Global number field 6.0.404353609875.6

• Label: 6.0.404353609875.6
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{8}\cdot 5^{3}\cdot 79^{3}$
• Root discriminant: $85.99$
• Ramified primes: $3, 5, 79$
• Class number: $1176$
• Class group: $[1176]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 294 x^{4} - 581 x^{3} + 29697 x^{2} - 30006 x + 1029999 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-404353609875=-\,3^{8}\cdot 5^{3}\cdot 79^{3}\)
Root discriminant:		$85.99$
Ramified primes:		$3, 5, 79$
This field is Galois and abelian over $\Q$.
Conductor:		\(3555=3^{2}\cdot 5\cdot 79\)
Dirichlet character group:		$\lbrace$$\chi_{3555}(1,·)$, $\chi_{3555}(1186,·)$, $\chi_{3555}(2371,·)$, $\chi_{3555}(394,·)$, $\chi_{3555}(1579,·)$, $\chi_{3555}(2764,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{64470401} a^{5} - \frac{17899448}{64470401} a^{4} + \frac{14309084}{64470401} a^{3} - \frac{5427013}{64470401} a^{2} + \frac{1912336}{64470401} a - \frac{31787789}{64470401}$

Class group and class number

$C_{1176}$, which has order $1176$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{2388}{64470401} a^{5} - \frac{5961}{64470401} a^{4} + \frac{780062}{64470401} a^{3} - \frac{1156443}{64470401} a^{2} + \frac{53730298}{64470401} a - \frac{27578155}{64470401} \),  \( \frac{2406}{64470401} a^{5} + \frac{155980}{64470401} a^{4} + \frac{461970}{64470401} a^{3} + \frac{30098125}{64470401} a^{2} + \frac{23681945}{64470401} a + \frac{1592235277}{64470401} \)
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

\(\Q(\sqrt{-395}) \), \(\Q(\zeta_{9})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-395}) \) $\times$ \(\Q(\zeta_{9})^+\)

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	R	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/37.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$3$	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
	3.3.4.2	$x^{3} - 3 x^{2} + 3$	$3$	$1$	$4$	$C_3$	$[2]$
$5$	5.6.3.1	$x^{6} - 10 x^{4} + 25 x^{2} - 500$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$79$	79.6.3.2	$x^{6} - 6241 x^{2} + 1972156$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5_79.2t1.1c1	$1$	$ 5 \cdot 79 $	$x^{2} - x + 99$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_5_79.6t1.1c1	$1$	$ 3^{2} \cdot 5 \cdot 79 $	$x^{6} - 3 x^{5} + 294 x^{4} - 581 x^{3} + 29697 x^{2} - 30006 x + 1029999$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_5_79.6t1.1c2	$1$	$ 3^{2} \cdot 5 \cdot 79 $	$x^{6} - 3 x^{5} + 294 x^{4} - 581 x^{3} + 29697 x^{2} - 30006 x + 1029999$	$C_6$ (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 *.