Global number field 6.0.4467667023.1

• Label: 6.0.4467667023.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{9}\cdot 61^{3}$
• Root discriminant: $40.58$
• Ramified primes: $3, 61$
• Class number: $104$
• Class group: $[104]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} + 90 x^{4} - 46 x^{3} + 2025 x^{2} - 2070 x + 12241 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-4467667023=-\,3^{9}\cdot 61^{3}\)
Root discriminant:		$40.58$
Ramified primes:		$3, 61$
This field is Galois and abelian over $\Q$.
Conductor:		\(549=3^{2}\cdot 61\)
Dirichlet character group:		$\lbrace$$\chi_{549}(1,·)$, $\chi_{549}(548,·)$, $\chi_{549}(182,·)$, $\chi_{549}(184,·)$, $\chi_{549}(365,·)$, $\chi_{549}(367,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{16} a^{3} - \frac{3}{16} a + \frac{1}{16}$, $\frac{1}{16} a^{4} - \frac{3}{16} a^{2} + \frac{1}{16} a$, $\frac{1}{87856} a^{5} - \frac{363}{87856} a^{4} + \frac{75}{87856} a^{3} - \frac{5399}{43928} a^{2} + \frac{12107}{87856} a - \frac{4801}{43928}$

Class group and class number

$C_{104}$, which has order $104$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{15}{87856} a^{5} + \frac{23}{43928} a^{4} + \frac{1125}{87856} a^{3} - \frac{2731}{87856} a^{2} + \frac{1423}{10982} a - \frac{28087}{43928} \),  \( \frac{31}{87856} a^{5} - \frac{271}{87856} a^{4} + \frac{2325}{87856} a^{3} - \frac{4065}{21964} a^{2} + \frac{34875}{87856} a - \frac{60975}{43928} \)
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

\(\Q(\sqrt{-183}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q(\zeta_{9})^+\) $\times$ \(\Q(\sqrt{-183}) \) $\times$ \(\Q\)

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	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.9.10	$x^{6} + 6 x^{4} + 3$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
$61$	61.6.3.1	$x^{6} - 122 x^{4} + 3721 x^{2} - 22698100$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_61.2t1.1c1	$1$	$ 3 \cdot 61 $	$x^{2} - x + 46$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_61.6t1.4c1	$1$	$ 3^{2} \cdot 61 $	$x^{6} + 90 x^{4} - 46 x^{3} + 2025 x^{2} - 2070 x + 12241$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_61.6t1.4c2	$1$	$ 3^{2} \cdot 61 $	$x^{6} + 90 x^{4} - 46 x^{3} + 2025 x^{2} - 2070 x + 12241$	$C_6$ (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 *.