Global number field 6.2.634429136.1

• Label: 6.2.634429136.1
• Degree: $6$
• Signature: $[2, 2]$
• Discriminant: $2^{4}\cdot 11^{3}\cdot 31^{3}$
• Root discriminant: $29.31$
• Ramified primes: $2, 11, 31$
• Class number: $1$
• Class group: Trivial
• Galois group: $S_3^2$ (as 6T9)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 3 x^{4} + 34 x^{3} - 35 x^{2} + 36 x - 17 \)

Invariants

Degree:		$6$
Signature:		$[2, 2]$
Discriminant:		\(634429136=2^{4}\cdot 11^{3}\cdot 31^{3}\)
Root discriminant:		$29.31$
Ramified primes:		$2, 11, 31$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{146} a^{5} - \frac{11}{73} a^{4} + \frac{5}{146} a^{3} + \frac{7}{146} a^{2} + \frac{22}{73} a - \frac{41}{146}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$3$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{2}{73} a^{5} - \frac{15}{146} a^{4} + \frac{10}{73} a^{3} + \frac{87}{73} a^{2} - \frac{131}{73} a + \frac{201}{146} \),  \( \frac{6}{73} a^{5} - \frac{45}{146} a^{4} + \frac{30}{73} a^{3} + \frac{188}{73} a^{2} - \frac{612}{73} a + \frac{603}{146} \),  \( \frac{82}{73} a^{5} - \frac{469}{146} a^{4} - \frac{1735}{146} a^{3} + \frac{1805}{146} a^{2} - \frac{2055}{146} a + \frac{507}{73} \)
Regulator:		\( 221.810232795 \)

Galois group

$S_3^2$ (as 6T9):

A solvable group of order 36

The 9 conjugacy class representatives for $S_3^2$

Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{341}) \)

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:	3.1.31.1 $\times$ 3.1.44.1
Degree 9 sibling:	9.1.2537716544.1
Degree 12 sibling:	Deg 12
Degree 18 siblings:	Deg 18, Deg 18, Deg 18

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/7.6.0.1}{6} }$	R	${\href{/LocalNumberField/13.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$	R	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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
$2$	2.6.4.2	$x^{6} - 2 x^{3} + 4$	$3$	$2$	$4$	$S_3\times C_3$	$[\ ]_{3}^{6}$
$11$	11.2.1.1	$x^{2} - 11$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.4.2.1	$x^{4} + 143 x^{2} + 5929$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
$31$	31.6.3.2	$x^{6} - 961 x^{2} + 268119$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

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_31.2t1.1c1	$1$	$ 11 \cdot 31 $	$x^{2} - x - 85$	$C_2$ (as 2T1)	$1$	$1$
	1.31.2t1.1c1	$1$	$ 31 $	$x^{2} - x + 8$	$C_2$ (as 2T1)	$1$	$-1$
	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
	2.2e2_11_31e2.6t3.2c1	$2$	$ 2^{2} \cdot 11 \cdot 31^{2}$	$x^{6} - x^{5} + 25 x^{4} - 19 x^{3} + 186 x^{2} - 4 x + 386$	$D_{6}$ (as 6T3)	$1$	$0$
	2.2e2_11.3t2.1c1	$2$	$ 2^{2} \cdot 11 $	$x^{3} - x^{2} + x + 1$	$S_3$ (as 3T2)	$1$	$0$
	2.31.3t2.1c1	$2$	$ 31 $	$x^{3} + x - 1$	$S_3$ (as 3T2)	$1$	$0$
	2.11e2_31.6t3.1c1	$2$	$ 11^{2} \cdot 31 $	$x^{6} - 2 x^{5} + 7 x^{4} + 19 x^{3} - 16 x^{2} + 75 x - 611$	$D_{6}$ (as 6T3)	$1$	$0$
*	4.2e4_11e2_31e2.6t9.1c1	$4$	$ 2^{4} \cdot 11^{2} \cdot 31^{2}$	$x^{6} - 2 x^{5} + 3 x^{4} + 34 x^{3} - 35 x^{2} + 36 x - 17$	$S_3^2$ (as 6T9)	$1$	$0$

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 *.