Global number field 6.0.1042568000.1

• Label: 6.0.1042568000.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{6}\cdot 5^{3}\cdot 19^{4}$
• Root discriminant: $31.84$
• Ramified primes: $2, 5, 19$
• Class number: $14$
• Class group: $[14]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 4 x^{4} + 6 x^{3} + 107 x^{2} - 284 x + 749 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-1042568000=-\,2^{6}\cdot 5^{3}\cdot 19^{4}\)
Root discriminant:		$31.84$
Ramified primes:		$2, 5, 19$
This field is Galois and abelian over $\Q$.
Conductor:		\(380=2^{2}\cdot 5\cdot 19\)
Dirichlet character group:		$\lbrace$$\chi_{380}(1,·)$, $\chi_{380}(121,·)$, $\chi_{380}(39,·)$, $\chi_{380}(201,·)$, $\chi_{380}(159,·)$, $\chi_{380}(239,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{30781} a^{5} - \frac{2534}{30781} a^{4} + \frac{13644}{30781} a^{3} - \frac{10320}{30781} a^{2} - \frac{2722}{30781} a - \frac{3124}{30781}$

Class group and class number

$C_{14}$, which has order $14$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{194}{30781} a^{5} + \frac{900}{30781} a^{4} - \frac{230}{30781} a^{3} - \frac{1315}{30781} a^{2} + \frac{25990}{30781} a + \frac{71126}{30781} \),  \( \frac{158}{30781} a^{5} - \frac{219}{30781} a^{4} + \frac{1082}{30781} a^{3} + \frac{833}{30781} a^{2} + \frac{858}{30781} a - \frac{1096}{30781} \)
Regulator:		\( 7.80862678603 \)

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

\(\Q(\sqrt{-5}) \), 3.3.361.1

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

Sibling algebras

Twin sextic algebra:

3.3.361.1 $\times$ \(\Q(\sqrt{-5}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/7.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.6.0.1}{6} }$	R	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.6.0.1}{6} }$

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
$2$	2.6.6.5	$x^{6} - 2 x^{4} + x^{2} - 3$	$2$	$3$	$6$	$C_6$	$[2]^{3}$
$5$	5.6.3.1	$x^{6} - 10 x^{4} + 25 x^{2} - 500$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$19$	19.6.4.3	$x^{6} + 95 x^{3} + 2888$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$

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_5.2t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{2} + 5$	$C_2$ (as 2T1)	$1$	$-1$
*	1.19.3t1.1c1	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_5_19.6t1.2c1	$1$	$ 2^{2} \cdot 5 \cdot 19 $	$x^{6} - 2 x^{5} + 4 x^{4} + 6 x^{3} + 107 x^{2} - 284 x + 749$	$C_6$ (as 6T1)	$0$	$-1$
*	1.19.3t1.1c2	$1$	$ 19 $	$x^{3} - x^{2} - 6 x + 7$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_5_19.6t1.2c2	$1$	$ 2^{2} \cdot 5 \cdot 19 $	$x^{6} - 2 x^{5} + 4 x^{4} + 6 x^{3} + 107 x^{2} - 284 x + 749$	$C_6$ (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 *.