Global number field 6.0.296019302976.3

• Label: 6.0.296019302976.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 3^{8}\cdot 89^{3}$
• Root discriminant: $81.64$
• Ramified primes: $2, 3, 89$
• Class number: $1008$
• Class group: $[2, 2, 252]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} + 261 x^{4} - 2 x^{3} + 23772 x^{2} + 540 x + 753297 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-296019302976=-\,2^{6}\cdot 3^{8}\cdot 89^{3}\)
Root discriminant:		$81.64$
Ramified primes:		$2, 3, 89$
This field is Galois and abelian over $\Q$.
Conductor:		\(3204=2^{2}\cdot 3^{2}\cdot 89\)
Dirichlet character group:		$\lbrace$$\chi_{3204}(1,·)$, $\chi_{3204}(355,·)$, $\chi_{3204}(2137,·)$, $\chi_{3204}(2491,·)$, $\chi_{3204}(1069,·)$, $\chi_{3204}(1423,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{47428181} a^{5} + \frac{21071852}{47428181} a^{4} - \frac{15809102}{47428181} a^{3} - \frac{3897305}{47428181} a^{2} - \frac{21031796}{47428181} a - \frac{9423642}{47428181}$

Class group and class number

$C_{2}\times C_{2}\times C_{252}$, which has order $1008$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{2154}{47428181} a^{5} - \frac{9}{47428181} a^{4} + \frac{628250}{47428181} a^{3} - \frac{6933}{47428181} a^{2} + \frac{38852452}{47428181} a + \frac{736600}{47428181} \),  \( \frac{18}{47428181} a^{5} - \frac{132112}{47428181} a^{4} + \frac{5250}{47428181} a^{3} - \frac{22723309}{47428181} a^{2} + \frac{853120}{47428181} a - \frac{1070760995}{47428181} \)
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

\(\Q(\sqrt{-89}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-89}) \) $\times$ \(\Q(\zeta_{9})^+\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.6.3	$x^{6} + 2 x^{4} + x^{2} - 7$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$3$	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
$89$	89.2.1.1	$x^{2} - 89$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.2.1.1	$x^{2} - 89$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	89.2.1.1	$x^{2} - 89$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_89.2t1.1c1	$1$	$ 2^{2} \cdot 89 $	$x^{2} + 89$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3e2_89.6t1.1c1	$1$	$ 2^{2} \cdot 3^{2} \cdot 89 $	$x^{6} + 261 x^{4} - 2 x^{3} + 23772 x^{2} + 540 x + 753297$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3e2_89.6t1.1c2	$1$	$ 2^{2} \cdot 3^{2} \cdot 89 $	$x^{6} + 261 x^{4} - 2 x^{3} + 23772 x^{2} + 540 x + 753297$	$C_6$ (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 *.