Global number field 6.0.70963776.10

• Label: 6.0.70963776.10
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 3^{8}\cdot 13^{2}$
• Root discriminant: $20.35$
• Ramified primes: $2, 3, 13$
• Class number: $3$
• Class group: $[3]$
• Galois group: $S_3\times C_3$ (as 6T5)

Normalized defining polynomial

\( x^{6} - 18 x^{4} - 22 x^{3} + 117 x^{2} + 390 x + 377 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-70963776=-\,2^{6}\cdot 3^{8}\cdot 13^{2}\)
Root discriminant:		$20.35$
Ramified primes:		$2, 3, 13$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{1610} a^{5} + \frac{106}{805} a^{4} - \frac{11}{115} a^{3} + \frac{67}{322} a^{2} + \frac{297}{1610} a - \frac{241}{1610}$

Class group and class number

$C_{3}$, which has order $3$

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{9}{322} a^{5} + \frac{12}{161} a^{4} + \frac{7}{23} a^{3} - \frac{117}{322} a^{2} - \frac{741}{322} a - \frac{1051}{322} \) (order $4$)
Fundamental units:		\( \frac{47}{805} a^{5} - \frac{197}{1610} a^{4} - \frac{114}{115} a^{3} + \frac{341}{322} a^{2} + \frac{11013}{1610} a + \frac{3968}{805} \),  \( \frac{12}{161} a^{5} - \frac{32}{161} a^{4} - \frac{45}{46} a^{3} + \frac{156}{161} a^{2} + \frac{2781}{322} a + \frac{4359}{322} \)
Regulator:		\( 31.1040695808 \)

Galois group

$C_3\times S_3$ (as 6T5):

A solvable group of order 18

The 9 conjugacy class representatives for $S_3\times C_3$

Character table for $S_3\times C_3$

Intermediate fields

\(\Q(\sqrt{-1}) \)

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:	3.1.676.1 $\times$ 3.3.13689.2
Degree 9 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	${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	R	${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$	2.6.6.3	$x^{6} + 2 x^{4} + x^{2} - 7$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$3$	3.6.8.6	$x^{6} + 18 x^{2} + 36$	$3$	$2$	$8$	$C_6$	$[2]^{2}$
$13$	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{13}$	$x + 2$	$1$	$1$	$0$	Trivial	$[\ ]$
	13.3.2.3	$x^{3} - 52$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2.2t1.1c1	$1$	$ 2^{2}$	$x^{2} + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3e2_13.3t1.1c1	$1$	$ 3^{2} \cdot 13 $	$x^{3} - 39 x - 91$	$C_3$ (as 3T1)	$0$	$1$
	1.2e2_3e2_13.6t1.8c1	$1$	$ 2^{2} \cdot 3^{2} \cdot 13 $	$x^{6} + 78 x^{4} + 1521 x^{2} + 8281$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e2_3e2_13.6t1.8c2	$1$	$ 2^{2} \cdot 3^{2} \cdot 13 $	$x^{6} + 78 x^{4} + 1521 x^{2} + 8281$	$C_6$ (as 6T1)	$0$	$-1$
	1.3e2_13.3t1.1c2	$1$	$ 3^{2} \cdot 13 $	$x^{3} - 39 x - 91$	$C_3$ (as 3T1)	$0$	$1$
	2.2e2_13e2.3t2.1c1	$2$	$ 2^{2} \cdot 13^{2}$	$x^{3} - x^{2} - 4 x + 12$	$S_3$ (as 3T2)	$1$	$0$
*	2.2e2_3e4_13.6t5.8c1	$2$	$ 2^{2} \cdot 3^{4} \cdot 13 $	$x^{6} - 18 x^{4} - 22 x^{3} + 117 x^{2} + 390 x + 377$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.2e2_3e4_13.6t5.8c2	$2$	$ 2^{2} \cdot 3^{4} \cdot 13 $	$x^{6} - 18 x^{4} - 22 x^{3} + 117 x^{2} + 390 x + 377$	$S_3\times C_3$ (as 6T5)	$0$	$0$

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 *.