Global number field 6.0.541760081751.3

• Label: 6.0.541760081751.3
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{8}\cdot 7^{5}\cdot 17^{3}$
• Root discriminant: $90.29$
• Ramified primes: $3, 7, 17$
• Class number: $3510$
• Class group: $[3, 3, 390]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 51 x^{4} - 41 x^{3} + 3084 x^{2} - 9252 x + 81264 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-541760081751=-\,3^{8}\cdot 7^{5}\cdot 17^{3}\)
Root discriminant:		$90.29$
Ramified primes:		$3, 7, 17$
This field is Galois and abelian over $\Q$.
Conductor:		\(1071=3^{2}\cdot 7\cdot 17\)
Dirichlet character group:		$\lbrace$$\chi_{1071}(1,·)$, $\chi_{1071}(562,·)$, $\chi_{1071}(118,·)$, $\chi_{1071}(934,·)$, $\chi_{1071}(985,·)$, $\chi_{1071}(970,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{3} + \frac{1}{7} a^{2} - \frac{1}{14} a + \frac{3}{7}$, $\frac{1}{56} a^{4} - \frac{1}{28} a^{3} - \frac{9}{56} a^{2} + \frac{5}{28} a - \frac{3}{7}$, $\frac{1}{70672} a^{5} - \frac{507}{70672} a^{4} - \frac{607}{70672} a^{3} + \frac{10579}{70672} a^{2} + \frac{12967}{35336} a + \frac{869}{8834}$

Class group and class number

$C_{3}\times C_{3}\times C_{390}$, which has order $3510$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{15}{35336} a^{5} - \frac{33}{35336} a^{4} + \frac{991}{35336} a^{3} - \frac{327}{35336} a^{2} + \frac{7729}{17668} a + \frac{4832}{4417} \),  \( \frac{3}{8834} a^{5} + \frac{113}{17668} a^{4} + \frac{36}{4417} a^{3} + \frac{2267}{17668} a^{2} + \frac{1289}{1262} a + \frac{55229}{4417} \)
Regulator:		\( 50.3767558277 \)

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

\(\Q(\sqrt{-119}) \), 3.3.3969.1

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-119}) \) $\times$ 3.3.3969.1

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.1.0.1}{1} }^{6}$	R	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.6.0.1}{6} }$	R	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.6.0.1}{6} }$	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.3.4.1	$x^{3} - 3 x^{2} + 21$	$3$	$1$	$4$	$C_3$	$[2]$
	3.3.4.1	$x^{3} - 3 x^{2} + 21$	$3$	$1$	$4$	$C_3$	$[2]$
$7$	7.6.5.1	$x^{6} - 28$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
$17$	17.6.3.2	$x^{6} - 289 x^{2} + 14739$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.7_17.2t1.1c1	$1$	$ 7 \cdot 17 $	$x^{2} - x + 30$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3e2_7.3t1.2c1	$1$	$ 3^{2} \cdot 7 $	$x^{3} - 21 x - 28$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_7_17.6t1.2c1	$1$	$ 3^{2} \cdot 7 \cdot 17 $	$x^{6} - 3 x^{5} + 51 x^{4} - 41 x^{3} + 3084 x^{2} - 9252 x + 81264$	$C_6$ (as 6T1)	$0$	$-1$
*	1.3e2_7.3t1.2c2	$1$	$ 3^{2} \cdot 7 $	$x^{3} - 21 x - 28$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_7_17.6t1.2c2	$1$	$ 3^{2} \cdot 7 \cdot 17 $	$x^{6} - 3 x^{5} + 51 x^{4} - 41 x^{3} + 3084 x^{2} - 9252 x + 81264$	$C_6$ (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 *.