Global number field 12.0.941480149401.1

• Label: 12.0.941480149401.1
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $3^{12}\cdot 11^{6}$
• Root discriminant: $9.95$
• Ramified primes: $3, 11$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_6\times S_3$ (as 12T18)

Normalized defining polynomial

\( x^{12} + x^{10} - 4 x^{9} - 2 x^{8} - 3 x^{7} + 9 x^{6} + 4 x^{5} + 12 x^{4} - 7 x^{3} - 2 x^{2} - x + 1 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(941480149401=3^{12}\cdot 11^{6}\)
Root discriminant:		$9.95$
Ramified primes:		$3, 11$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{6}$, $\frac{1}{30} a^{10} + \frac{1}{15} a^{9} - \frac{7}{15} a^{8} + \frac{1}{6} a^{7} + \frac{2}{15} a^{6} + \frac{4}{15} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{13}{30} a + \frac{2}{15}$, $\frac{1}{90} a^{11} + \frac{1}{90} a^{10} - \frac{1}{90} a^{9} + \frac{19}{90} a^{8} - \frac{31}{90} a^{7} - \frac{19}{90} a^{6} - \frac{11}{45} a^{5} + \frac{2}{15} a^{4} + \frac{1}{3} a^{3} - \frac{37}{90} a^{2} - \frac{13}{30} a + \frac{41}{90}$

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

Unit group

Rank:		$5$
Torsion generator:		\( \frac{1}{2} a^{11} + \frac{1}{2} a^{10} + \frac{2}{3} a^{9} - \frac{3}{2} a^{8} - \frac{17}{6} a^{7} - 3 a^{6} + \frac{8}{3} a^{5} + \frac{17}{3} a^{4} + 9 a^{3} + \frac{17}{6} a^{2} - \frac{11}{6} a - \frac{2}{3} \) (order $6$)
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 (assuming GRH)
Regulator:		\( 27.6923649209 \) (assuming GRH)

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

\(\Q(\sqrt{-11}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-3}, \sqrt{-11})\), 6.0.107811.1

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	${\href{/LocalNumberField/2.6.0.1}{6} }^{2}$	R	${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$	R	${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/17.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/19.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$	${\href{/LocalNumberField/37.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/43.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$	${\href{/LocalNumberField/53.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$3$	3.6.9.11	$x^{6} + 6 x^{4} + 12$	$6$	$1$	$9$	$C_6$	$[2]_{2}$
	3.6.3.2	$x^{6} - 9 x^{2} + 27$	$2$	$3$	$3$	$C_6$	$[\ ]_{2}^{3}$
$11$	11.12.6.1	$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$	$2$	$6$	$6$	$C_6\times C_2$	$[\ ]_{2}^{6}$

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_11.2t1.1c1	$1$	$ 3 \cdot 11 $	$x^{2} - x - 8$	$C_2$ (as 2T1)	$1$	$1$
*	1.11.2t1.1c1	$1$	$ 11 $	$x^{2} - x + 3$	$C_2$ (as 2T1)	$1$	$-1$
*	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
	1.3e2_11.6t1.2c1	$1$	$ 3^{2} \cdot 11 $	$x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$	$C_6$ (as 6T1)	$0$	$-1$
	1.3e2.6t1.1c1	$1$	$ 3^{2}$	$x^{6} - x^{3} + 1$	$C_6$ (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)	$0$	$1$
	1.3e2_11.6t1.1c1	$1$	$ 3^{2} \cdot 11 $	$x^{6} - 18 x^{4} - 8 x^{3} + 81 x^{2} + 72 x - 17$	$C_6$ (as 6T1)	$0$	$1$
	1.3e2_11.6t1.2c2	$1$	$ 3^{2} \cdot 11 $	$x^{6} - 3 x^{5} + 6 x^{4} - 5 x^{3} + 33 x^{2} - 54 x + 111$	$C_6$ (as 6T1)	$0$	$-1$
	1.3e2_11.6t1.1c2	$1$	$ 3^{2} \cdot 11 $	$x^{6} - 18 x^{4} - 8 x^{3} + 81 x^{2} + 72 x - 17$	$C_6$ (as 6T1)	$0$	$1$
	1.3e2.6t1.1c2	$1$	$ 3^{2}$	$x^{6} - x^{3} + 1$	$C_6$ (as 6T1)	$0$	$-1$
	2.3e4_11.3t2.1c1	$2$	$ 3^{4} \cdot 11 $	$x^{3} + 6 x - 1$	$S_3$ (as 3T2)	$1$	$0$
	2.3e4_11.6t3.2c1	$2$	$ 3^{4} \cdot 11 $	$x^{6} - x^{3} - 8$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.3e2_11.6t5.1c1	$2$	$ 3^{2} \cdot 11 $	$x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.3e2_11.6t5.1c2	$2$	$ 3^{2} \cdot 11 $	$x^{6} - x^{4} - 2 x^{3} + 3 x^{2} + x + 1$	$S_3\times C_3$ (as 6T5)	$0$	$0$
*	2.3e3_11.12t18.2c1	$2$	$ 3^{3} \cdot 11 $	$x^{12} + x^{10} - 4 x^{9} - 2 x^{8} - 3 x^{7} + 9 x^{6} + 4 x^{5} + 12 x^{4} - 7 x^{3} - 2 x^{2} - x + 1$	$C_6\times S_3$ (as 12T18)	$0$	$0$
*	2.3e3_11.12t18.2c2	$2$	$ 3^{3} \cdot 11 $	$x^{12} + x^{10} - 4 x^{9} - 2 x^{8} - 3 x^{7} + 9 x^{6} + 4 x^{5} + 12 x^{4} - 7 x^{3} - 2 x^{2} - x + 1$	$C_6\times S_3$ (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 *.