Global number field 22.0.4978518112499354698647829163838661251242411.1

• Label: 22.0.49785181124...2411.1
• Degree: $22$
• Signature: $[0, 11]$
• Discriminant: $-\,11^{41}$
• Root discriminant: $87.25$
• Ramified prime: $11$
• Class number: $23651$ (GRH)
• Class group: $[23651]$ (GRH)
• Galois group: $C_{22}$ (as 22T1)

Normalized defining polynomial

\( x^{22} - 11 x^{19} + 33 x^{18} + 462 x^{17} + 583 x^{16} + 1023 x^{15} + 5214 x^{14} + 8239 x^{13} + 31647 x^{12} - 7140 x^{11} + 18271 x^{10} + 320529 x^{9} - 276177 x^{8} - 359084 x^{7} + 861894 x^{6} - 236907 x^{5} - 210870 x^{4} - 339471 x^{3} + 277992 x^{2} - 96228 x + 177147 \)

Invariants

Degree:		$22$
Signature:		$[0, 11]$
Discriminant:		\(-4978518112499354698647829163838661251242411=-\,11^{41}\)
Root discriminant:		$87.25$
Ramified primes:		$11$
This field is Galois and abelian over $\Q$.
Conductor:		\(121=11^{2}\)
Dirichlet character group:		$\lbrace$$\chi_{121}(1,·)$, $\chi_{121}(67,·)$, $\chi_{121}(65,·)$, $\chi_{121}(76,·)$, $\chi_{121}(10,·)$, $\chi_{121}(87,·)$, $\chi_{121}(12,·)$, $\chi_{121}(98,·)$, $\chi_{121}(78,·)$, $\chi_{121}(109,·)$, $\chi_{121}(120,·)$, $\chi_{121}(21,·)$, $\chi_{121}(23,·)$, $\chi_{121}(89,·)$, $\chi_{121}(32,·)$, $\chi_{121}(34,·)$, $\chi_{121}(100,·)$, $\chi_{121}(43,·)$, $\chi_{121}(45,·)$, $\chi_{121}(111,·)$, $\chi_{121}(54,·)$, $\chi_{121}(56,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{5} - \frac{1}{3} a$, $\frac{1}{9} a^{6} + \frac{1}{9} a^{4} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{5} + \frac{1}{9} a^{3} - \frac{1}{3} a$, $\frac{1}{27} a^{8} + \frac{1}{27} a^{7} + \frac{1}{27} a^{6} + \frac{1}{27} a^{5} - \frac{2}{27} a^{4} - \frac{2}{27} a^{3}$, $\frac{1}{81} a^{9} - \frac{1}{27} a^{7} + \frac{1}{27} a^{5} - \frac{10}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{81} a^{10} + \frac{1}{27} a^{7} - \frac{1}{27} a^{6} + \frac{1}{27} a^{5} + \frac{2}{81} a^{4} - \frac{2}{27} a^{3}$, $\frac{1}{81} a^{11} + \frac{1}{27} a^{7} + \frac{8}{81} a^{5} - \frac{4}{27} a^{3}$, $\frac{1}{243} a^{12} - \frac{1}{243} a^{10} - \frac{1}{81} a^{8} - \frac{4}{243} a^{6} - \frac{11}{243} a^{4} + \frac{2}{27} a^{2}$, $\frac{1}{243} a^{13} - \frac{1}{243} a^{11} - \frac{13}{243} a^{7} - \frac{2}{243} a^{5} - \frac{4}{81} a^{3} + \frac{1}{9} a$, $\frac{1}{729} a^{14} + \frac{1}{729} a^{13} - \frac{1}{729} a^{12} + \frac{2}{729} a^{11} - \frac{1}{243} a^{9} + \frac{5}{729} a^{8} + \frac{23}{729} a^{7} + \frac{16}{729} a^{6} - \frac{50}{729} a^{5} + \frac{11}{243} a^{4} + \frac{4}{27} a^{3} - \frac{2}{27} a^{2} - \frac{1}{9} a$, $\frac{1}{729} a^{15} + \frac{1}{729} a^{13} + \frac{4}{729} a^{11} - \frac{1}{729} a^{9} - \frac{19}{729} a^{7} + \frac{95}{729} a^{5} + \frac{1}{9} a^{3} - \frac{2}{9} a$, $\frac{1}{6561} a^{16} - \frac{2}{6561} a^{15} + \frac{1}{6561} a^{14} + \frac{1}{6561} a^{13} - \frac{2}{6561} a^{12} - \frac{2}{6561} a^{11} + \frac{32}{6561} a^{10} + \frac{11}{6561} a^{9} + \frac{107}{6561} a^{8} + \frac{188}{6561} a^{7} - \frac{178}{6561} a^{6} - \frac{151}{6561} a^{5} - \frac{113}{2187} a^{4} - \frac{59}{729} a^{3} + \frac{14}{243} a^{2} + \frac{2}{27} a$, $\frac{1}{6561} a^{17} - \frac{1}{2187} a^{15} + \frac{1}{2187} a^{14} - \frac{2}{2187} a^{12} + \frac{28}{6561} a^{11} - \frac{2}{2187} a^{10} - \frac{11}{2187} a^{9} - \frac{28}{2187} a^{8} - \frac{5}{729} a^{7} - \frac{7}{2187} a^{6} + \frac{331}{6561} a^{5} + \frac{110}{2187} a^{4} + \frac{23}{729} a^{3} - \frac{8}{243} a^{2} - \frac{2}{27} a$, $\frac{1}{59049} a^{18} - \frac{1}{59049} a^{17} - \frac{4}{59049} a^{16} - \frac{1}{59049} a^{15} + \frac{5}{59049} a^{14} + \frac{20}{59049} a^{13} + \frac{4}{59049} a^{11} + \frac{130}{59049} a^{10} - \frac{161}{59049} a^{9} - \frac{428}{59049} a^{8} + \frac{2536}{59049} a^{7} - \frac{919}{59049} a^{6} - \frac{3184}{19683} a^{5} - \frac{37}{729} a^{4} - \frac{86}{2187} a^{3} + \frac{133}{729} a^{2} + \frac{22}{81} a$, $\frac{1}{59049} a^{19} + \frac{4}{59049} a^{17} + \frac{4}{59049} a^{16} + \frac{40}{59049} a^{15} - \frac{20}{59049} a^{14} + \frac{29}{59049} a^{13} + \frac{13}{59049} a^{12} - \frac{199}{59049} a^{11} + \frac{203}{59049} a^{10} + \frac{104}{59049} a^{9} - \frac{277}{59049} a^{8} - \frac{166}{19683} a^{7} - \frac{2623}{59049} a^{6} + \frac{515}{19683} a^{5} + \frac{349}{2187} a^{4} - \frac{146}{2187} a^{3} + \frac{79}{729} a^{2} + \frac{22}{81} a$, $\frac{1}{177147} a^{20} - \frac{1}{177147} a^{19} + \frac{1}{177147} a^{18} - \frac{2}{59049} a^{17} + \frac{4}{59049} a^{16} - \frac{13}{59049} a^{15} + \frac{52}{177147} a^{14} + \frac{131}{177147} a^{13} + \frac{76}{177147} a^{12} + \frac{178}{59049} a^{11} + \frac{362}{59049} a^{10} - \frac{296}{59049} a^{9} - \frac{170}{177147} a^{8} - \frac{544}{177147} a^{7} - \frac{5837}{177147} a^{6} - \frac{5903}{59049} a^{5} - \frac{28}{729} a^{4} - \frac{1069}{6561} a^{3} - \frac{829}{2187} a^{2} + \frac{119}{243} a - \frac{1}{3}$, $\frac{1}{1002303565823607364444691632347} a^{21} - \frac{2148105224947381733017216}{1002303565823607364444691632347} a^{20} + \frac{5088325457916537612553426}{1002303565823607364444691632347} a^{19} + \frac{311523895308447740185529}{111367062869289707160521292483} a^{18} - \frac{6034587813638496963817378}{111367062869289707160521292483} a^{17} + \frac{12497984911937183507842231}{334101188607869121481563877449} a^{16} - \frac{274632272470308520763853305}{1002303565823607364444691632347} a^{15} - \frac{46865271380954464658923864}{1002303565823607364444691632347} a^{14} + \frac{1096184337880849682933231680}{1002303565823607364444691632347} a^{13} - \frac{98863680331072943248414330}{334101188607869121481563877449} a^{12} - \frac{123399038779061815282140418}{37122354289763235720173764161} a^{11} + \frac{1725204888936712842085151168}{334101188607869121481563877449} a^{10} + \frac{4509742277241888036625640830}{1002303565823607364444691632347} a^{9} - \frac{214422933760949567182772674}{1002303565823607364444691632347} a^{8} - \frac{12235050208413840290699585555}{1002303565823607364444691632347} a^{7} - \frac{2963532871329777297635171228}{334101188607869121481563877449} a^{6} + \frac{1715269695236237679228007030}{37122354289763235720173764161} a^{5} + \frac{4310597420732826281760849554}{37122354289763235720173764161} a^{4} - \frac{1596582629367751306552608181}{12374118096587745240057921387} a^{3} + \frac{682721839337214867240447259}{1374902010731971693339769043} a^{2} + \frac{48412053677462777069096696}{152766890081330188148863227} a + \frac{787900824193337265175886}{1886010988658397384553867}$

Class group and class number

$C_{23651}$, which has order $23651$ (assuming GRH)

Unit group

Rank:		$10$
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:		\( 285114946276 \) (assuming GRH)

Galois group

$C_{22}$ (as 22T1):

A cyclic group of order 22

The 22 conjugacy class representatives for $C_{22}$

Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-11}) \), 11.11.672749994932560009201.1

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	$22$	${\href{/LocalNumberField/3.1.0.1}{1} }^{22}$	${\href{/LocalNumberField/5.11.0.1}{11} }^{2}$	$22$	R	$22$	$22$	$22$	${\href{/LocalNumberField/23.11.0.1}{11} }^{2}$	$22$	${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$	$22$	$22$	${\href{/LocalNumberField/47.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$	${\href{/LocalNumberField/59.11.0.1}{11} }^{2}$

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
11	Data not computed