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Properties

Label 22.0.44146235673...4347.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,3^{11}\cdot 331^{20}$
Root discriminant $338.31$
Ramified primes $3, 331$
Class number $263659$ (GRH)
Class group $[11, 23969]$ (GRH)
Galois group $C_{22}$ (as 22T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5233186963456, -9185162559488, 11744248303616, -9253854241792, 6127699221760, -3019035195648, 1408047471744, -530942481728, 203712968048, -60845873872, 18785461384, -4321709316, 1133460253, -205415593, 44834779, -5403558, 995289, -75763, 16325, -654, 151, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + 151*x^20 - 654*x^19 + 16325*x^18 - 75763*x^17 + 995289*x^16 - 5403558*x^15 + 44834779*x^14 - 205415593*x^13 + 1133460253*x^12 - 4321709316*x^11 + 18785461384*x^10 - 60845873872*x^9 + 203712968048*x^8 - 530942481728*x^7 + 1408047471744*x^6 - 3019035195648*x^5 + 6127699221760*x^4 - 9253854241792*x^3 + 11744248303616*x^2 - 9185162559488*x + 5233186963456)
 
gp: K = bnfinit(x^22 - x^21 + 151*x^20 - 654*x^19 + 16325*x^18 - 75763*x^17 + 995289*x^16 - 5403558*x^15 + 44834779*x^14 - 205415593*x^13 + 1133460253*x^12 - 4321709316*x^11 + 18785461384*x^10 - 60845873872*x^9 + 203712968048*x^8 - 530942481728*x^7 + 1408047471744*x^6 - 3019035195648*x^5 + 6127699221760*x^4 - 9253854241792*x^3 + 11744248303616*x^2 - 9185162559488*x + 5233186963456, 1)
 

Normalized defining polynomial

\( x^{22} - x^{21} + 151 x^{20} - 654 x^{19} + 16325 x^{18} - 75763 x^{17} + 995289 x^{16} - 5403558 x^{15} + 44834779 x^{14} - 205415593 x^{13} + 1133460253 x^{12} - 4321709316 x^{11} + 18785461384 x^{10} - 60845873872 x^{9} + 203712968048 x^{8} - 530942481728 x^{7} + 1408047471744 x^{6} - 3019035195648 x^{5} + 6127699221760 x^{4} - 9253854241792 x^{3} + 11744248303616 x^{2} - 9185162559488 x + 5233186963456 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

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:  \(-44146235673740062627077157283789263729923191836638934347=-\,3^{11}\cdot 331^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $338.31$
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:  $3, 331$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(993=3\cdot 331\)
Dirichlet character group:    $\lbrace$$\chi_{993}(1,·)$, $\chi_{993}(451,·)$, $\chi_{993}(773,·)$, $\chi_{993}(74,·)$, $\chi_{993}(332,·)$, $\chi_{993}(782,·)$, $\chi_{993}(80,·)$, $\chi_{993}(274,·)$, $\chi_{993}(85,·)$, $\chi_{993}(601,·)$, $\chi_{993}(736,·)$, $\chi_{993}(605,·)$, $\chi_{993}(416,·)$, $\chi_{993}(932,·)$, $\chi_{993}(293,·)$, $\chi_{993}(742,·)$, $\chi_{993}(167,·)$, $\chi_{993}(829,·)$, $\chi_{993}(442,·)$, $\chi_{993}(955,·)$, $\chi_{993}(842,·)$, $\chi_{993}(511,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} + \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{8} a^{4} - \frac{1}{4} a^{2}$, $\frac{1}{16} a^{9} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{10} - \frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{1}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{7} - \frac{1}{8} a^{6} + \frac{1}{8} a^{5} - \frac{1}{8} a^{4} + \frac{3}{16} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{64} a^{8} + \frac{1}{32} a^{7} + \frac{1}{32} a^{6} + \frac{1}{16} a^{5} + \frac{5}{64} a^{4} + \frac{1}{32} a^{3} + \frac{1}{16} a^{2}$, $\frac{1}{128} a^{13} + \frac{1}{64} a^{11} + \frac{3}{128} a^{9} - \frac{3}{64} a^{7} + \frac{1}{16} a^{6} + \frac{25}{128} a^{5} + \frac{1}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} + \frac{1}{4} a$, $\frac{1}{256} a^{14} - \frac{1}{128} a^{12} + \frac{3}{256} a^{10} + \frac{3}{128} a^{8} - \frac{1}{16} a^{7} - \frac{7}{256} a^{6} + \frac{3}{16} a^{5} - \frac{3}{64} a^{4} + \frac{3}{16} a^{3} - \frac{1}{16} a^{2} - \frac{1}{4} a$, $\frac{1}{1024} a^{15} - \frac{1}{512} a^{13} - \frac{13}{1024} a^{11} - \frac{1}{32} a^{10} - \frac{5}{512} a^{9} - \frac{1}{32} a^{8} + \frac{9}{1024} a^{7} - \frac{1}{32} a^{6} + \frac{5}{256} a^{5} + \frac{1}{64} a^{3} + \frac{3}{8} a^{2} - \frac{1}{2} a$, $\frac{1}{31744} a^{16} - \frac{1}{31744} a^{15} + \frac{29}{15872} a^{14} - \frac{11}{15872} a^{13} - \frac{117}{31744} a^{12} - \frac{419}{31744} a^{11} + \frac{117}{15872} a^{10} + \frac{177}{15872} a^{9} - \frac{895}{31744} a^{8} + \frac{1607}{31744} a^{7} - \frac{169}{1984} a^{6} - \frac{1611}{7936} a^{5} - \frac{415}{1984} a^{4} + \frac{277}{1984} a^{3} + \frac{147}{496} a^{2} - \frac{43}{124} a + \frac{4}{31}$, $\frac{1}{126976} a^{17} - \frac{1}{126976} a^{16} + \frac{27}{126976} a^{15} + \frac{113}{63488} a^{14} + \frac{193}{126976} a^{13} - \frac{915}{126976} a^{12} - \frac{851}{126976} a^{11} - \frac{939}{63488} a^{10} + \frac{2143}{126976} a^{9} + \frac{6071}{126976} a^{8} - \frac{2487}{126976} a^{7} + \frac{1179}{31744} a^{6} - \frac{3241}{31744} a^{5} + \frac{1083}{7936} a^{4} - \frac{1055}{7936} a^{3} + \frac{479}{1984} a^{2} - \frac{15}{496} a$, $\frac{1}{21078016} a^{18} - \frac{15}{21078016} a^{17} - \frac{63}{21078016} a^{16} + \frac{453}{5269504} a^{15} + \frac{28693}{21078016} a^{14} - \frac{79449}{21078016} a^{13} - \frac{75073}{21078016} a^{12} - \frac{37737}{2634752} a^{11} + \frac{129091}{21078016} a^{10} + \frac{506925}{21078016} a^{9} - \frac{145185}{21078016} a^{8} - \frac{617891}{10539008} a^{7} + \frac{541101}{5269504} a^{6} + \frac{215079}{2634752} a^{5} - \frac{54897}{1317376} a^{4} - \frac{163107}{658688} a^{3} - \frac{81273}{164672} a^{2} - \frac{10369}{41168} a + \frac{1250}{2573}$, $\frac{1}{84312064} a^{19} + \frac{1}{84312064} a^{18} + \frac{29}{84312064} a^{17} + \frac{71}{5269504} a^{16} - \frac{445}{1015808} a^{15} + \frac{81503}{84312064} a^{14} - \frac{267589}{84312064} a^{13} + \frac{146387}{21078016} a^{12} - \frac{1618625}{84312064} a^{11} + \frac{869221}{84312064} a^{10} + \frac{1872419}{84312064} a^{9} - \frac{5259}{1359872} a^{8} + \frac{35571}{2634752} a^{7} + \frac{593917}{10539008} a^{6} - \frac{45731}{1317376} a^{5} - \frac{436781}{2634752} a^{4} + \frac{307553}{1317376} a^{3} + \frac{3501}{10624} a^{2} + \frac{37023}{82336} a - \frac{63}{2573}$, $\frac{1}{4240559570944} a^{20} - \frac{16987}{4240559570944} a^{19} + \frac{17137}{4240559570944} a^{18} - \frac{2256351}{1060139892736} a^{17} - \frac{24374407}{4240559570944} a^{16} - \frac{1210852237}{4240559570944} a^{15} + \frac{3163546343}{4240559570944} a^{14} - \frac{595741729}{530069946368} a^{13} + \frac{14843885615}{4240559570944} a^{12} - \frac{30340875}{1648099328} a^{11} - \frac{88867748281}{4240559570944} a^{10} - \frac{33623388655}{2120279785472} a^{9} + \frac{8515040371}{530069946368} a^{8} - \frac{32310119025}{530069946368} a^{7} - \frac{6378132953}{132517486592} a^{6} + \frac{29016778009}{132517486592} a^{5} - \frac{15762104417}{66258743296} a^{4} - \frac{24369105}{199574528} a^{3} + \frac{612346281}{4141171456} a^{2} - \frac{28198113}{64705804} a - \frac{78357}{194897}$, $\frac{1}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{21} + \frac{13579890775733212982810008915000298423380254077899106443349621613154983}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{20} + \frac{204515485547710422838336497591245594560484478594396445149677051933575337835}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{19} - \frac{802167166253016121806513298189351241749044269208701935010918946685124812021}{57690888733004754362422008706655627426746177904738651870204804457106046219929255936} a^{18} - \frac{106805931095680703136582587205935894325381694001486771225316683538104747672207}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{17} - \frac{659175606782898928797136479705698026176259778951296205162081607201090437592859}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{16} - \frac{10001977382168893588078055443355539530692090894086845266731731447627413276958339}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{15} + \frac{1849574641998522237172868406447144114927920056674605161997936200567409929450645}{1860996410742088850400709958279213787959554125959311350651767885713098265159008256} a^{14} + \frac{167693587649163466782737419437673536711735384858775116327468386465085480923675055}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{13} + \frac{170627701990714339591433917925913173133596453439744027999152536176670072264253583}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{12} + \frac{1410738818226025308450558536044271727779516845169491135589200371110209495754195449}{115381777466009508724844017413311254853492355809477303740409608914212092439858511872} a^{11} + \frac{36485619057970528558968336025053999655892704952147500110448888865523801796308193}{3605680545812797147651375544165976714171636119046165741887800278569127888745578496} a^{10} + \frac{128159517712060598248539798498239923680003489650971161944555151882111138157999227}{28845444366502377181211004353327813713373088952369325935102402228553023109964627968} a^{9} + \frac{384208427100966690128546483819127019440934341119020906132771143797039855643472457}{14422722183251188590605502176663906856686544476184662967551201114276511554982313984} a^{8} + \frac{319478946374104334330103663084019345209072558779163277710078933523380056804322957}{7211361091625594295302751088331953428343272238092331483775600557138255777491156992} a^{7} - \frac{435461057721558625863664500609198110245278603818031902925053675420530554233246605}{3605680545812797147651375544165976714171636119046165741887800278569127888745578496} a^{6} - \frac{21822259767641908143970711383907133203985552052490774559799204915576080428224549}{225355034113299821728210971510373544635727257440385358867987517410570493046598656} a^{5} + \frac{126640730029163279628272293788098615764796006400611310408245857686397496705606093}{901420136453199286912843886041494178542909029761541435471950069642281972186394624} a^{4} - \frac{45112562817522738415639216260597373799322146384332324854563783822939366779374083}{225355034113299821728210971510373544635727257440385358867987517410570493046598656} a^{3} - \frac{8098004887563583227416317082052441184365193012487909036157305031414434457982541}{56338758528324955432052742877593386158931814360096339716996879352642623261649664} a^{2} - \frac{3036219287592184518404779975329058804652237259400134369262960762313961248575711}{7042344816040619429006592859699173269866476795012042464624609919080327907706208} a + \frac{14663456377171923430260982251787745587120630144373902174181887274794094572}{197021732767474804974445861115129064174867860200650248003150456554395924007}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{11}\times C_{23969}$, which has order $263659$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{8815248526247631401978101625574288076939349004012262085793318075}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{21} - \frac{1312057150936880192547563212486117180268748318393017675244608603}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{20} + \frac{1339580941924289005673380202892646833528503038830184327343916609901}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{19} - \frac{2327782641932279975277333889418339075732209482355664656336108076877}{27639213797932985814303184763863291927090984219080226682236247897797351112704} a^{18} + \frac{141444782665498295376731252969806441789949835454203005855775327433255}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{17} - \frac{556760867532462663682530100162341961632074865478646909333409158584225}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{16} + \frac{8473909008051144384332354074137191512720570433460681409361021419548131}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{15} - \frac{20740321713749298331637748854140090340625603088795348878246125375215793}{27639213797932985814303184763863291927090984219080226682236247897797351112704} a^{14} + \frac{371348976724144426839675825539262160144686947082636143312961584683033657}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{13} - \frac{1566303315506507474615358876943410021847090263298345469003628941250721875}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{12} + \frac{9210478988412497808143734606499700434996503519679277179921451936389425487}{55278427595865971628606369527726583854181968438160453364472495795594702225408} a^{11} - \frac{8278277659813304614923543518952046485343102152867655187485930464034559523}{13819606898966492907151592381931645963545492109540113341118123948898675556352} a^{10} + \frac{19069730522760820561784612156378073169244026356282478062666011853554659627}{6909803449483246453575796190965822981772746054770056670559061974449337778176} a^{9} - \frac{29229327765989958959778578865682399363181398414868368235739626482342265135}{3454901724741623226787898095482911490886373027385028335279530987224668889088} a^{8} + \frac{103153751620041233303051937533956217805222331986022102198442859219079448301}{3454901724741623226787898095482911490886373027385028335279530987224668889088} a^{7} - \frac{64393734653819970335063894007301426668955711884558038279680180330223353295}{863725431185405806696974523870727872721593256846257083819882746806167222272} a^{6} + \frac{91249944792869841685287379948058434349017156262804402098897028952492432643}{431862715592702903348487261935363936360796628423128541909941373403083611136} a^{5} - \frac{92908773177835975824201066159106925093596185892273699351009608695561319635}{215931357796351451674243630967681968180398314211564270954970686701541805568} a^{4} + \frac{199374067678155653574635914894276120230873012456973289954383866525501605199}{215931357796351451674243630967681968180398314211564270954970686701541805568} a^{3} - \frac{67034091828814740827420113600486089828891668854553707200903661506415571851}{53982839449087862918560907741920492045099578552891067738742671675385451392} a^{2} + \frac{26125971599825809652920947422927986563484956041605792002574096878776138627}{13495709862271965729640226935480123011274894638222766934685667918846362848} a - \frac{210444359910027455112781467105932482086894036738341564557658625833144}{377565741446731359938457557505598786125640516960126648799397602922067} \) (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:  \( 1372307132022027.0 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 22
The 22 conjugacy class representatives for $C_{22}$
Character table for $C_{22}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.11.15786284949774657045043801.1

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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{11}$ R $22$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/19.11.0.1}{11} }^{2}$ $22$ $22$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{22}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ $22$ $22$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
331Data not computed