Global number field 6.0.60003090875.1

• Label: 6.0.60003090875.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,5^{3}\cdot 7^{5}\cdot 13^{4}$
• Root discriminant: $62.57$
• Ramified primes: $5, 7, 13$
• Class number: $18$
• Class group: $[3, 6]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - x^{5} - 34 x^{4} - 85 x^{3} + 1198 x^{2} + 5711 x + 15289 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-60003090875=-\,5^{3}\cdot 7^{5}\cdot 13^{4}\)
Root discriminant:		$62.57$
Ramified primes:		$5, 7, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(455=5\cdot 7\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{455}(256,·)$, $\chi_{455}(1,·)$, $\chi_{455}(16,·)$, $\chi_{455}(209,·)$, $\chi_{455}(269,·)$, $\chi_{455}(159,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{3} + \frac{3}{14} a^{2} + \frac{3}{14} a - \frac{3}{7}$, $\frac{1}{28} a^{4} + \frac{1}{28} a^{2} - \frac{2}{7} a - \frac{3}{28}$, $\frac{1}{21028} a^{5} - \frac{163}{21028} a^{4} + \frac{87}{21028} a^{3} - \frac{3665}{21028} a^{2} + \frac{887}{21028} a + \frac{9213}{21028}$

Class group and class number

$C_{3}\times C_{6}$, which has order $18$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{751} a^{5} - \frac{29}{10514} a^{4} - \frac{142}{5257} a^{3} - \frac{993}{10514} a^{2} + \frac{201}{5257} a - \frac{36989}{10514} \),  \( \frac{43}{10514} a^{5} - \frac{125}{5257} a^{4} - \frac{765}{10514} a^{3} + \frac{1935}{5257} a^{2} + \frac{4805}{1502} a + \frac{49006}{5257} \)
Regulator:		\( 62.4891977899 \)

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

\(\Q(\sqrt{-35}) \), 3.3.8281.1

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

Sibling algebras

Twin sextic algebra:

3.3.8281.1 $\times$ \(\Q(\sqrt{-35}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$	R	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/17.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/19.6.0.1}{6} }$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }$	${\href{/LocalNumberField/43.6.0.1}{6} }$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$5$	5.6.3.2	$x^{6} - 25 x^{2} + 250$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$7$	7.6.5.3	$x^{6} - 112$	$6$	$1$	$5$	$C_6$	$[\ ]_{6}$
$13$	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$
	13.3.2.2	$x^{3} - 13$	$3$	$1$	$2$	$C_3$	$[\ ]_{3}$

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_7.2t1.1c1	$1$	$ 5 \cdot 7 $	$x^{2} - x + 9$	$C_2$ (as 2T1)	$1$	$-1$
*	1.7_13.3t1.2c1	$1$	$ 7 \cdot 13 $	$x^{3} - x^{2} - 30 x + 64$	$C_3$ (as 3T1)	$0$	$1$
*	1.5_7_13.6t1.4c1	$1$	$ 5 \cdot 7 \cdot 13 $	$x^{6} - x^{5} - 34 x^{4} - 85 x^{3} + 1198 x^{2} + 5711 x + 15289$	$C_6$ (as 6T1)	$0$	$-1$
*	1.7_13.3t1.2c2	$1$	$ 7 \cdot 13 $	$x^{3} - x^{2} - 30 x + 64$	$C_3$ (as 3T1)	$0$	$1$
*	1.5_7_13.6t1.4c2	$1$	$ 5 \cdot 7 \cdot 13 $	$x^{6} - x^{5} - 34 x^{4} - 85 x^{3} + 1198 x^{2} + 5711 x + 15289$	$C_6$ (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 *.