Global number field 6.6.66854673.1

• Label: 6.6.66854673.1
• Degree: $6$
• Signature: $[6, 0]$
• Discriminant: $3^{3}\cdot 19^{5}$
• Root discriminant: $20.15$
• Ramified primes: $3, 19$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} - 17 x^{4} - 11 x^{3} + 37 x^{2} + 33 x + 7 \)

Invariants

Degree:		$6$
Signature:		$[6, 0]$
Discriminant:		\(66854673=3^{3}\cdot 19^{5}\)
Root discriminant:		$20.15$
Ramified primes:		$3, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(57=3\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{57}(1,·)$, $\chi_{57}(50,·)$, $\chi_{57}(56,·)$, $\chi_{57}(49,·)$, $\chi_{57}(7,·)$, $\chi_{57}(8,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{7} a^{5} - \frac{3}{7} a^{3} + \frac{2}{7} a$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$5$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{4}{7} a^{5} - a^{4} - \frac{61}{7} a^{3} - a^{2} + \frac{141}{7} a + 9 \),  \( \frac{15}{7} a^{5} - 3 a^{4} - \frac{248}{7} a^{3} - 9 a^{2} + \frac{597}{7} a + 37 \),  \( \frac{12}{7} a^{5} - 3 a^{4} - \frac{183}{7} a^{3} - 2 a^{2} + \frac{409}{7} a + 25 \),  \( \frac{4}{7} a^{5} - a^{4} - \frac{61}{7} a^{3} - a^{2} + \frac{148}{7} a + 8 \),  \( \frac{1}{7} a^{5} - \frac{24}{7} a^{3} - 2 a^{2} + \frac{86}{7} a + 5 \)
Regulator:		\( 120.272162571 \)

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

\(\Q(\sqrt{57}) \), 3.3.361.1

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

Sibling algebras

Twin sextic algebra:

3.3.361.1 $\times$ \(\Q(\sqrt{57}) \) $\times$ \(\Q\)

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	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.6.0.1}{6} }$	R	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.3.1	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$19$	19.6.5.5	$x^{6} + 1216$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$

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_19.2t1.1c1	$1$	$ 3 \cdot 19 $	$x^{2} - x - 14$	$C_2$ (as 2T1)	$1$	$1$
*	1.19.3t1.1c1	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_19.6t1.2c1	$1$	$ 3 \cdot 19 $	$x^{6} - x^{5} - 17 x^{4} - 11 x^{3} + 37 x^{2} + 33 x + 7$	$C_6$ (as 6T1)	$0$	$1$
*	1.19.3t1.1c2	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_19.6t1.2c2	$1$	$ 3 \cdot 19 $	$x^{6} - x^{5} - 17 x^{4} - 11 x^{3} + 37 x^{2} + 33 x + 7$	$C_6$ (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 *.