Global number field 6.6.36622433792.1

• Label: 6.6.36622433792.1
• Degree: $6$
• Signature: $[6, 0]$
• Discriminant: $2^{9}\cdot 7^{4}\cdot 31^{3}$
• Root discriminant: $57.63$
• Ramified primes: $2, 7, 31$
• Class number: $7$
• Class group: $[7]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 189 x^{4} + 254 x^{3} + 11410 x^{2} - 7568 x - 219479 \)

Invariants

Degree:		$6$
Signature:		$[6, 0]$
Discriminant:		\(36622433792=2^{9}\cdot 7^{4}\cdot 31^{3}\)
Root discriminant:		$57.63$
Ramified primes:		$2, 7, 31$
This field is Galois and abelian over $\Q$.
Conductor:		\(1736=2^{3}\cdot 7\cdot 31\)
Dirichlet character group:		$\lbrace$$\chi_{1736}(1,·)$, $\chi_{1736}(1115,·)$, $\chi_{1736}(1611,·)$, $\chi_{1736}(249,·)$, $\chi_{1736}(1241,·)$, $\chi_{1736}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{14404039} a^{5} + \frac{6185722}{14404039} a^{4} + \frac{6156198}{14404039} a^{3} - \frac{6752293}{14404039} a^{2} + \frac{3155748}{14404039} a + \frac{155120}{1108003}$

Class group and class number

$C_{7}$, which has order $7$

Unit group

Rank:		$5$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
Regulator:		\( 313.228190532 \)

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

\(\Q(\sqrt{62}) \), \(\Q(\zeta_{7})^+\)

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

Sibling algebras

Twin sextic algebra:

\(\Q\) $\times$ \(\Q(\sqrt{62}) \) $\times$ \(\Q(\zeta_{7})^+\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.6.0.1}{6} }$	${\href{/LocalNumberField/5.6.0.1}{6} }$	R	${\href{/LocalNumberField/11.6.0.1}{6} }$	${\href{/LocalNumberField/13.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/17.6.0.1}{6} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$	R	${\href{/LocalNumberField/37.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/41.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.9.5	$x^{6} - 4 x^{4} + 4 x^{2} + 8$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
$7$	7.6.4.3	$x^{6} + 56 x^{3} + 1323$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$
$31$	31.6.3.1	$x^{6} - 62 x^{4} + 961 x^{2} - 2413071$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e3_31.2t1.1c1	$1$	$ 2^{3} \cdot 31 $	$x^{2} - 62$	$C_2$ (as 2T1)	$1$	$1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e3_7_31.6t1.1c1	$1$	$ 2^{3} \cdot 7 \cdot 31 $	$x^{6} - 2 x^{5} - 189 x^{4} + 254 x^{3} + 11410 x^{2} - 7568 x - 219479$	$C_6$ (as 6T1)	$0$	$1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e3_7_31.6t1.1c2	$1$	$ 2^{3} \cdot 7 \cdot 31 $	$x^{6} - 2 x^{5} - 189 x^{4} + 254 x^{3} + 11410 x^{2} - 7568 x - 219479$	$C_6$ (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 *.