Global number field 6.0.59869095867.4

• Label: 6.0.59869095867.4
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{3}\cdot 7^{4}\cdot 31^{4}$
• Root discriminant: $62.55$
• Ramified primes: $3, 7, 31$
• Class number: $36$
• Class group: $[6, 6]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} + 73 x^{4} - 378 x^{3} + 5409 x^{2} - 16200 x + 50625 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-59869095867=-\,3^{3}\cdot 7^{4}\cdot 31^{4}\)
Root discriminant:		$62.55$
Ramified primes:		$3, 7, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(651=3\cdot 7\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{651}(1,·)$, $\chi_{651}(242,·)$, $\chi_{651}(625,·)$, $\chi_{651}(25,·)$, $\chi_{651}(218,·)$, $\chi_{651}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{4} - \frac{1}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{28399275} a^{5} - \frac{1514701}{28399275} a^{4} - \frac{3023927}{28399275} a^{3} - \frac{238614}{1051825} a^{2} + \frac{1558501}{3155475} a - \frac{1752}{42073}$

Class group and class number

$C_{6}\times C_{6}$, which has order $36$

Unit group

Rank:		$2$
Torsion generator:		\( -\frac{584}{9466425} a^{5} + \frac{559}{9466425} a^{4} - \frac{40807}{9466425} a^{3} + \frac{87527}{9466425} a^{2} - \frac{335959}{1051825} a + \frac{40248}{42073} \) (order $6$)
Fundamental units:		\( \frac{71056}{9466425} a^{5} + \frac{450694}{9466425} a^{4} + \frac{4965038}{9466425} a^{3} - \frac{10649518}{9466425} a^{2} + \frac{7866531}{1051825} a + \frac{222050}{42073} \),  \( \frac{265}{378657} a^{5} - \frac{19345}{378657} a^{4} - \frac{102443}{378657} a^{3} - \frac{159265}{42073} a^{2} + \frac{477000}{42073} a - \frac{2551009}{42073} \)
Regulator:		\( 112.817848262 \)

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

\(\Q(\sqrt{-3}) \), 3.3.47089.2

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-3}) \) $\times$ 3.3.47089.2

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	${\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	3.2.1.2	$x^{2} + 3$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
$7$	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	7.3.2.1	$x^{3} + 14$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$31$	31.3.2.3	$x^{3} - 1519$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	31.3.2.3	$x^{3} - 1519$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

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)	$1$	$-1$
*	1.7_31.3t1.2c1	$1$	$ 7 \cdot 31 $	$x^{3} - x^{2} - 72 x + 225$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_7_31.6t1.3c1	$1$	$ 3 \cdot 7 \cdot 31 $	$x^{6} - x^{5} + 73 x^{4} - 378 x^{3} + 5409 x^{2} - 16200 x + 50625$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7_31.3t1.2c2	$1$	$ 7 \cdot 31 $	$x^{3} - x^{2} - 72 x + 225$	$C_3$ (as 3T1)	$0$	$1$
*	1.3_7_31.6t1.3c2	$1$	$ 3 \cdot 7 \cdot 31 $	$x^{6} - x^{5} + 73 x^{4} - 378 x^{3} + 5409 x^{2} - 16200 x + 50625$	$C_6$ (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 *.