Global number field 12.12.46118408000000000.1

• Label: 12.12.46118408000000000.1
• Degree: $12$
• Signature: $[12, 0]$
• Discriminant: $2^{12}\cdot 5^{9}\cdot 7^{8}$
• Root discriminant: $24.47$
• Ramified primes: $2, 5, 7$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_{12}$ (as 12T1)

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1 \)

Invariants

Degree:		$12$
Signature:		$[12, 0]$
Discriminant:		\(46118408000000000=2^{12}\cdot 5^{9}\cdot 7^{8}\)
Root discriminant:		$24.47$
Ramified primes:		$2, 5, 7$
This field is Galois and abelian over $\Q$.
Conductor:		\(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:		$\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(67,·)$, $\chi_{140}(9,·)$, $\chi_{140}(107,·)$, $\chi_{140}(109,·)$, $\chi_{140}(81,·)$, $\chi_{140}(43,·)$, $\chi_{140}(23,·)$, $\chi_{140}(121,·)$, $\chi_{140}(123,·)$, $\chi_{140}(29,·)$, $\chi_{140}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{239} a^{10} - \frac{44}{239} a^{9} + \frac{74}{239} a^{8} + \frac{45}{239} a^{7} + \frac{4}{239} a^{6} + \frac{86}{239} a^{5} - \frac{101}{239} a^{4} + \frac{99}{239} a^{3} + \frac{2}{239} a^{2} - \frac{8}{239} a - \frac{60}{239}$, $\frac{1}{6715661} a^{11} + \frac{7910}{6715661} a^{10} + \frac{1933982}{6715661} a^{9} - \frac{2998271}{6715661} a^{8} + \frac{1626068}{6715661} a^{7} - \frac{1904715}{6715661} a^{6} + \frac{1845244}{6715661} a^{5} - \frac{2486054}{6715661} a^{4} - \frac{1903453}{6715661} a^{3} - \frac{2807168}{6715661} a^{2} - \frac{970458}{6715661} a - \frac{2405731}{6715661}$

Class group and class number

Trivial group, which has order $1$

Unit group

Rank:		$11$
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:		\( 17863.7216242 \)

Galois group

$C_{12}$ (as 12T1):

A cyclic group of order 12

The 12 conjugacy class representatives for $C_{12}$

Character table for $C_{12}$

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{20})^+\), 6.6.300125.1

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

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	R	${\href{/LocalNumberField/3.12.0.1}{12} }$	R	R	${\href{/LocalNumberField/11.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/13.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/17.12.0.1}{12} }$	${\href{/LocalNumberField/19.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/23.12.0.1}{12} }$	${\href{/LocalNumberField/29.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/37.12.0.1}{12} }$	${\href{/LocalNumberField/41.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/47.12.0.1}{12} }$	${\href{/LocalNumberField/53.12.0.1}{12} }$	${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$2$	2.12.12.25	$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$	$2$	$6$	$12$	$C_{12}$	$[2]^{6}$
$5$	5.12.9.2	$x^{12} - 10 x^{8} + 25 x^{4} - 500$	$4$	$3$	$9$	$C_{12}$	$[\ ]_{4}^{3}$
$7$	7.12.8.1	$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$

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_5.4t1.1c1	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e2_5.4t1.1c2	$1$	$ 2^{2} \cdot 5 $	$x^{4} - 5 x^{2} + 5$	$C_4$ (as 4T1)	$0$	$1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_5_7.12t1.1c1	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.5_7.6t1.1c1	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.2e2_5_7.12t1.1c2	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e2_5_7.12t1.1c3	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.5_7.6t1.1c2	$1$	$ 5 \cdot 7 $	$x^{6} - x^{5} - 7 x^{4} + 2 x^{3} + 7 x^{2} - 2 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.2e2_5_7.12t1.1c4	$1$	$ 2^{2} \cdot 5 \cdot 7 $	$x^{12} - 4 x^{11} - 17 x^{10} + 74 x^{9} + 74 x^{8} - 412 x^{7} - 23 x^{6} + 734 x^{5} - 175 x^{4} - 324 x^{3} + 90 x^{2} + 22 x + 1$	$C_{12}$ (as 12T1)	$0$	$1$

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 *.