Global number field 12.12.9891413435408203125.1

• Label: 12.12.9891413435...3125.1
• Degree: $12$
• Signature: $[12, 0]$
• Discriminant: $3^{16}\cdot 5^{9}\cdot 7^{6}$
• Root discriminant: $38.28$
• Ramified primes: $3, 5, 7$
• Class number: $1$ (GRH)
• Class group: Trivial (GRH)
• Galois group: $C_{12}$ (as 12T1)

Normalized defining polynomial

\( x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + 441 x^{4} - 6643 x^{3} + 1926 x^{2} + 2865 x - 1349 \)

Invariants

Degree:		$12$
Signature:		$[12, 0]$
Discriminant:		\(9891413435408203125=3^{16}\cdot 5^{9}\cdot 7^{6}\)
Root discriminant:		$38.28$
Ramified primes:		$3, 5, 7$
This field is Galois and abelian over $\Q$.
Conductor:		\(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:		$\lbrace$$\chi_{315}(64,·)$, $\chi_{315}(1,·)$, $\chi_{315}(211,·)$, $\chi_{315}(97,·)$, $\chi_{315}(169,·)$, $\chi_{315}(202,·)$, $\chi_{315}(13,·)$, $\chi_{315}(274,·)$, $\chi_{315}(307,·)$, $\chi_{315}(118,·)$, $\chi_{315}(106,·)$, $\chi_{315}(223,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} + \frac{4}{19} a^{8} + \frac{1}{19} a^{7} + \frac{8}{19} a^{6} + \frac{8}{19} a^{5} + \frac{1}{19} a^{4} - \frac{5}{19} a^{3} - \frac{8}{19} a^{2} + \frac{9}{19} a$, $\frac{1}{19} a^{10} + \frac{4}{19} a^{8} + \frac{4}{19} a^{7} - \frac{5}{19} a^{6} + \frac{7}{19} a^{5} - \frac{9}{19} a^{4} - \frac{7}{19} a^{3} + \frac{3}{19} a^{2} + \frac{2}{19} a$, $\frac{1}{1045156909429} a^{11} - \frac{2242936008}{1045156909429} a^{10} - \frac{24144692793}{1045156909429} a^{9} + \frac{33669755457}{1045156909429} a^{8} + \frac{316103098293}{1045156909429} a^{7} + \frac{99583078814}{1045156909429} a^{6} - \frac{5713060868}{11743336061} a^{5} - \frac{443185672267}{1045156909429} a^{4} + \frac{71685496487}{1045156909429} a^{3} + \frac{420826743567}{1045156909429} a^{2} - \frac{234702917662}{1045156909429} a + \frac{20409816750}{55008258391}$

Class group and class number

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

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 (assuming GRH)
Regulator:		\( 413348.51768 \) (assuming GRH)

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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.4.6125.1, 6.6.820125.1

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	${\href{/LocalNumberField/2.12.0.1}{12} }$	R	R	R	${\href{/LocalNumberField/11.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/13.12.0.1}{12} }$	${\href{/LocalNumberField/17.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/19.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/23.12.0.1}{12} }$	${\href{/LocalNumberField/29.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/31.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/37.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/41.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/43.12.0.1}{12} }$	${\href{/LocalNumberField/47.12.0.1}{12} }$	${\href{/LocalNumberField/53.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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
$3$	3.12.16.14	$x^{12} + 72 x^{11} - 36 x^{10} + 108 x^{9} - 108 x^{8} + 54 x^{7} + 72 x^{6} - 81 x^{5} - 81 x^{4} - 81 x^{3} + 81 x^{2} - 81$	$3$	$4$	$16$	$C_{12}$	$[2]^{4}$
$5$	5.12.9.1	$x^{12} - 10 x^{8} - 375 x^{4} - 2000$	$4$	$3$	$9$	$C_{12}$	$[\ ]_{4}^{3}$
$7$	7.12.6.2	$x^{12} + 7203 x^{4} - 16807 x^{2} + 588245$	$2$	$6$	$6$	$C_{12}$	$[\ ]_{2}^{6}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.5_7.4t1.1c1	$1$	$ 5 \cdot 7 $	$x^{4} - x^{3} - 9 x^{2} + 9 x + 11$	$C_4$ (as 4T1)	$0$	$1$
*	1.5.2t1.1c1	$1$	$ 5 $	$x^{2} - x - 1$	$C_2$ (as 2T1)	$1$	$1$
*	1.5_7.4t1.1c2	$1$	$ 5 \cdot 7 $	$x^{4} - x^{3} - 9 x^{2} + 9 x + 11$	$C_4$ (as 4T1)	$0$	$1$
*	1.3e2.3t1.1c1	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_5_7.12t1.2c1	$1$	$ 3^{2} \cdot 5 \cdot 7 $	$x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + 441 x^{4} - 6643 x^{3} + 1926 x^{2} + 2865 x - 1349$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.3e2_5.6t1.1c1	$1$	$ 3^{2} \cdot 5 $	$x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.3e2_5_7.12t1.2c2	$1$	$ 3^{2} \cdot 5 \cdot 7 $	$x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + 441 x^{4} - 6643 x^{3} + 1926 x^{2} + 2865 x - 1349$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.3e2.3t1.1c2	$1$	$ 3^{2}$	$x^{3} - 3 x - 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.3e2_5_7.12t1.2c3	$1$	$ 3^{2} \cdot 5 \cdot 7 $	$x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + 441 x^{4} - 6643 x^{3} + 1926 x^{2} + 2865 x - 1349$	$C_{12}$ (as 12T1)	$0$	$1$
*	1.3e2_5.6t1.1c2	$1$	$ 3^{2} \cdot 5 $	$x^{6} - 9 x^{4} - 4 x^{3} + 9 x^{2} + 3 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.3e2_5_7.12t1.2c4	$1$	$ 3^{2} \cdot 5 \cdot 7 $	$x^{12} - 3 x^{11} - 36 x^{10} + 106 x^{9} + 393 x^{8} - 1164 x^{7} - 1350 x^{6} + 4794 x^{5} + 441 x^{4} - 6643 x^{3} + 1926 x^{2} + 2865 x - 1349$	$C_{12}$ (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 *.