Global number field 6.0.3811270347.2

• Label: 6.0.3811270347.2
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 109^{4}$
• Root discriminant: $39.52$
• Ramified primes: $3, 109$
• Class number: $3$
• Class group: $[3]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 37 x^{4} + 28 x^{3} + 1300 x^{2} - 144 x + 16 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-3811270347=-\,3^{3}\cdot 109^{4}\)
Root discriminant:		$39.52$
Ramified primes:		$3, 109$
This field is Galois and abelian over $\Q$.
Conductor:		\(327=3\cdot 109\)
Dirichlet character group:		$\lbrace$$\chi_{327}(1,·)$, $\chi_{327}(263,·)$, $\chi_{327}(281,·)$, $\chi_{327}(154,·)$, $\chi_{327}(172,·)$, $\chi_{327}(110,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{4} + \frac{1}{8} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{191824} a^{5} - \frac{37}{191824} a^{4} - \frac{46587}{191824} a^{3} + \frac{23653}{47956} a^{2} - \frac{11971}{23978} a + \frac{11323}{23978}$

Class group and class number

$C_{3}$, which has order $3$

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{333}{47956} a^{5} + \frac{83}{11989} a^{4} - \frac{3071}{11989} a^{3} - \frac{10693}{47956} a^{2} - \frac{107900}{11989} a + \frac{11952}{11989} \) (order $6$)
Fundamental units:		\( \frac{1683}{23978} a^{5} + \frac{9663}{23978} a^{4} + \frac{31042}{11989} a^{3} + \frac{54043}{23978} a^{2} + \frac{254341}{23978} a + \frac{374}{11989} \),  \( \frac{111147}{11989} a^{5} - \frac{444441}{47956} a^{4} + \frac{16444317}{47956} a^{3} + \frac{12457883}{47956} a^{2} + \frac{144443325}{11989} a - \frac{15999876}{11989} \)
Regulator:		\( 212.765371498 \)

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

\(\Q(\sqrt{-3}) \), 3.3.11881.1

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

Sibling algebras

Twin sextic algebra:

3.3.11881.1 $\times$ \(\Q(\sqrt{-3}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$109$	109.3.2.1	$x^{3} - 109$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	109.3.2.1	$x^{3} - 109$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

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)	$1$	$-1$
*	1.109.3t1.1c1	$1$	$ 109 $	$x^{3} - x^{2} - 36 x + 4$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_109.6t1.1c1	$1$	$ 3 \cdot 109 $	$x^{6} - x^{5} + 37 x^{4} + 28 x^{3} + 1300 x^{2} - 144 x + 16$	$C_6$ (as 6T1)	$0$	$-1$
*	1.109.3t1.1c2	$1$	$ 109 $	$x^{3} - x^{2} - 36 x + 4$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_109.6t1.1c2	$1$	$ 3 \cdot 109 $	$x^{6} - x^{5} + 37 x^{4} + 28 x^{3} + 1300 x^{2} - 144 x + 16$	$C_6$ (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 *.