Global number field 6.2.150006330000.2

• Label: 6.2.150006330000.2
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 3^{7}\cdot 5^{4}\cdot 19^{3}$
• Root discriminant: $72.89$
• Ramified primes: $2, 3, 5, 19$
• Class number: $45$
• Class group: $[3, 15]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 24 x^{4} + 42 x^{3} - 447 x^{2} + 873 x - 1002 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(150006330000=2^{4}\cdot 3^{7}\cdot 5^{4}\cdot 19^{3}\)
Root discriminant:		$72.89$
Ramified primes:		$2, 3, 5, 19$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{3} - \frac{3}{8} a + \frac{1}{4}$, $\frac{1}{64} a^{4} + \frac{3}{64} a^{3} - \frac{3}{64} a^{2} + \frac{17}{64} a - \frac{9}{32}$, $\frac{1}{128} a^{5} + \frac{1}{32} a^{3} + \frac{13}{64} a^{2} - \frac{53}{128} a + \frac{11}{64}$

Class group and class number

$C_{3}\times C_{15}$, which has order $45$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{64} a^{5} + \frac{5}{16} a^{3} + \frac{61}{32} a^{2} - \frac{261}{64} a - \frac{293}{32} \),  \( \frac{1}{128} a^{5} - \frac{1}{64} a^{4} + \frac{7}{64} a^{3} + \frac{3}{4} a^{2} - \frac{583}{128} a + \frac{301}{64} \),  \( \frac{13}{128} a^{5} - \frac{23}{64} a^{4} + \frac{141}{64} a^{3} + \frac{103}{32} a^{2} - \frac{6991}{128} a + \frac{4637}{64} \)
Regulator:		\( 693.54289438 \)

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

\(\Q(\sqrt{57}) \), 3.1.51300.1

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:	\(\Q\) $\times$ \(\Q(\sqrt{-1}) \) $\times$ 3.1.51300.1
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	R	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.2.2.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
	2.2.2.1	$x^{2} + 2 x + 2$	$2$	$1$	$2$	$C_2$	$[2]$
$3$	3.6.7.2	$x^{6} + 3 x^{2} + 6$	$6$	$1$	$7$	$D_{6}$	$[3/2]_{2}^{2}$
$5$	5.6.4.1	$x^{6} + 25 x^{3} + 200$	$3$	$2$	$4$	$S_3$	$[\ ]_{3}^{2}$
$19$	19.2.1.2	$x^{2} + 76$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.4.2.1	$x^{4} + 57 x^{2} + 1444$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$

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_3_19.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 19 $	$x^{2} + 57$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3_19.2t1.1c1	$1$	$ 3 \cdot 19 $	$x^{2} - x - 14$	$C_2$ (as 2T1)	$1$	$1$
*	2.2e2_3e3_5e2_19.3t2.1c1	$2$	$ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 19 $	$x^{3} + 45 x - 130$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3e3_5e2_19.6t3.4c1	$2$	$ 2^{2} \cdot 3^{3} \cdot 5^{2} \cdot 19 $	$x^{6} - 3 x^{5} + 24 x^{4} + 42 x^{3} - 447 x^{2} + 873 x - 1002$	$D_{6}$ (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 *.