Global number field 6.0.10744731.1

• Label: 6.0.10744731.1
• Degree: $6$
• Signature: $[0, 3]$
• Discriminant: $-\,3^{7}\cdot 17^{3}$
• Root discriminant: $14.85$
• Ramified primes: $3, 17$
• Class number: $2$
• Class group: $[2]$
• Galois group: $S_3$ (as 6T2)

Normalized defining polynomial

\( x^{6} - 3 x^{5} + 9 x^{4} - 3 x^{2} + 27 x + 33 \)

Invariants

Degree:		$6$
Signature:		$[0, 3]$
Discriminant:		\(-10744731=-\,3^{7}\cdot 17^{3}\)
Root discriminant:		$14.85$
Ramified primes:		$3, 17$
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} + \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{7}{16} a^{2} + \frac{1}{4} a + \frac{7}{16}$

Class group and class number

$C_{2}$, which has order $2$

Unit group

Rank:		$2$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		\( \frac{1}{16} a^{5} - \frac{3}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{16} a^{2} - 2 a - \frac{29}{16} \),  \( \frac{1}{16} a^{5} - \frac{3}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{16} a^{2} - 4 a - \frac{61}{16} \)
Regulator:		\( 13.4648151135 \)

Galois group

$S_3$ (as 6T2):

A solvable group of order 6

The 3 conjugacy class representatives for $S_3$

Character table for $S_3$

Intermediate fields

\(\Q(\sqrt{-51}) \), 3.1.459.1 x3

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

Sibling algebras

Twin sextic algebra:	3.1.459.1 $\times$ \(\Q\) $\times$ \(\Q\) $\times$ \(\Q\)
Degree 3 sibling:	3.1.459.1

Multiplicative Galois module structure

$U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A'$

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	${\href{/LocalNumberField/2.2.0.1}{2} }^{3}$	R	${\href{/LocalNumberField/5.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/7.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/11.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/13.3.0.1}{3} }^{2}$	R	${\href{/LocalNumberField/19.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/23.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/29.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/31.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/41.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/43.3.0.1}{3} }^{2}$	${\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.6.7.1	$x^{6} + 6 x^{2} + 6$	$6$	$1$	$7$	$S_3$	$[3/2]_{2}$
$17$	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	17.2.1.2	$x^{2} + 51$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.3_17.2t1.1c1	$1$	$ 3 \cdot 17 $	$x^{2} - x + 13$	$C_2$ (as 2T1)	$1$	$-1$
*^2	2.3e3_17.3t2.2c1	$2$	$ 3^{3} \cdot 17 $	$x^{6} - 3 x^{5} + 9 x^{4} - 3 x^{2} + 27 x + 33$	$S_3$ (as 6T2)	$1$	$0$

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 *.