Global number field 12.0.1037970703125.1

• Label: 12.0.1037970703125.1
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $3^{12}\cdot 5^{9}$
• Root discriminant: $10.03$
• Ramified primes: $3, 5$
• Class number: $1$
• Class group: Trivial
• Galois group: $S_3 \times C_4$ (as 12T11)

Normalized defining polynomial

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

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(1037970703125=3^{12}\cdot 5^{9}\)
Root discriminant:		$10.03$
Ramified primes:		$3, 5$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$5$
Torsion generator:		\( a^{9} - a^{6} + 2 a^{3} \) (order $10$)
Fundamental units:		\( a \),  \( 2 a^{11} - 3 a^{8} + 6 a^{5} - 3 a^{2} - 1 \),  \( a^{11} - a^{9} - a^{8} + a^{6} + 3 a^{5} - 3 a^{3} + 1 \),  \( a^{10} - a^{7} + 3 a^{4} - a + 1 \),  \( a^{11} + a^{10} - 2 a^{8} - a^{7} + 3 a^{5} + 2 a^{4} - 2 a^{2} \)
Regulator:		\( 46.5914285344 \)

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

\(\Q(\sqrt{5}) \), 3.1.135.1, \(\Q(\zeta_{5})\), 6.2.91125.1

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	R	${\href{/LocalNumberField/7.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/17.12.0.1}{12} }$	${\href{/LocalNumberField/19.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/23.12.0.1}{12} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/47.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/53.12.0.1}{12} }$	${\href{/LocalNumberField/59.2.0.1}{2} }^{6}$

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
$3$	3.12.12.5	$x^{12} + 33 x^{11} - 63 x^{10} - 36 x^{9} - 90 x^{8} - 54 x^{7} - 54 x^{6} - 108 x^{4} - 27 x^{3} - 81 x^{2} + 81 x - 81$	$3$	$4$	$12$	$S_3 \times C_4$	$[3/2]_{2}^{4}$
$5$	5.4.3.2	$x^{4} - 20$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	5.8.6.1	$x^{8} - 5 x^{4} + 400$	$4$	$2$	$6$	$C_4\times C_2$	$[\ ]_{4}^{2}$

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_5.2t1.1c1	$1$	$ 3 \cdot 5 $	$x^{2} - x + 4$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
	1.3.2t1.1c1	$1$	$ 3 $	$x^{2} - x + 1$	$C_2$ (as 2T1)	$1$	$-1$
*	1.5.4t1.1c1	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	1.3_5.4t1.1c1	$1$	$ 3 \cdot 5 $	$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.4t1.1c2	$1$	$ 5 $	$x^{4} - x^{3} + x^{2} - x + 1$	$C_4$ (as 4T1)	$0$	$-1$
	1.3_5.4t1.1c2	$1$	$ 3 \cdot 5 $	$x^{4} - x^{3} - 4 x^{2} + 4 x + 1$	$C_4$ (as 4T1)	$0$	$1$
*	2.3e3_5.3t2.1c1	$2$	$ 3^{3} \cdot 5 $	$x^{3} + 3 x - 1$	$S_3$ (as 3T2)	$1$	$0$
*	2.3e3_5.6t3.1c1	$2$	$ 3^{3} \cdot 5 $	$x^{6} - x^{3} - 1$	$D_{6}$ (as 6T3)	$1$	$0$
*	2.3e3_5e2.12t11.1c1	$2$	$ 3^{3} \cdot 5^{2}$	$x^{12} - 2 x^{9} + 4 x^{6} - 3 x^{3} + 1$	$S_3 \times C_4$ (as 12T11)	$0$	$0$
*	2.3e3_5e2.12t11.1c2	$2$	$ 3^{3} \cdot 5^{2}$	$x^{12} - 2 x^{9} + 4 x^{6} - 3 x^{3} + 1$	$S_3 \times C_4$ (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 *.