Global number field 6.0.1872286839.1

• Label: 6.0.1872286839.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 37^{5}$
• Root discriminant: $35.11$
• Ramified primes: $3, 37$
• Class number: $8$
• Class group: $[8]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 22 x^{4} - 83 x^{3} + 52 x^{2} + 36 x + 184 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-1872286839=-\,3^{3}\cdot 37^{5}\)
Root discriminant:		$35.11$
Ramified primes:		$3, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(111=3\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(100,·)$, $\chi_{111}(101,·)$, $\chi_{111}(10,·)$, $\chi_{111}(11,·)$, $\chi_{111}(110,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{25868} a^{5} + \frac{4445}{25868} a^{4} - \frac{165}{6467} a^{3} - \frac{11359}{25868} a^{2} - \frac{3863}{12934} a + \frac{736}{6467}$

Class group and class number

$C_{8}$, which has order $8$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{323}{25868} a^{5} + \frac{61}{25868} a^{4} + \frac{3349}{12934} a^{3} - \frac{21569}{25868} a^{2} - \frac{6276}{6467} a - \frac{8018}{6467} \),  \( \frac{24}{6467} a^{5} - \frac{51}{12934} a^{4} + \frac{655}{12934} a^{3} - \frac{1002}{6467} a^{2} - \frac{2229}{12934} a + \frac{5986}{6467} \)
Regulator:		\( 12.5049207637 \)

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

\(\Q(\sqrt{-111}) \), 3.3.1369.1

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

Sibling algebras

Twin sextic algebra:

3.3.1369.1 $\times$ \(\Q(\sqrt{-111}) \) $\times$ \(\Q\)

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	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$37$	37.6.5.1	$x^{6} - 37$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$

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_37.2t1.1c1	$1$	$ 3 \cdot 37 $	$x^{2} - x + 28$	$C_2$ (as 2T1)	$1$	$-1$
*	1.37.3t1.1c1	$1$	$ 37 $	$x^{3} - x^{2} - 12 x - 11$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_37.6t1.2c1	$1$	$ 3 \cdot 37 $	$x^{6} - x^{5} + 22 x^{4} - 83 x^{3} + 52 x^{2} + 36 x + 184$	$C_6$ (as 6T1)	$0$	$-1$
*	1.37.3t1.1c2	$1$	$ 37 $	$x^{3} - x^{2} - 12 x - 11$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_37.6t1.2c2	$1$	$ 3 \cdot 37 $	$x^{6} - x^{5} + 22 x^{4} - 83 x^{3} + 52 x^{2} + 36 x + 184$	$C_6$ (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 *.