Global number field 12.0.61132828589969773.1

• Label: 12.0.61132828589969773.1
• Degree: $12$
• Signature: $[0, 6]$
• Discriminant: $7^{8}\cdot 13^{9}$
• Root discriminant: $25.05$
• Ramified primes: $7, 13$
• Class number: $4$
• Class group: $[2, 2]$
• Galois group: $C_{12}$ (as 12T1)

Normalized defining polynomial

\( x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601 \)

Invariants

Degree:		$12$
Signature:		$[0, 6]$
Discriminant:		\(61132828589969773=7^{8}\cdot 13^{9}\)
Root discriminant:		$25.05$
Ramified primes:		$7, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(91=7\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{91}(64,·)$, $\chi_{91}(1,·)$, $\chi_{91}(8,·)$, $\chi_{91}(44,·)$, $\chi_{91}(79,·)$, $\chi_{91}(18,·)$, $\chi_{91}(51,·)$, $\chi_{91}(53,·)$, $\chi_{91}(86,·)$, $\chi_{91}(25,·)$, $\chi_{91}(57,·)$, $\chi_{91}(60,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{57088767249011397} a^{11} - \frac{1169367241308461}{19029589083003799} a^{10} - \frac{1610397135385953}{19029589083003799} a^{9} - \frac{7466144941468694}{19029589083003799} a^{8} + \frac{6376119561762270}{19029589083003799} a^{7} - \frac{8633952934697097}{19029589083003799} a^{6} + \frac{10512530373335212}{57088767249011397} a^{5} - \frac{3846971296255146}{19029589083003799} a^{4} - \frac{5120076750511115}{57088767249011397} a^{3} - \frac{15265976468250832}{57088767249011397} a^{2} + \frac{4682440286535021}{19029589083003799} a - \frac{5382215146921500}{19029589083003799}$

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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:		\( 562.775330001 \)

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

\(\Q(\sqrt{13}) \), \(\Q(\zeta_{7})^+\), 4.0.2197.1, 6.6.5274997.1

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	${\href{/LocalNumberField/2.12.0.1}{12} }$	${\href{/LocalNumberField/3.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/5.12.0.1}{12} }$	R	${\href{/LocalNumberField/11.12.0.1}{12} }$	R	${\href{/LocalNumberField/17.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/19.12.0.1}{12} }$	${\href{/LocalNumberField/23.6.0.1}{6} }^{2}$	${\href{/LocalNumberField/29.1.0.1}{1} }^{12}$	${\href{/LocalNumberField/31.12.0.1}{12} }$	${\href{/LocalNumberField/37.12.0.1}{12} }$	${\href{/LocalNumberField/41.4.0.1}{4} }^{3}$	${\href{/LocalNumberField/43.2.0.1}{2} }^{6}$	${\href{/LocalNumberField/47.12.0.1}{12} }$	${\href{/LocalNumberField/53.3.0.1}{3} }^{4}$	${\href{/LocalNumberField/59.12.0.1}{12} }$

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
$7$	7.12.8.1	$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$	$3$	$4$	$8$	$C_{12}$	$[\ ]_{3}^{4}$
$13$	13.4.3.2	$x^{4} - 52$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	13.4.3.2	$x^{4} - 52$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$
	13.4.3.2	$x^{4} - 52$	$4$	$1$	$3$	$C_4$	$[\ ]_{4}$

Artin representations

	Label	Dimension	Conductor	Defining polynomial of Artin field	$G$	Ind	$\chi(c)$
*	1.1.1t1.1c1	$1$	$1$	$x$	$C_1$	$1$	$1$
*	1.13.4t1.1c1	$1$	$ 13 $	$x^{4} - x^{3} + 2 x^{2} + 4 x + 3$	$C_4$ (as 4T1)	$0$	$-1$
*	1.13.2t1.1c1	$1$	$ 13 $	$x^{2} - x - 3$	$C_2$ (as 2T1)	$1$	$1$
*	1.13.4t1.1c2	$1$	$ 13 $	$x^{4} - x^{3} + 2 x^{2} + 4 x + 3$	$C_4$ (as 4T1)	$0$	$-1$
*	1.7.3t1.1c1	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.7_13.12t1.1c1	$1$	$ 7 \cdot 13 $	$x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601$	$C_{12}$ (as 12T1)	$0$	$-1$
*	1.7_13.6t1.3c1	$1$	$ 7 \cdot 13 $	$x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.7_13.12t1.1c2	$1$	$ 7 \cdot 13 $	$x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601$	$C_{12}$ (as 12T1)	$0$	$-1$
*	1.7.3t1.1c2	$1$	$ 7 $	$x^{3} - x^{2} - 2 x + 1$	$C_3$ (as 3T1)	$0$	$1$
*	1.7_13.12t1.1c3	$1$	$ 7 \cdot 13 $	$x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601$	$C_{12}$ (as 12T1)	$0$	$-1$
*	1.7_13.6t1.3c2	$1$	$ 7 \cdot 13 $	$x^{6} - x^{5} - 14 x^{4} + 9 x^{3} + 35 x^{2} - 16 x - 1$	$C_6$ (as 6T1)	$0$	$1$
*	1.7_13.12t1.1c4	$1$	$ 7 \cdot 13 $	$x^{12} - x^{11} - 4 x^{10} - 10 x^{9} + 39 x^{8} - 78 x^{7} + 214 x^{6} - 280 x^{5} + 693 x^{4} - 573 x^{3} - 222 x^{2} + 123 x + 601$	$C_{12}$ (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 *.