Global number field 6.0.1585615607.1

• Label: 6.0.1585615607.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,19^{4}\cdot 23^{3}$
• Root discriminant: $34.15$
• Ramified primes: $19, 23$
• Class number: $9$
• Class group: $[9]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 5 x^{4} - 13 x^{3} + 145 x^{2} + 185 x + 829 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-1585615607=-\,19^{4}\cdot 23^{3}\)
Root discriminant:		$34.15$
Ramified primes:		$19, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(437=19\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{437}(1,·)$, $\chi_{437}(68,·)$, $\chi_{437}(229,·)$, $\chi_{437}(45,·)$, $\chi_{437}(254,·)$, $\chi_{437}(277,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{40933} a^{5} + \frac{8139}{40933} a^{4} - \frac{19062}{40933} a^{3} + \frac{12310}{40933} a^{2} - \frac{439}{40933} a - \frac{12104}{40933}$

Class group and class number

$C_{9}$, which has order $9$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{206}{40933} a^{5} - \frac{1619}{40933} a^{4} + \frac{2796}{40933} a^{3} - \frac{1986}{40933} a^{2} + \frac{32365}{40933} a - \frac{119310}{40933} \),  \( \frac{382}{40933} a^{5} - \frac{1810}{40933} a^{4} + \frac{4390}{40933} a^{3} - \frac{4875}{40933} a^{2} + \frac{36967}{40933} a - \frac{39232}{40933} \)
Regulator:		\( 7.80862678603 \)

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

\(\Q(\sqrt{-23}) \), 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 $\times$ \(\Q(\sqrt{-23}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	R	R	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$19$	19.6.4.3	$x^{6} + 95 x^{3} + 2888$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$23$	23.6.3.2	$x^{6} - 529 x^{2} + 48668$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.23.2t1.1c1	$1$	$ 23 $	$x^{2} - x + 6$	$C_2$ (as 2T1)	$1$	$-1$
*	1.19.3t1.1c1	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.19_23.6t1.2c1	$1$	$ 19 \cdot 23 $	$x^{6} - x^{5} + 5 x^{4} - 13 x^{3} + 145 x^{2} + 185 x + 829$	$C_6$ (as 6T1)	$0$	$-1$
*	1.19.3t1.1c2	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.19_23.6t1.2c2	$1$	$ 19 \cdot 23 $	$x^{6} - x^{5} + 5 x^{4} - 13 x^{3} + 145 x^{2} + 185 x + 829$	$C_6$ (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 *.