Global number field 6.0.28400541651.3

• Label: 6.0.28400541651.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{9}\cdot 113^{3}$
• Root discriminant: $55.24$
• Ramified primes: $3, 113$
• Class number: $126$
• Class group: $[126]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} + 168 x^{4} - 85 x^{3} + 7056 x^{2} - 7140 x + 73081 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-28400541651=-\,3^{9}\cdot 113^{3}\)
Root discriminant:		$55.24$
Ramified primes:		$3, 113$
This field is Galois and abelian over $\Q$.
Conductor:		\(1017=3^{2}\cdot 113\)
Dirichlet character group:		$\lbrace$$\chi_{1017}(1,·)$, $\chi_{1017}(338,·)$, $\chi_{1017}(340,·)$, $\chi_{1017}(677,·)$, $\chi_{1017}(679,·)$, $\chi_{1017}(1016,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{29} a^{3} - \frac{3}{29} a + \frac{1}{29}$, $\frac{1}{29} a^{4} - \frac{3}{29} a^{2} + \frac{1}{29} a$, $\frac{1}{846133} a^{5} - \frac{1039}{846133} a^{4} + \frac{140}{846133} a^{3} - \frac{58014}{846133} a^{2} + \frac{62274}{846133} a - \frac{53594}{846133}$

Class group and class number

$C_{126}$, which has order $126$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{28}{846133} a^{5} + \frac{85}{846133} a^{4} + \frac{3920}{846133} a^{3} - \frac{19657}{846133} a^{2} + \frac{80583}{846133} a - \frac{1500632}{846133} \),  \( \frac{57}{846133} a^{5} - \frac{869}{846133} a^{4} + \frac{7980}{846133} a^{3} - \frac{97328}{846133} a^{2} + \frac{223440}{846133} a - \frac{1362592}{846133} \)
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

\(\Q(\sqrt{-339}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-339}) \) $\times$ \(\Q(\zeta_{9})^+\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.6.0.1}{6} }$	R	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$3$	3.6.9.3	$x^{6} + 3 x^{4} + 24$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
$113$	113.6.3.2	$x^{6} - 12769 x^{2} + 7214485$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

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_113.2t1.1c1	$1$	$ 3 \cdot 113 $	$x^{2} - x + 85$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_113.6t1.1c1	$1$	$ 3^{2} \cdot 113 $	$x^{6} + 168 x^{4} - 85 x^{3} + 7056 x^{2} - 7140 x + 73081$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_113.6t1.1c2	$1$	$ 3^{2} \cdot 113 $	$x^{6} + 168 x^{4} - 85 x^{3} + 7056 x^{2} - 7140 x + 73081$	$C_6$ (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 *.