Global number field 12.0.8916100448256.2

• Label: 12.0.8916100448256.2
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $2^{24}\cdot 3^{12}$
• Root discriminant: $12.00$
• Ramified primes: $2, 3$
• Class number: $2$
• Class group: $[2]$
• Galois group: $C_6\times S_3$ (as 12T18)

Normalized defining polynomial

\( x^{12} - 4 x^{11} + 4 x^{10} + 12 x^{9} - 37 x^{8} + 32 x^{7} + 22 x^{6} - 80 x^{5} + 88 x^{4} - 48 x^{3} + 10 x^{2} + 1 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(8916100448256=2^{24}\cdot 3^{12}\)
Root discriminant:		$12.00$
Ramified primes:		$2, 3$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{11} a^{10} + \frac{5}{11} a^{9} - \frac{2}{11} a^{8} + \frac{3}{11} a^{7} + \frac{4}{11} a^{6} + \frac{3}{11} a^{5} - \frac{1}{11} a^{4} - \frac{4}{11} a^{2} + \frac{4}{11} a - \frac{3}{11}$, $\frac{1}{1397} a^{11} - \frac{58}{1397} a^{10} + \frac{596}{1397} a^{9} - \frac{168}{1397} a^{8} + \frac{145}{1397} a^{7} - \frac{51}{1397} a^{6} - \frac{399}{1397} a^{5} - \frac{124}{1397} a^{4} - \frac{455}{1397} a^{3} - \frac{624}{1397} a^{2} + \frac{559}{1397} a + \frac{167}{1397}$

Class group and class number

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

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:		\( 20.761295793 \)

Galois group

$C_6\times S_3$ (as 12T18):

A solvable group of order 36

The 18 conjugacy class representatives for $C_6\times S_3$

Character table for $C_6\times S_3$

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-2}, \sqrt{3})\), 6.0.41472.1

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

Sibling fields

Galois closure:	data not computed
Degree 18 siblings:	data not computed

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	R	${\href{/LocalNumberField/5.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/37.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/47.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/53.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
$2$	2.12.24.307	$x^{12} + 28 x^{11} - 2 x^{10} + 16 x^{9} + 26 x^{8} + 8 x^{7} + 20 x^{6} - 24 x^{5} - 8 x^{4} + 32 x^{3} + 32 x^{2} + 32 x + 24$	$4$	$3$	$24$	$C_6\times C_2$	$[2, 3]^{3}$
$3$	3.6.9.2	$x^{6} + 3 x^{4} + 6$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
	3.6.3.1	$x^{6} - 6 x^{4} + 9 x^{2} - 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e2_3.2t1.1c1	$1$	$ 2^{2} \cdot 3 $	$x^{2} - 3$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e3.2t1.2c1	$1$	$ 2^{3}$	$x^{2} + 2$	$C_2$ (as 2T1)	$1$	$-1$
*	1.2e3_3.2t1.2c1	$1$	$ 2^{3} \cdot 3 $	$x^{2} + 6$	$C_2$ (as 2T1)	$1$	$-1$
	1.2e3_3e2.6t1.4c1	$1$	$ 2^{3} \cdot 3^{2}$	$x^{6} + 12 x^{4} + 36 x^{2} + 24$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e2_3e2.6t1.1c1	$1$	$ 2^{2} \cdot 3^{2}$	$x^{6} - 6 x^{4} + 9 x^{2} - 3$	$C_6$ (as 6T1)	$0$	$1$
	1.2e2_3e2.6t1.1c2	$1$	$ 2^{2} \cdot 3^{2}$	$x^{6} - 6 x^{4} + 9 x^{2} - 3$	$C_6$ (as 6T1)	$0$	$1$
	1.2e3_3e2.6t1.3c1	$1$	$ 2^{3} \cdot 3^{2}$	$x^{6} + 12 x^{4} + 36 x^{2} + 8$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e3_3e2.6t1.4c2	$1$	$ 2^{3} \cdot 3^{2}$	$x^{6} + 12 x^{4} + 36 x^{2} + 24$	$C_6$ (as 6T1)	$0$	$-1$
	1.2e3_3e2.6t1.3c2	$1$	$ 2^{3} \cdot 3^{2}$	$x^{6} + 12 x^{4} + 36 x^{2} + 8$	$C_6$ (as 6T1)	$0$	$-1$
	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
	2.2e3_3e4.3t2.1c1	$2$	$ 2^{3} \cdot 3^{4}$	$x^{3} - 3 x - 10$	$S_3$ (as 3T2)	$1$	$0$
	2.2e5_3e4.6t3.3c1	$2$	$ 2^{5} \cdot 3^{4}$	$x^{6} - 6 x^{4} + 9 x^{2} - 12$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.2e3_3e2.6t5.1c1	$2$	$ 2^{3} \cdot 3^{2}$	$x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.2e3_3e2.6t5.1c2	$2$	$ 2^{3} \cdot 3^{2}$	$x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 2 x^{2} + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.2e5_3e3.12t18.2c1	$2$	$ 2^{5} \cdot 3^{3}$	$x^{12} - 4 x^{11} + 4 x^{10} + 12 x^{9} - 37 x^{8} + 32 x^{7} + 22 x^{6} - 80 x^{5} + 88 x^{4} - 48 x^{3} + 10 x^{2} + 1$	$C_6\times S_3$ (as 12T18)	$0$	$0$
*	2.2e5_3e3.12t18.2c2	$2$	$ 2^{5} \cdot 3^{3}$	$x^{12} - 4 x^{11} + 4 x^{10} + 12 x^{9} - 37 x^{8} + 32 x^{7} + 22 x^{6} - 80 x^{5} + 88 x^{4} - 48 x^{3} + 10 x^{2} + 1$	$C_6\times S_3$ (as 12T18)	$0$	$0$

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 *.