Global number field 6.0.959570432.1

• Label: 6.0.959570432.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,2^{9}\cdot 37^{4}$
• Root discriminant: $31.41$
• Ramified primes: $2, 37$
• Class number: $7$
• Class group: $[7]$
• Galois group: $C_6$ (as 6T1)

Normalized defining polynomial

\( x^{6} - 2 x^{5} - 17 x^{4} - 6 x^{3} + 182 x^{2} + 436 x + 473 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-959570432=-\,2^{9}\cdot 37^{4}\)
Root discriminant:		$31.41$
Ramified primes:		$2, 37$
This field is Galois and abelian over $\Q$.
Conductor:		\(296=2^{3}\cdot 37\)
Dirichlet character group:		$\lbrace$$\chi_{296}(1,·)$, $\chi_{296}(195,·)$, $\chi_{296}(211,·)$, $\chi_{296}(121,·)$, $\chi_{296}(137,·)$, $\chi_{296}(75,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{17569} a^{5} - \frac{3891}{17569} a^{4} + \frac{5173}{17569} a^{3} - \frac{1298}{17569} a^{2} + \frac{5801}{17569} a - \frac{1057}{17569}$

Class group and class number

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

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{434}{17569} a^{5} - \frac{2070}{17569} a^{4} - \frac{3750}{17569} a^{3} + \frac{16445}{17569} a^{2} + \frac{57974}{17569} a - \frac{19513}{17569} \),  \( \frac{122}{17569} a^{5} - \frac{339}{17569} a^{4} - \frac{1378}{17569} a^{3} - \frac{235}{17569} a^{2} + \frac{4962}{17569} a - \frac{5971}{17569} \)
Regulator:		\( 12.5049207637 \)

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

\(\Q(\sqrt{-2}) \), 3.3.1369.1

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

Sibling algebras

Twin sextic algebra:

3.3.1369.1 $\times$ \(\Q(\sqrt{-2}) \) $\times$ \(\Q\)

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/5.6.0.1}{6} }$	${\href{/LocalNumberField/7.6.0.1}{6} }$	${\href{/LocalNumberField/11.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/13.6.0.1}{6} }$	${\href{/LocalNumberField/17.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/29.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.6.0.1}{6} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$2$	2.6.9.5	$x^{6} - 4 x^{4} + 4 x^{2} + 8$	$2$	$3$	$9$	$C_6$	$[3]^{3}$
$37$	37.6.4.1	$x^{6} + 518 x^{3} + 171125$	$3$	$2$	$4$	$C_6$	$[\ ]_{3}^{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	1.37.3t1.1c1	$1$	$ 37 $	$x^{3} - x^{2} - 12 x - 11$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e3_37.6t1.4c1	$1$	$ 2^{3} \cdot 37 $	$x^{6} - 2 x^{5} - 17 x^{4} - 6 x^{3} + 182 x^{2} + 436 x + 473$	$C_6$ (as 6T1)	$0$	$-1$
*	1.37.3t1.1c2	$1$	$ 37 $	$x^{3} - x^{2} - 12 x - 11$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e3_37.6t1.4c2	$1$	$ 2^{3} \cdot 37 $	$x^{6} - 2 x^{5} - 17 x^{4} - 6 x^{3} + 182 x^{2} + 436 x + 473$	$C_6$ (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 *.