Normalized defining polynomial
\( x^{22} - x^{21} + 91 x^{20} - 140 x^{19} + 5866 x^{18} - 6595 x^{17} + 171038 x^{16} + 105438 x^{15} + 3197216 x^{14} + 5217907 x^{13} + 46857656 x^{12} + 117294960 x^{11} + 496574452 x^{10} + 1287274232 x^{9} + 4044314069 x^{8} + 9224730377 x^{7} + 18936907588 x^{6} + 27784386063 x^{5} + 32870636944 x^{4} + 26173775028 x^{3} + 15505412182 x^{2} + 4564257710 x + 939238609 \)
Invariants
| Degree: | $22$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 11]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-1680332897640332021620975129530177197841148968789147=-\,3^{11}\cdot 199^{20}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $213.02$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 199$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(597=3\cdot 199\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{597}(512,·)$, $\chi_{597}(1,·)$, $\chi_{597}(139,·)$, $\chi_{597}(260,·)$, $\chi_{597}(262,·)$, $\chi_{597}(200,·)$, $\chi_{597}(586,·)$, $\chi_{597}(523,·)$, $\chi_{597}(460,·)$, $\chi_{597}(461,·)$, $\chi_{597}(320,·)$, $\chi_{597}(338,·)$, $\chi_{597}(121,·)$, $\chi_{597}(217,·)$, $\chi_{597}(416,·)$, $\chi_{597}(103,·)$, $\chi_{597}(302,·)$, $\chi_{597}(125,·)$, $\chi_{597}(313,·)$, $\chi_{597}(188,·)$, $\chi_{597}(61,·)$, $\chi_{597}(62,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{19} a^{15} + \frac{5}{19} a^{14} + \frac{3}{19} a^{12} - \frac{3}{19} a^{11} + \frac{2}{19} a^{10} - \frac{6}{19} a^{9} - \frac{6}{19} a^{8} - \frac{6}{19} a^{7} - \frac{7}{19} a^{6} + \frac{8}{19} a^{5} - \frac{6}{19} a^{4} - \frac{9}{19} a^{3} - \frac{3}{19} a^{2} - \frac{8}{19} a$, $\frac{1}{19} a^{16} - \frac{6}{19} a^{14} + \frac{3}{19} a^{13} + \frac{1}{19} a^{12} - \frac{2}{19} a^{11} + \frac{3}{19} a^{10} + \frac{5}{19} a^{9} + \frac{5}{19} a^{8} + \frac{4}{19} a^{7} + \frac{5}{19} a^{6} - \frac{8}{19} a^{5} + \frac{2}{19} a^{4} + \frac{4}{19} a^{3} + \frac{7}{19} a^{2} + \frac{2}{19} a$, $\frac{1}{19} a^{17} - \frac{5}{19} a^{14} + \frac{1}{19} a^{13} - \frac{3}{19} a^{12} + \frac{4}{19} a^{11} - \frac{2}{19} a^{10} + \frac{7}{19} a^{9} + \frac{6}{19} a^{8} + \frac{7}{19} a^{7} + \frac{7}{19} a^{6} - \frac{7}{19} a^{5} + \frac{6}{19} a^{4} - \frac{9}{19} a^{3} + \frac{3}{19} a^{2} + \frac{9}{19} a$, $\frac{1}{19} a^{18} + \frac{7}{19} a^{14} - \frac{3}{19} a^{13} + \frac{2}{19} a^{11} - \frac{2}{19} a^{10} - \frac{5}{19} a^{9} - \frac{4}{19} a^{8} - \frac{4}{19} a^{7} - \frac{4}{19} a^{6} + \frac{8}{19} a^{5} - \frac{1}{19} a^{4} - \frac{4}{19} a^{3} - \frac{6}{19} a^{2} - \frac{2}{19} a$, $\frac{1}{30229} a^{19} - \frac{413}{30229} a^{18} - \frac{423}{30229} a^{17} - \frac{638}{30229} a^{16} + \frac{186}{30229} a^{15} + \frac{1493}{30229} a^{14} + \frac{12829}{30229} a^{13} - \frac{1737}{30229} a^{12} - \frac{13390}{30229} a^{11} + \frac{2144}{30229} a^{10} + \frac{10283}{30229} a^{9} + \frac{7234}{30229} a^{8} - \frac{7276}{30229} a^{7} + \frac{9874}{30229} a^{6} - \frac{14309}{30229} a^{5} + \frac{2608}{30229} a^{4} + \frac{9802}{30229} a^{3} + \frac{1163}{30229} a^{2} + \frac{2329}{30229} a + \frac{749}{1591}$, $\frac{1}{2145569749012917751} a^{20} + \frac{5764089518760}{2145569749012917751} a^{19} + \frac{50500116472462850}{2145569749012917751} a^{18} + \frac{45624375151523705}{2145569749012917751} a^{17} + \frac{1035011611393701}{57988371594943723} a^{16} - \frac{40328991994727056}{2145569749012917751} a^{15} + \frac{715446761393828495}{2145569749012917751} a^{14} - \frac{308427803831830163}{2145569749012917751} a^{13} + \frac{221921392051476007}{2145569749012917751} a^{12} - \frac{4080298968667184}{2145569749012917751} a^{11} - \frac{960341180313020105}{2145569749012917751} a^{10} - \frac{870165653477738729}{2145569749012917751} a^{9} - \frac{632083057748737877}{2145569749012917751} a^{8} - \frac{267536619899087572}{2145569749012917751} a^{7} + \frac{1557322134695257}{49896970907277157} a^{6} + \frac{894844509653417546}{2145569749012917751} a^{5} + \frac{1023726376553540296}{2145569749012917751} a^{4} + \frac{98172127006318343}{2145569749012917751} a^{3} + \frac{701740401033368725}{2145569749012917751} a^{2} + \frac{32512894301728259}{112924723632258829} a + \frac{1061978757805027}{5943406506960991}$, $\frac{1}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{21} + \frac{20608505505954819447473564467968312295632231986929582310334935829}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{20} - \frac{1369309395323696901075557797004520118063244689051146327672986889600318165700180}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{19} - \frac{2058554974114611045943517156357025100227251308625621631951027128335921544038569200}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{18} - \frac{1356064350846320754323879925344608846401346681258438808373667773506384958248803200}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{17} - \frac{2147327669367999824208436080051660860594913225779609936847900028301381438382122694}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{16} + \frac{863269392538022614933564719066494505728600029252577035263918869732870973813381802}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{15} - \frac{16301258917219141036043978330384738317160820080134851455644272338584447519727974853}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{14} - \frac{33087525207047126137263214943714435767840559331847304078166232064137471210918641801}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{13} + \frac{14987253702365657881147792104670345594629265828319771717083950685759595515053673326}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{12} + \frac{45942155304839158451854195236075300751776642398086972980205923739193162419202984670}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{11} + \frac{26970725555211858079987486561055472046361999555555146227238954718162948446188895055}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{10} - \frac{18003971197785337963249138740997019728348903128538643169208917441216462624067950406}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{9} + \frac{26349155121354708017064915894582402239617330970646963304430875555802058489394809226}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{8} - \frac{22453701059653338410739793335595436781313155324073992117472984760028082960943242445}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{7} + \frac{32423586841637348677887166323799059544792436371438919569214128806173902314659435798}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{6} - \frac{10107887523481045510134436993966629478003753008740676616454569164011962720797552734}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{5} + \frac{15673272955435691247043727393750461193332545579962927011305189491584461847041043349}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{4} + \frac{46785043783303799932386940328916494286369972681842683671620484321139604904356533437}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{3} - \frac{35139296152369449433894056666156819367822107885198538584991736102556567500335776810}{94052522756969591809021241425594316034475029179385016574671846259306031353657689391} a^{2} - \frac{1255128606101782232322733843443987117102129854935491764437279726422455340263994556}{4950132776682610095211644285557595580761843641020264030245886645226633229139878389} a - \frac{75744693104027413148376213640011883465723705164094551018853265599285712333228}{161520957244839954814880552274532436478671440631065488636600210305303397694387}$
Class group and class number
$C_{3608231}$, which has order $3608231$ (assuming GRH)
Unit group
| Rank: | $10$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( \frac{10669988243800450223614884489530853471806538286583450531486988}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{21} - \frac{14317694213933402625266910294479851611485882076362766766583842}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{20} + \frac{977134797323728333957375526262063170346211586760004511615115162}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{19} - \frac{1830288589490429315019054020939366182578564267844096548052592719}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{18} + \frac{63334242708230348398443342049734789625971862355745155711074263580}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{17} - \frac{92306129905755691258291057651694553855035559913431574310765371538}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{16} + \frac{1864295116078175874643821941423541545415954452417935392350595798382}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{15} + \frac{472224007613804462387630352863063122941618713541550731733386695224}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{14} + \frac{34187404913866925295396047873190899675542630685513792828603950765945}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{13} + \frac{43910687106659668022198308169552720010318161908118557722789939581842}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{12} + \frac{489192467524947801844522088814051630384765880575867147192146210912882}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{11} + \frac{1087282099415167655668052651762966749882527508096275189791072981753030}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{10} + \frac{4985979480457454868096151774434807061941320065392498046677854371400165}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{9} + \frac{12132257305473249444543514981688720696825645673012939329370455974719753}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{8} + \frac{39584467715300153048041129550877838613837062121681516005847147684493155}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{7} + \frac{86122435767090156300952165778591249005109365744156346836286822171799533}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{6} + \frac{177165173925357983844829414156466263602208649469720180213383578492320029}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{5} + \frac{244888401789213451848225433678043679992082723480371466071922187874642622}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{4} + \frac{287248911176398054449770625463558221345986668975163941328269211262315160}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{3} + \frac{206334209656567240055517403282453224222308428808740992020658063840848831}{25177070343595253075827697601461659446708478239002979978722188158169991} a^{2} + \frac{7119328705822642183105528251943573519515837911383856308608442247918287}{1325108965452381740833036715866403128774130433631735788353799376745789} a + \frac{68400989483269467507143743966177301219120397155943997543637916083}{43237803551811979666298062318217219589980436376537207177009148587} \) (order $6$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 117135822355.96071 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
| A cyclic group of order 22 |
| The 22 conjugacy class representatives for $C_{22}$ |
| Character table for $C_{22}$ is not computed |
Intermediate fields
| \(\Q(\sqrt{-3}) \), 11.11.97393677359695041798001.1 |
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 | $22$ | R | $22$ | ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ | $22$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{22}$ | $22$ | $22$ | ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ | ${\href{/LocalNumberField/37.1.0.1}{1} }^{22}$ | $22$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{22}$ | $22$ | $22$ | $22$ |
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 | Data not computed | ||||||
| $199$ | 199.11.10.1 | $x^{11} - 199$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ |
| 199.11.10.1 | $x^{11} - 199$ | $11$ | $1$ | $10$ | $C_{11}$ | $[\ ]_{11}$ | |