.4.2.1	$x^{4} + 9 x^{2} + 36$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$5$	5.2.0.1	$x^{2} - x + 2$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	5.4.2.1	$x^{4} + 15 x^{2} + 100$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$71$	71.2.0.1	$x^{2} - x + 11$	$1$	$2$	$0$	$C_2$	$[\ ]^{2}$
	71.4.2.1	$x^{4} + 1491 x^{2} + 609961$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.5_71.2t1.1c1	$1$	$ 5 \cdot 71 $	$x^{2} - x + 89$	$C_2$ (as 2T1)	$1$	$-1$
	1.3_5_71.2t1.1c1	$1$	$ 3 \cdot 5 \cdot 71 $	$x^{2} - x - 266$	$C_2$ (as 2T1)	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	2.2e2_5_71.3t2.1c1	$2$	$ 2^{2} \cdot 5 \cdot 71 $	$x^{3} - x^{2} - 5 x + 17$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3e2_5_71.6t3.3c1	$2$	$ 2^{2} \cdot 3^{2} \cdot 5 \cdot 71 $	$x^{6} - x^{5} + 6 x^{4} - 29 x^{3} + 42 x^{2} - 85 x + 289$	$D_{6}$ (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 *.