Global number field 6.0.7308160119.2

• Label: 6.0.7308160119.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{9}\cdot 13^{5}$
• Root discriminant: $44.05$
• Ramified primes: $3, 13$
• Class number: $12$
• Class group: $[12]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 26 x^{3} + 351 x^{2} - 936 x + 793 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-7308160119=-\,3^{9}\cdot 13^{5}\)
Root discriminant:		$44.05$
Ramified primes:		$3, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(117=3^{2}\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{117}(1,·)$, $\chi_{117}(116,·)$, $\chi_{117}(23,·)$, $\chi_{117}(56,·)$, $\chi_{117}(61,·)$, $\chi_{117}(94,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{24682} a^{5} + \frac{2201}{12341} a^{4} + \frac{1117}{12341} a^{3} - \frac{1735}{24682} a^{2} - \frac{1483}{3526} a + \frac{625}{24682}$

Class group and class number

$C_{12}$, which has order $12$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{9}{1763} a^{5} - \frac{99}{3526} a^{4} - \frac{337}{3526} a^{3} - \frac{1259}{3526} a^{2} + \frac{3536}{1763} a - \frac{43403}{3526} \),  \( \frac{7}{1763} a^{5} - \frac{77}{3526} a^{4} - \frac{229}{1763} a^{3} - \frac{1371}{3526} a^{2} + \frac{8047}{3526} a - \frac{6203}{1763} \)
Regulator:		\( 22.4556846903 \)

Galois group

$C_6$ (as 6T1):

A cyclic group of order 6

The 6 conjugacy class representatives for $C_6$

Character table for $C_6$

Intermediate fields

\(\Q(\sqrt{-39}) \), 3.3.13689.2

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling algebras

Twin sextic algebra:

3.3.13689.2 $\times$ \(\Q(\sqrt{-39}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.9.9	$x^{6} + 6 x^{4} + 21$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
$13$	13.6.5.1	$x^{6} - 52$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_13.2t1.1c1	$1$	$ 3 \cdot 13 $	$x^{2} - x + 10$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2_13.3t1.1c1	$1$	$ 3^{2} \cdot 13 $	$x^{3} - 39 x - 91$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_13.6t1.8c1	$1$	$ 3^{2} \cdot 13 $	$x^{6} - 26 x^{3} + 351 x^{2} - 936 x + 793$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2_13.3t1.1c2	$1$	$ 3^{2} \cdot 13 $	$x^{3} - 39 x - 91$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_13.6t1.8c2	$1$	$ 3^{2} \cdot 13 $	$x^{6} - 26 x^{3} + 351 x^{2} - 936 x + 793$	$C_6$ (as 6T1)	$0$	$-1$

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 *.