Global number field 6.0.600482915136.7

• Label: 6.0.600482915136.7
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 3^{3}\cdot 13^{4}\cdot 23^{3}$
• Root discriminant: $91.85$
• Ramified primes: $2, 3, 13, 23$
• Class number: $1032$
• Class group: $[2, 516]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 200 x^{4} - 270 x^{3} + 14439 x^{2} - 8548 x + 372325 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-600482915136=-\,2^{6}\cdot 3^{3}\cdot 13^{4}\cdot 23^{3}\)
Root discriminant:		$91.85$
Ramified primes:		$2, 3, 13, 23$
This field is Galois and abelian over $\Q$.
Conductor:		\(3588=2^{2}\cdot 3\cdot 13\cdot 23\)
Dirichlet character group:		$\lbrace$$\chi_{3588}(1,·)$, $\chi_{3588}(1379,·)$, $\chi_{3588}(1381,·)$, $\chi_{3588}(2759,·)$, $\chi_{3588}(1933,·)$, $\chi_{3588}(3311,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{23051965} a^{5} + \frac{496124}{4610393} a^{4} - \frac{2198252}{4610393} a^{3} + \frac{2255812}{4610393} a^{2} - \frac{7609341}{23051965} a - \frac{925217}{4610393}$

Class group and class number

$C_{2}\times C_{516}$, which has order $1032$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{604}{23051965} a^{5} - \frac{16649}{4610393} a^{4} + \frac{48976}{4610393} a^{3} - \frac{2165880}{4610393} a^{2} + \frac{14351036}{23051965} a - \frac{83960589}{4610393} \),  \( \frac{1682}{23051965} a^{5} - \frac{565}{4610393} a^{4} + \frac{75322}{4610393} a^{3} - \frac{77655}{4610393} a^{2} + \frac{17980978}{23051965} a - \frac{7122946}{4610393} \)
Regulator:		\( 5.46019947038 \)

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

\(\Q(\sqrt{-69}) \), 3.3.169.1

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{-69}) \) $\times$ 3.3.169.1

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	${\href{/LocalNumberField/5.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/11.6.0.1}{6} }$	R	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/29.6.0.1}{6} }$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/53.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$3$	3.6.3.1	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$13$	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
$23$	23.6.3.2	$x^{6} - 529 x^{2} + 48668$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_3_23.2t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 23 $	$x^{2} + 69$	$C_2$ (as 2T1)	$1$	$-1$
*	1.13.3t1.1c1	$1$	$ 13 $	$x^{3} - x^{2} - 4 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3_13_23.6t1.1c1	$1$	$ 2^{2} \cdot 3 \cdot 13 \cdot 23 $	$x^{6} - 2 x^{5} + 200 x^{4} - 270 x^{3} + 14439 x^{2} - 8548 x + 372325$	$C_6$ (as 6T1)	$0$	$-1$
*	1.13.3t1.1c2	$1$	$ 13 $	$x^{3} - x^{2} - 4 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_3_13_23.6t1.1c2	$1$	$ 2^{2} \cdot 3 \cdot 13 \cdot 23 $	$x^{6} - 2 x^{5} + 200 x^{4} - 270 x^{3} + 14439 x^{2} - 8548 x + 372325$	$C_6$ (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 *.