Global number field 12.0.1175078824045389.1

• Label: 12.0.1175078824045389.1
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $3^{4}\cdot 29^{9}$
• Root discriminant: $18.02$
• Ramified primes: $3, 29$
• Class number: $2$
• Class group: $[2]$
• Galois group: $S_3 \times C_4$ (as 12T11)

Normalized defining polynomial

\( x^{12} - 2 x^{11} + 3 x^{9} - 3 x^{8} + 17 x^{7} + 22 x^{6} - 25 x^{5} - 20 x^{4} + 91 x^{3} + 230 x^{2} + 228 x + 83 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(1175078824045389=3^{4}\cdot 29^{9}\)
Root discriminant:		$18.02$
Ramified primes:		$3, 29$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{8} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{1}{5} a^{2} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{10} - \frac{1}{25} a^{8} + \frac{1}{25} a^{7} - \frac{9}{25} a^{5} + \frac{4}{25} a^{4} - \frac{8}{25} a^{3} + \frac{2}{5} a^{2} - \frac{6}{25} a + \frac{8}{25}$, $\frac{1}{233125} a^{11} + \frac{784}{233125} a^{10} + \frac{10099}{233125} a^{9} + \frac{11567}{233125} a^{8} - \frac{216}{233125} a^{7} + \frac{109991}{233125} a^{6} + \frac{103448}{233125} a^{5} - \frac{97147}{233125} a^{4} + \frac{107438}{233125} a^{3} - \frac{84766}{233125} a^{2} + \frac{94529}{233125} a + \frac{26397}{233125}$

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:		\( 142.483768915 \)

Galois group

$C_4\times S_3$ (as 12T11):

A solvable group of order 24

The 12 conjugacy class representatives for $S_3 \times C_4$

Character table for $S_3 \times C_4$

Intermediate fields

\(\Q(\sqrt{29}) \), 3.1.87.1, 4.0.24389.1, 6.2.219501.1

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 12 sibling:	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	${\href{/LocalNumberField/2.12.0.1}{12} }$	R	${\href{/LocalNumberField/5.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/7.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/11.12.0.1}{12} }$	${\href{/LocalNumberField/13.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/17.12.0.1}{12} }$	${\href{/LocalNumberField/19.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{4}$	R	${\href{/LocalNumberField/31.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/47.12.0.1}{12} }$	${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{4}$

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
$3$	3.4.0.1	$x^{4} - x + 2$	$1$	$4$	$0$	$C_4$	$[\ ]^{4}$
	3.8.4.1	$x^{8} + 36 x^{4} - 27 x^{2} + 324$	$2$	$4$	$4$	$C_4\times C_2$	$[\ ]_{2}^{4}$
$29$	29.4.3.2	$x^{4} - 116$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	29.8.6.1	$x^{8} - 203 x^{4} + 68121$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$

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_29.2t1.1c1	$1$	$ 3 \cdot 29 $	$x^{2} - x + 22$	$C_2$ (as 2T1)	$1$	$-1$
*	1.29.2t1.1c1	$1$	$ 29 $	$x^{2} - x - 7$	$C_2$ (as 2T1)	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.29.4t1.1c1	$1$	$ 29 $	$x^{4} - x^{3} + 4 x^{2} - 20 x + 23$	$C_4$ (as 4T1)	$0$	$-1$
	1.3_29.4t1.1c1	$1$	$ 3 \cdot 29 $	$x^{4} - x^{3} - 25 x^{2} + 67 x - 35$	$C_4$ (as 4T1)	$0$	$1$
*	1.29.4t1.1c2	$1$	$ 29 $	$x^{4} - x^{3} + 4 x^{2} - 20 x + 23$	$C_4$ (as 4T1)	$0$	$-1$
	1.3_29.4t1.1c2	$1$	$ 3 \cdot 29 $	$x^{4} - x^{3} - 25 x^{2} + 67 x - 35$	$C_4$ (as 4T1)	$0$	$1$
*	2.3_29.3t2.1c1	$2$	$ 3 \cdot 29 $	$x^{3} - x^{2} + 2 x + 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.3_29.6t3.1c1	$2$	$ 3 \cdot 29 $	$x^{6} - x^{5} + 4 x^{4} - 4 x^{3} + 5 x^{2} - 3 x + 1$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.3_29e2.12t11.2c1	$2$	$ 3 \cdot 29^{2}$	$x^{12} - 2 x^{11} + 3 x^{9} - 3 x^{8} + 17 x^{7} + 22 x^{6} - 25 x^{5} - 20 x^{4} + 91 x^{3} + 230 x^{2} + 228 x + 83$	$S_3 \times C_4$ (as 12T11)	$0$	$0$
*	2.3_29e2.12t11.2c2	$2$	$ 3 \cdot 29^{2}$	$x^{12} - 2 x^{11} + 3 x^{9} - 3 x^{8} + 17 x^{7} + 22 x^{6} - 25 x^{5} - 20 x^{4} + 91 x^{3} + 230 x^{2} + 228 x + 83$	$S_3 \times C_4$ (as 12T11)	$0$	$0$

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 *.