Global number field 6.0.28978644267.2

• Label: 6.0.28978644267.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 181^{4}$
• Root discriminant: $55.42$
• Ramified primes: $3, 181$
• Class number: $21$
• Class group: $[21]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 61 x^{4} - 74 x^{3} + 3667 x^{2} - 4020 x + 4489 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-28978644267=-\,3^{3}\cdot 181^{4}\)
Root discriminant:		$55.42$
Ramified primes:		$3, 181$
This field is Galois and abelian over $\Q$.
Conductor:		\(543=3\cdot 181\)
Dirichlet character group:		$\lbrace$$\chi_{543}(1,·)$, $\chi_{543}(229,·)$, $\chi_{543}(182,·)$, $\chi_{543}(313,·)$, $\chi_{543}(410,·)$, $\chi_{543}(494,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{1}{5}$, $\frac{1}{73588445} a^{5} + \frac{351455}{14717689} a^{4} - \frac{4169952}{73588445} a^{3} + \frac{3777539}{14717689} a^{2} + \frac{17138046}{73588445} a + \frac{435674}{1098335}$

Class group and class number

$C_{21}$, which has order $21$

Unit group

Rank:		$2$
Torsion generator:		\( \frac{3660}{14717689} a^{5} - \frac{3593}{14717689} a^{4} + \frac{219173}{14717689} a^{3} - \frac{21533}{14717689} a^{2} + \frac{13175531}{14717689} a + \frac{4087}{219667} \) (order $6$)
Fundamental units:		\( \frac{4}{22415} a^{5} - \frac{244}{22415} a^{4} + \frac{287}{4483} a^{3} - \frac{14668}{22415} a^{2} + \frac{3216}{4483} a - \frac{321879}{22415} \),  \( \frac{93753}{73588445} a^{5} - \frac{7591}{73588445} a^{4} + \frac{463051}{73588445} a^{3} - \frac{24835767}{73588445} a^{2} + \frac{27836197}{73588445} a - \frac{91092}{219667} \)
Regulator:		\( 129.910068447 \)

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

\(\Q(\sqrt{-3}) \), 3.3.32761.1

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

Sibling algebras

Twin sextic algebra:

3.3.32761.1 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.6.0.1}{6} }$	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$181$	181.3.2.1	$x^{3} - 181$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	181.3.2.1	$x^{3} - 181$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.181.3t1.1c1	$1$	$ 181 $	$x^{3} - x^{2} - 60 x + 67$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_181.6t1.1c1	$1$	$ 3 \cdot 181 $	$x^{6} - x^{5} + 61 x^{4} - 74 x^{3} + 3667 x^{2} - 4020 x + 4489$	$C_6$ (as 6T1)	$0$	$-1$
*	1.181.3t1.1c2	$1$	$ 181 $	$x^{3} - x^{2} - 60 x + 67$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_181.6t1.1c2	$1$	$ 3 \cdot 181 $	$x^{6} - x^{5} + 61 x^{4} - 74 x^{3} + 3667 x^{2} - 4020 x + 4489$	$C_6$ (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 *.