Global number field 12.12.7007073538075000832.1

• Label: 12.12.7007073538...0832.1
• Degree: $12$
• Signature: $[12, 0]$
• Discriminant: $2^{33}\cdot 13^{8}$
• Root discriminant: $37.19$
• Ramified primes: $2, 13$
• Class number: $1$
• Class group: Trivial
• Galois group: $C_{12}$ (as 12T1)

Normalized defining polynomial

\( x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79 \)

Invariants

Degree:		$12$
Signature:		$[12, 0]$
Discriminant:		\(7007073538075000832=2^{33}\cdot 13^{8}\)
Root discriminant:		$37.19$
Ramified primes:		$2, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(208=2^{4}\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{208}(1,·)$, $\chi_{208}(165,·)$, $\chi_{208}(81,·)$, $\chi_{208}(9,·)$, $\chi_{208}(29,·)$, $\chi_{208}(157,·)$, $\chi_{208}(113,·)$, $\chi_{208}(53,·)$, $\chi_{208}(105,·)$, $\chi_{208}(185,·)$, $\chi_{208}(61,·)$, $\chi_{208}(133,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{463} a^{10} + \frac{10}{463} a^{9} + \frac{200}{463} a^{8} - \frac{4}{463} a^{7} - \frac{165}{463} a^{6} + \frac{95}{463} a^{5} - \frac{105}{463} a^{4} + \frac{188}{463} a^{3} - \frac{228}{463} a^{2} + \frac{12}{463} a + \frac{176}{463}$, $\frac{1}{7333457} a^{11} + \frac{3752}{7333457} a^{10} + \frac{3274453}{7333457} a^{9} + \frac{2459644}{7333457} a^{8} - \frac{1970382}{7333457} a^{7} - \frac{2025781}{7333457} a^{6} - \frac{1092879}{7333457} a^{5} + \frac{854137}{7333457} a^{4} + \frac{2626107}{7333457} a^{3} - \frac{2266240}{7333457} a^{2} - \frac{2716252}{7333457} a + \frac{782213}{7333457}$

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:		\( 329963.498245 \)

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

\(\Q(\sqrt{2}) \), 3.3.169.1, \(\Q(\zeta_{16})^+\), 6.6.14623232.1

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	${\href{/LocalNumberField/3.12.0.1}{12} }$	${\href{/LocalNumberField/5.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/7.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/11.12.0.1}{12} }$	R	${\href{/LocalNumberField/17.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/19.12.0.1}{12} }$	${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/29.12.0.1}{12} }$	${\href{/LocalNumberField/31.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/37.12.0.1}{12} }$	${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/43.12.0.1}{12} }$	${\href{/LocalNumberField/47.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$2$	2.12.33.375	$x^{12} - 4 x^{10} + 26 x^{8} + 8 x^{6} - 24 x^{4} + 32 x^{2} + 8$	$4$	$3$	$33$	$C_{12}$	$[3, 4]^{3}$
$13$	13.12.8.1	$x^{12} - 39 x^{9} - 338 x^{6} + 10985 x^{3} + 228488$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.2e4.4t1.1c1	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
*	1.2e3.2t1.1c1	$1$	$ 2^{3}$	$x^{2} - 2$	$C_2$ (as 2T1)	$1$	$1$
*	1.2e4.4t1.1c2	$1$	$ 2^{4}$	$x^{4} - 4 x^{2} + 2$	$C_4$ (as 4T1)	$0$	$1$
*	1.13.3t1.1c1	$1$	$ 13 $	$x^{3} - x^{2} - 4 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e4_13.12t1.2c1	$1$	$ 2^{4} \cdot 13 $	$x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.2e3_13.6t1.2c1	$1$	$ 2^{3} \cdot 13 $	$x^{6} - 2 x^{5} - 13 x^{4} + 14 x^{3} + 26 x^{2} - 28 x + 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.2e4_13.12t1.2c2	$1$	$ 2^{4} \cdot 13 $	$x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.13.3t1.1c2	$1$	$ 13 $	$x^{3} - x^{2} - 4 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.2e4_13.12t1.2c3	$1$	$ 2^{4} \cdot 13 $	$x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.2e3_13.6t1.2c2	$1$	$ 2^{3} \cdot 13 $	$x^{6} - 2 x^{5} - 13 x^{4} + 14 x^{3} + 26 x^{2} - 28 x + 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.2e4_13.12t1.2c4	$1$	$ 2^{4} \cdot 13 $	$x^{12} - 4 x^{11} - 22 x^{10} + 80 x^{9} + 159 x^{8} - 508 x^{7} - 406 x^{6} + 1276 x^{5} + 194 x^{4} - 1100 x^{3} + 184 x^{2} + 256 x - 79$	$C_{12}$ (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 *.