Global number field 6.0.568803016375.3

• Label: 6.0.568803016375.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,5^{3}\cdot 11^{3}\cdot 43^{4}$
• Root discriminant: $91.02$
• Ramified primes: $5, 11, 43$
• Class number: $1008$
• Class group: $[2, 6, 84]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 13 x^{4} + 17 x^{3} + 762 x^{2} - 1672 x + 11264 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-568803016375=-\,5^{3}\cdot 11^{3}\cdot 43^{4}\)
Root discriminant:		$91.02$
Ramified primes:		$5, 11, 43$
This field is Galois and abelian over $\Q$.
Conductor:		\(2365=5\cdot 11\cdot 43\)
Dirichlet character group:		$\lbrace$$\chi_{2365}(1,·)$, $\chi_{2365}(1154,·)$, $\chi_{2365}(1541,·)$, $\chi_{2365}(2199,·)$, $\chi_{2365}(1979,·)$, $\chi_{2365}(221,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{66952} a^{5} + \frac{4023}{66952} a^{4} - \frac{13819}{66952} a^{3} - \frac{4003}{66952} a^{2} - \frac{2677}{33476} a + \frac{1547}{8369}$

Class group and class number

$C_{2}\times C_{6}\times C_{84}$, which has order $1008$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{23}{33476} a^{5} + \frac{235}{16738} a^{4} + \frac{185}{33476} a^{3} - \frac{5}{16738} a^{2} + \frac{6875}{8369} a - \frac{20897}{8369} \),  \( \frac{75}{33476} a^{5} + \frac{441}{33476} a^{4} + \frac{1331}{33476} a^{3} + \frac{1059}{33476} a^{2} + \frac{16819}{16738} a + \frac{106515}{8369} \)
Regulator:		\( 75.6874575318 \)

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

\(\Q(\sqrt{-55}) \), 3.3.1849.1

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-55}) \) $\times$ 3.3.1849.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/3.6.0.1}{6} }$	R	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$5$	5.6.3.1	$x^{6} - 10 x^{4} + 25 x^{2} - 500$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$11$	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	11.2.1.2	$x^{2} + 33$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$43$	43.3.2.1	$x^{3} - 43$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	43.3.2.1	$x^{3} - 43$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5_11.2t1.1c1	$1$	$ 5 \cdot 11 $	$x^{2} - x + 14$	$C_2$ (as 2T1)	$1$	$-1$
*	1.43.3t1.1c1	$1$	$ 43 $	$x^{3} - x^{2} - 14 x - 8$	$C_3$ (as 3T1)	$0$	$1$
*	1.5_11_43.6t1.1c1	$1$	$ 5 \cdot 11 \cdot 43 $	$x^{6} - x^{5} + 13 x^{4} + 17 x^{3} + 762 x^{2} - 1672 x + 11264$	$C_6$ (as 6T1)	$0$	$-1$
*	1.43.3t1.1c2	$1$	$ 43 $	$x^{3} - x^{2} - 14 x - 8$	$C_3$ (as 3T1)	$0$	$1$
*	1.5_11_43.6t1.1c2	$1$	$ 5 \cdot 11 \cdot 43 $	$x^{6} - x^{5} + 13 x^{4} + 17 x^{3} + 762 x^{2} - 1672 x + 11264$	$C_6$ (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 *.