Number field 12.2.9621842244611119611904.1

• Label: 12.2.962...904.1
• Degree: $12$
• Signature: $[2, 5]$
• Discriminant: $-9.622\times 10^{21}$
• Root discriminant: $67.91$
• Ramified primes: $2,17,19$
• Class number: $2$ (GRH)
• Class group: [2] (GRH)
• Galois group: $D_{12}$ (as 12T12)

Normalized defining polynomial

x^12 - 4*x^11 + 5*x^10 - 16*x^9 + 139*x^8 - 620*x^7 + 9067*x^6 + 12152*x^5 + 29456*x^4 + 238584*x^3 - 142428*x^2 - 88432*x - 936752

Invariants

Degree:		$12$
Signature:		$[2, 5]$
Discriminant:		$-9621842244611119611904$ $\medspace = -\,2^{15}\cdot 17^{9}\cdot 19^{5}$
Root discriminant:		$67.91$
Galois root discriminant:		$2^{3/2}17^{3/4}19^{1/2}\approx 103.21872376945974$
Ramified primes:		$2$, $17$, $19$
Discriminant root field:		$\Q(\sqrt{-646})$
$\Aut(K/\Q)$:		$C_2$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{4}$, $\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{8}a^{5}-\frac{1}{8}a^{4}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{80}a^{8}-\frac{1}{16}a^{7}-\frac{9}{80}a^{6}+\frac{1}{16}a^{5}-\frac{1}{10}a^{4}-\frac{1}{4}a^{3}+\frac{1}{5}a^{2}-\frac{1}{4}a-\frac{3}{10}$, $\frac{1}{160}a^{9}-\frac{1}{40}a^{7}+\frac{1}{16}a^{6}+\frac{7}{160}a^{5}+\frac{3}{16}a^{4}-\frac{3}{20}a^{3}-\frac{1}{2}a^{2}+\frac{9}{40}a-\frac{1}{4}$, $\frac{1}{640}a^{10}+\frac{1}{640}a^{9}-\frac{1}{160}a^{8}+\frac{3}{320}a^{7}+\frac{17}{640}a^{6}+\frac{37}{640}a^{5}+\frac{43}{320}a^{4}+\frac{7}{80}a^{3}+\frac{9}{160}a^{2}+\frac{39}{160}a+\frac{3}{16}$, $\frac{1}{11\!\cdots\!40}a^{11}+\frac{38\!\cdots\!47}{10\!\cdots\!40}a^{10}+\frac{38\!\cdots\!41}{28\!\cdots\!60}a^{9}-\frac{15\!\cdots\!31}{30\!\cdots\!80}a^{8}-\frac{68\!\cdots\!59}{22\!\cdots\!08}a^{7}+\frac{42\!\cdots\!99}{10\!\cdots\!40}a^{6}-\frac{13\!\cdots\!29}{57\!\cdots\!20}a^{5}+\frac{24\!\cdots\!19}{14\!\cdots\!80}a^{4}-\frac{36\!\cdots\!01}{15\!\cdots\!40}a^{3}-\frac{14\!\cdots\!57}{28\!\cdots\!60}a^{2}+\frac{11\!\cdots\!31}{14\!\cdots\!80}a+\frac{14\!\cdots\!23}{17\!\cdots\!86}$

Monogenic:		No
Index:		Not computed
Inessential primes:		$2$

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

Unit group

Rank:		$6$
Torsion generator:		$-1$  (order $2$)
Fundamental units:		$\frac{59\!\cdots\!33}{22\!\cdots\!08}a^{11}-\frac{68\!\cdots\!11}{20\!\cdots\!28}a^{10}+\frac{54\!\cdots\!81}{11\!\cdots\!04}a^{9}+\frac{17\!\cdots\!49}{60\!\cdots\!16}a^{8}+\frac{51\!\cdots\!17}{22\!\cdots\!08}a^{7}-\frac{19\!\cdots\!11}{20\!\cdots\!28}a^{6}+\frac{21\!\cdots\!11}{57\!\cdots\!52}a^{5}-\frac{38\!\cdots\!41}{57\!\cdots\!52}a^{4}-\frac{19\!\cdots\!25}{30\!\cdots\!08}a^{3}+\frac{62\!\cdots\!41}{57\!\cdots\!52}a^{2}+\frac{23\!\cdots\!83}{17\!\cdots\!86}a+\frac{51\!\cdots\!13}{14\!\cdots\!88}$, $\frac{6011878497}{26\!\cdots\!20}a^{11}-\frac{359402039}{243013929158720}a^{10}-\frac{27187430451}{13\!\cdots\!60}a^{9}+\frac{49725169699}{13\!\cdots\!60}a^{8}+\frac{6999772993}{534630644149184}a^{7}-\frac{34926629531}{243013929158720}a^{6}+\frac{10291797253731}{668288305186480}a^{5}+\frac{12981597790247}{133657661037296}a^{4}+\frac{90297184295917}{668288305186480}a^{3}+\frac{754983958747657}{668288305186480}a^{2}+\frac{51314924953889}{41768019074155}a-\frac{10\!\cdots\!83}{167072076296620}$, $\frac{12\!\cdots\!43}{14\!\cdots\!80}a^{11}-\frac{14\!\cdots\!57}{51\!\cdots\!20}a^{10}-\frac{63\!\cdots\!67}{57\!\cdots\!20}a^{9}+\frac{15\!\cdots\!41}{75\!\cdots\!20}a^{8}+\frac{18\!\cdots\!01}{57\!\cdots\!52}a^{7}-\frac{27\!\cdots\!21}{51\!\cdots\!20}a^{6}+\frac{18\!\cdots\!09}{57\!\cdots\!20}a^{5}+\frac{23\!\cdots\!07}{28\!\cdots\!60}a^{4}-\frac{78\!\cdots\!81}{37\!\cdots\!60}a^{3}-\frac{36\!\cdots\!31}{14\!\cdots\!80}a^{2}-\frac{49\!\cdots\!01}{14\!\cdots\!80}a+\frac{44\!\cdots\!11}{71\!\cdots\!40}$, $\frac{34\!\cdots\!89}{22\!\cdots\!08}a^{11}-\frac{58\!\cdots\!47}{10\!\cdots\!40}a^{10}+\frac{36\!\cdots\!39}{57\!\cdots\!20}a^{9}+\frac{28\!\cdots\!37}{30\!\cdots\!80}a^{8}+\frac{22\!\cdots\!53}{11\!\cdots\!40}a^{7}-\frac{14\!\cdots\!51}{10\!\cdots\!40}a^{6}+\frac{17\!\cdots\!07}{14\!\cdots\!80}a^{5}+\frac{87\!\cdots\!41}{28\!\cdots\!60}a^{4}+\frac{11\!\cdots\!43}{15\!\cdots\!40}a^{3}+\frac{15\!\cdots\!29}{28\!\cdots\!60}a^{2}+\frac{48\!\cdots\!93}{71\!\cdots\!40}a+\frac{72\!\cdots\!43}{71\!\cdots\!40}$, $\frac{21\!\cdots\!31}{11\!\cdots\!40}a^{11}-\frac{93\!\cdots\!85}{20\!\cdots\!28}a^{10}-\frac{59\!\cdots\!83}{57\!\cdots\!20}a^{9}-\frac{29\!\cdots\!69}{60\!\cdots\!16}a^{8}+\frac{11\!\cdots\!15}{22\!\cdots\!08}a^{7}-\frac{13\!\cdots\!85}{20\!\cdots\!28}a^{6}+\frac{96\!\cdots\!37}{71\!\cdots\!40}a^{5}+\frac{21\!\cdots\!51}{57\!\cdots\!52}a^{4}+\frac{12\!\cdots\!49}{15\!\cdots\!40}a^{3}-\frac{41\!\cdots\!93}{57\!\cdots\!52}a^{2}-\frac{10\!\cdots\!67}{71\!\cdots\!40}a-\frac{45\!\cdots\!71}{14\!\cdots\!88}$, $\frac{70\!\cdots\!23}{28\!\cdots\!60}a^{11}-\frac{33\!\cdots\!03}{51\!\cdots\!20}a^{10}+\frac{10\!\cdots\!69}{57\!\cdots\!20}a^{9}-\frac{78\!\cdots\!73}{75\!\cdots\!52}a^{8}+\frac{17\!\cdots\!97}{28\!\cdots\!76}a^{7}-\frac{10\!\cdots\!23}{51\!\cdots\!20}a^{6}+\frac{52\!\cdots\!37}{57\!\cdots\!20}a^{5}-\frac{36\!\cdots\!81}{28\!\cdots\!60}a^{4}+\frac{24\!\cdots\!91}{18\!\cdots\!80}a^{3}-\frac{20\!\cdots\!29}{28\!\cdots\!76}a^{2}+\frac{59\!\cdots\!07}{14\!\cdots\!80}a-\frac{30\!\cdots\!09}{71\!\cdots\!40}$ (assuming GRH)
Regulator:		$79246391.31530055$ (assuming GRH)

Class number formula

$\begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{5}\cdot 79246391.31530055 \cdot 2}{2\cdot\sqrt{9621842244611119611904}}\cr\approx \mathstrut & 31.6453360337721 \end{aligned}$ (assuming GRH)

Galois group

$D_{12}$ (as 12T12):

A solvable group of order 24

The 9 conjugacy class representatives for $D_{12}$

Character table for $D_{12}$

Intermediate fields

$\Q(\sqrt{17})$, 3.1.152.1, 4.2.746776.1, 6.2.113509952.3

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

Sibling fields

Galois closure:	deg 24
Degree 12 sibling:	12.0.1462520021180890181009408.1
Minimal sibling:	This field is its own minimal sibling

Frobenius cycle types

$p$	$2$	$3$	$5$	$7$	$11$	$13$	$17$	$19$	$23$	$29$	$31$	$37$	$41$	$43$	$47$	$53$	$59$
Cycle type	R	${\href{/padicField/3.12.0.1}{12} }$	${\href{/padicField/5.2.0.1}{2} }^{6}$	${\href{/padicField/7.12.0.1}{12} }$	${\href{/padicField/11.2.0.1}{2} }^{6}$	${\href{/padicField/13.3.0.1}{3} }^{4}$	R	R	${\href{/padicField/23.12.0.1}{12} }$	${\href{/padicField/29.12.0.1}{12} }$	${\href{/padicField/31.2.0.1}{2} }^{6}$	${\href{/padicField/37.4.0.1}{4} }^{3}$	${\href{/padicField/41.2.0.1}{2} }^{6}$	${\href{/padicField/43.2.0.1}{2} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$	${\href{/padicField/47.2.0.1}{2} }^{6}$	${\href{/padicField/53.3.0.1}{3} }^{4}$	${\href{/padicField/59.6.0.1}{6} }^{2}$

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
$2$	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{2}$	$x + 1$	$1$	$1$	$0$	Trivial	$[\ ]$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
	2.1.2.3a1.4	$x^{2} + 4 x + 10$	$2$	$1$	$3$	$C_2$	$[3]$
$17$	17.3.4.9a1.3	$x^{12} + 4 x^{10} + 56 x^{9} + 6 x^{8} + 168 x^{7} + 1180 x^{6} + 168 x^{5} + 2353 x^{4} + 11032 x^{3} + 1176 x^{2} + 10976 x + 38433$	$4$	$3$	$9$	$C_{12}$	$[\ ]_{4}^{3}$
$19$	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$[\ ]$
	$\Q_{19}$	$x + 17$	$1$	$1$	$0$	Trivial	$[\ ]$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$
	19.1.2.1a1.2	$x^{2} + 38$	$2$	$1$	$1$	$C_2$	$[\ ]_{2}$

Artin representations

	Label	Dimension	Conductor	Artin stem field	$G$	Ind	$\chi(c)$
*	1.1.1t1.a.a	$1$	$1$	$\Q$	$C_1$	$1$	$1$
	1.152.2t1.b.a	$1$	$2^{3} \cdot 19$	$\Q(\sqrt{-38})$	$C_2$ (as 2T1)	$1$	$-1$
	1.2584.2t1.b.a	$1$	$2^{3} \cdot 17 \cdot 19$	$\Q(\sqrt{-646})$	$C_2$ (as 2T1)	$1$	$-1$
*	1.17.2t1.a.a	$1$	$17$	$\Q(\sqrt{17})$	$C_2$ (as 2T1)	$1$	$1$
*	2.152.3t2.b.a	$2$	$2^{3} \cdot 19$	3.1.152.1	$S_3$ (as 3T2)	$1$	$0$
*	2.43928.6t3.b.a	$2$	$2^{3} \cdot 17^{2} \cdot 19$	6.2.113509952.3	$D_{6}$ (as 6T3)	$1$	$0$
*	2.43928.4t3.b.a	$2$	$2^{3} \cdot 17^{2} \cdot 19$	4.2.746776.1	$D_{4}$ (as 4T3)	$1$	$0$
*	2.43928.12t12.b.b	$2$	$2^{3} \cdot 17^{2} \cdot 19$	12.2.9621842244611119611904.1	$D_{12}$ (as 12T12)	$1$	$0$
*	2.43928.12t12.b.a	$2$	$2^{3} \cdot 17^{2} \cdot 19$	12.2.9621842244611119611904.1	$D_{12}$ (as 12T12)	$1$	$0$

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 *.

Spectrum of ring of integers