Global number field 6.0.48827236375.1

• Label: 6.0.48827236375.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,5^{3}\cdot 17^{3}\cdot 43^{3}$
• Root discriminant: $60.46$
• Ramified primes: $5, 17, 43$
• Class number: $80$
• Class group: $[2, 2, 2, 10]$
• Galois group: $D_{6}$ (as 6T3)

Normalized defining polynomial

\( x^{6} - 2 x^{5} + 23 x^{4} + 25 x^{3} + 74 x^{2} + 517 x + 1466 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-48827236375=-\,5^{3}\cdot 17^{3}\cdot 43^{3}\)
Root discriminant:		$60.46$
Ramified primes:		$5, 17, 43$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{4} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{11056} a^{5} + \frac{1121}{11056} a^{4} - \frac{739}{5528} a^{3} - \frac{1369}{11056} a^{2} - \frac{529}{11056} a + \frac{1737}{5528}$

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{10}$, which has order $80$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{11}{5528} a^{5} - \frac{107}{5528} a^{4} + \frac{163}{2764} a^{3} - \frac{1239}{5528} a^{2} - \frac{1673}{5528} a + \frac{3905}{2764} \),  \( \frac{5}{691} a^{5} + \frac{77}{691} a^{4} + \frac{211}{691} a^{3} + \frac{1447}{691} a^{2} + \frac{4956}{691} a + \frac{7005}{691} \)
Regulator:		\( 32.6106094218 \)

Galois group

$D_6$ (as 6T3):

A solvable group of order 12

The 6 conjugacy class representatives for $D_{6}$

Character table for $D_{6}$

Intermediate fields

\(\Q(\sqrt{-3655}) \), 3.1.731.1

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

Sibling algebras

Galois closure:	Deg 12
Twin sextic algebra:	3.1.731.1 $\times$ \(\Q(\sqrt{5}) \) $\times$ \(\Q\)
Degree 6 sibling:	6.2.66795125.2

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/3.6.0.1}{6} }$	R	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	R	${\href{/LocalNumberField/19.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.6.0.1}{6} }$	${\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/47.6.0.1}{6} }$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$5$	5.6.3.1	$x^{6} - 10 x^{4} + 25 x^{2} - 500$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$17$	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.4.2.1	$x^{4} + 85 x^{2} + 2601$	$2$	$2$	$2$	$C_2^2$	$[\ ]_{2}^{2}$
43	Data not computed

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
	1.17_43.2t1.1c1	$1$	$ 17 \cdot 43 $	$x^{2} - x + 183$	$C_2$ (as 2T1)	$1$	$-1$
	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.5_17_43.2t1.1c1	$1$	$ 5 \cdot 17 \cdot 43 $	$x^{2} - x + 914$	$C_2$ (as 2T1)	$1$	$-1$
*	2.17_43.3t2.1c1	$2$	$ 17 \cdot 43 $	$x^{3} - x^{2} + 3 x + 4$	$S_3$ (as 3T2)	$1$	$0$
*	2.5e2_17_43.6t3.2c1	$2$	$ 5^{2} \cdot 17 \cdot 43 $	$x^{6} - 2 x^{5} + 23 x^{4} + 25 x^{3} + 74 x^{2} + 517 x + 1466$	$D_{6}$ (as 6T3)	$1$	$0$

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 *.