Global number field 6.0.997002999.1

• Label: 6.0.997002999.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{9}\cdot 37^{3}$
• Root discriminant: $31.61$
• Ramified primes: $3, 37$
• Class number: $32$
• Class group: $[2, 2, 8]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} + 54 x^{4} - 28 x^{3} + 729 x^{2} - 756 x + 2971 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-997002999=-\,3^{9}\cdot 37^{3}\)
Root discriminant:		$31.61$
Ramified primes:		$3, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(333=3^{2}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{333}(112,·)$, $\chi_{333}(1,·)$, $\chi_{333}(332,·)$, $\chi_{333}(221,·)$, $\chi_{333}(110,·)$, $\chi_{333}(223,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{10} a^{3} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{10} a^{4} - \frac{3}{10} a^{2} + \frac{1}{10} a$, $\frac{1}{15130} a^{5} - \frac{33}{3026} a^{4} + \frac{9}{3026} a^{3} - \frac{1457}{7565} a^{2} + \frac{3431}{15130} a - \frac{1261}{7565}$

Class group and class number

$C_{2}\times C_{2}\times C_{8}$, which has order $32$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{19}{15130} a^{5} - \frac{109}{15130} a^{4} + \frac{171}{3026} a^{3} - \frac{1962}{7565} a^{2} + \frac{1539}{3026} a - \frac{8829}{7565} \),  \( \frac{14}{7565} a^{5} - \frac{81}{15130} a^{4} + \frac{126}{1513} a^{3} - \frac{4429}{15130} a^{2} + \frac{9827}{15130} a - \frac{20178}{7565} \)
Regulator:		\( 3.39714980258 \)

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

\(\Q(\sqrt{-111}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-111}) \) $\times$ \(\Q\)

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	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	R	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.9.10	$x^{6} + 6 x^{4} + 3$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
$37$	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	37.2.1.1	$x^{2} - 37$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_37.2t1.1c1	$1$	$ 3 \cdot 37 $	$x^{2} - x + 28$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_37.6t1.3c1	$1$	$ 3^{2} \cdot 37 $	$x^{6} + 54 x^{4} - 28 x^{3} + 729 x^{2} - 756 x + 2971$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_37.6t1.3c2	$1$	$ 3^{2} \cdot 37 $	$x^{6} + 54 x^{4} - 28 x^{3} + 729 x^{2} - 756 x + 2971$	$C_6$ (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 *.