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Properties

Label 22.6.46598165082...2736.1
Degree $22$
Signature $[6, 8]$
Discriminant $2^{70}\cdot 3^{22}\cdot 67^{10}\cdot 337^{8}\cdot 2851^{10}\cdot 310501^{8}\cdot 818717^{10}$
Root discriminant $2{,}750{,}877{,}871.52$
Ramified primes $2, 3, 67, 337, 2851, 310501, 818717$
Class number Not computed
Class group Not computed
Galois group 22T43

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-51673851153110341911101636537143872524838733712511652216813045532878318310449694003880719633962468700454539909239644900000000000000000000, 0, -198251395036319826765137880268363506213709892664773937814853717831634869375711794380750206495940054252200460000000000000000000, 0, -215482514009663496075841906438049192600636056827517073619648074832500266214861368952421191535957390000000000000000, 0, 29163452098994916756535843681511546182067427493883528300264411710594464387592978997173200000000000000, 0, 172961783005919316733286895245898258178180718110899199234964810104670395301160000000000000, 0, 73155017113516899879646403705978634052064727703137418176016931767440000000000, 0, -14327363001042340287658373126532762994707342768400250803200000000, 0, -14033625357672757890962248232195405624570831712000000, 0, -1918326833766516877145612304640202610000, 0, 209644616157290336872450800, 0, 44093388187521, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 + 44093388187521*x^20 + 209644616157290336872450800*x^18 - 1918326833766516877145612304640202610000*x^16 - 14033625357672757890962248232195405624570831712000000*x^14 - 14327363001042340287658373126532762994707342768400250803200000000*x^12 + 73155017113516899879646403705978634052064727703137418176016931767440000000000*x^10 + 172961783005919316733286895245898258178180718110899199234964810104670395301160000000000000*x^8 + 29163452098994916756535843681511546182067427493883528300264411710594464387592978997173200000000000000*x^6 - 215482514009663496075841906438049192600636056827517073619648074832500266214861368952421191535957390000000000000000*x^4 - 198251395036319826765137880268363506213709892664773937814853717831634869375711794380750206495940054252200460000000000000000000*x^2 - 51673851153110341911101636537143872524838733712511652216813045532878318310449694003880719633962468700454539909239644900000000000000000000)
 
gp: K = bnfinit(x^22 + 44093388187521*x^20 + 209644616157290336872450800*x^18 - 1918326833766516877145612304640202610000*x^16 - 14033625357672757890962248232195405624570831712000000*x^14 - 14327363001042340287658373126532762994707342768400250803200000000*x^12 + 73155017113516899879646403705978634052064727703137418176016931767440000000000*x^10 + 172961783005919316733286895245898258178180718110899199234964810104670395301160000000000000*x^8 + 29163452098994916756535843681511546182067427493883528300264411710594464387592978997173200000000000000*x^6 - 215482514009663496075841906438049192600636056827517073619648074832500266214861368952421191535957390000000000000000*x^4 - 198251395036319826765137880268363506213709892664773937814853717831634869375711794380750206495940054252200460000000000000000000*x^2 - 51673851153110341911101636537143872524838733712511652216813045532878318310449694003880719633962468700454539909239644900000000000000000000, 1)
 

Normalized defining polynomial

\( x^{22} + 44093388187521 x^{20} + 209644616157290336872450800 x^{18} - 1918326833766516877145612304640202610000 x^{16} - 14033625357672757890962248232195405624570831712000000 x^{14} - 14327363001042340287658373126532762994707342768400250803200000000 x^{12} + 73155017113516899879646403705978634052064727703137418176016931767440000000000 x^{10} + 172961783005919316733286895245898258178180718110899199234964810104670395301160000000000000 x^{8} + 29163452098994916756535843681511546182067427493883528300264411710594464387592978997173200000000000000 x^{6} - 215482514009663496075841906438049192600636056827517073619648074832500266214861368952421191535957390000000000000000 x^{4} - 198251395036319826765137880268363506213709892664773937814853717831634869375711794380750206495940054252200460000000000000000000 x^{2} - 51673851153110341911101636537143872524838733712511652216813045532878318310449694003880719633962468700454539909239644900000000000000000000 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4659816508200478508412033889999092351551762621968614936542784442804549758984043866125318034750187275475080556854253986865982314800422915936162973675231354319646594465684660980065285087986232564698737841012736=2^{70}\cdot 3^{22}\cdot 67^{10}\cdot 337^{8}\cdot 2851^{10}\cdot 310501^{8}\cdot 818717^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2{,}750{,}877{,}871.52$
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:  $2, 3, 67, 337, 2851, 310501, 818717$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{30} a^{3} - \frac{3}{10} a$, $\frac{1}{46916659556700} a^{4} - \frac{941090456393}{15638886518900} a^{2}$, $\frac{1}{1407499786701000} a^{5} - \frac{941090456393}{469166595567000} a^{3} + \frac{1}{10} a$, $\frac{1}{2201172943959289440514890000} a^{6} - \frac{941090456393}{733724314653096480171630000} a^{4} + \frac{2430572857301}{15638886518900} a^{2}$, $\frac{1}{66035188318778683215446700000} a^{7} - \frac{941090456393}{22011729439592894405148900000} a^{5} + \frac{2430572857301}{469166595567000} a^{3} - \frac{2}{5} a$, $\frac{1}{103271681637157070465074310087149263000000} a^{8} - \frac{941090456393}{34423893879052356821691436695716421000000} a^{6} + \frac{2430572857301}{733724314653096480171630000} a^{4} - \frac{680268388971}{3909721629725} a^{2}$, $\frac{1}{3098150449114712113952229302614477890000000} a^{9} - \frac{941090456393}{1032716816371570704650743100871492630000000} a^{7} + \frac{2430572857301}{22011729439592894405148900000} a^{5} - \frac{226756129657}{39097216297250} a^{3} - \frac{2}{5} a$, $\frac{1}{4845162329218405171854203943925910679788311712100000000} a^{10} - \frac{941090456393}{1615054109739468390618067981308636893262770570700000000} a^{8} + \frac{2430572857301}{34423893879052356821691436695716421000000} a^{6} - \frac{226756129657}{61143692887758040014302500} a^{4} + \frac{668944192819}{3909721629725} a^{2}$, $\frac{1}{48451623292184051718542039439259106797883117121000000000} a^{11} - \frac{941090456393}{16150541097394683906180679813086368932627705707000000000} a^{9} + \frac{2430572857301}{344238938790523568216914366957164210000000} a^{7} + \frac{467229115703}{1375733089974555900321806250} a^{5} + \frac{1417247971487}{93833319113400} a^{3} + \frac{1}{10} a$, $\frac{1}{227318831496887520647606243324862408211640888220455004026070000000000} a^{12} - \frac{941090456393}{75772943832295840215868747774954136070546962740151668008690000000000} a^{10} + \frac{2430572857301}{1615054109739468390618067981308636893262770570700000000} a^{8} - \frac{226756129657}{2868657823254363068474286391309701750000} a^{6} + \frac{668944192819}{183431078663274120042907500} a^{4} - \frac{133838122647}{781944325945} a^{2}$, $\frac{1}{2273188314968875206476062433248624082116408882204550040260700000000000} a^{13} - \frac{941090456393}{757729438322958402158687477749541360705469627401516680086900000000000} a^{11} + \frac{2430572857301}{16150541097394683906180679813086368932627705707000000000} a^{9} + \frac{467229115703}{64544801023223169040671443804468289375000} a^{7} + \frac{1417247971487}{4402345887918578881029780000} a^{5} - \frac{5599714501519}{469166595567000} a^{3} - \frac{1}{5} a$, $\frac{1}{10665040228166324861896484028865991486935258034516686721062054041443169000000000000} a^{14} - \frac{941090456393}{3555013409388774953965494676288663828978419344838895573687351347147723000000000000} a^{12} + \frac{2430572857301}{75772943832295840215868747774954136070546962740151668008690000000000} a^{10} - \frac{226756129657}{134587842478289032551505665109053074438564214225000000} a^{8} + \frac{668944192819}{8605973469763089205422859173929105250000} a^{6} - \frac{44612707549}{12228738577551608002860500} a^{4} + \frac{669217431301}{3909721629725} a^{2}$, $\frac{1}{106650402281663248618964840288659914869352580345166867210620540414431690000000000000} a^{15} - \frac{941090456393}{35550134093887749539654946762886638289784193448388955736873513471477230000000000000} a^{13} + \frac{2430572857301}{757729438322958402158687477749541360705469627401516680086900000000000} a^{11} + \frac{467229115703}{3028226455761503232408877464953694174867694820062500000} a^{9} + \frac{1417247971487}{206543363274314140930148620174298526000000} a^{7} - \frac{5599714501519}{22011729439592894405148900000} a^{5} + \frac{221230650822}{19548608148625} a^{3} + \frac{2}{5} a$, $\frac{1}{500368061543389553849012485099024538159141917059287768517336004917206512723783182300000000000000} a^{16} - \frac{941090456393}{166789353847796517949670828366341512719713972353095922839112001639068837574594394100000000000000} a^{14} + \frac{2430572857301}{3555013409388774953965494676288663828978419344838895573687351347147723000000000000} a^{12} - \frac{226756129657}{6314411986024653351322395647912844672545580228345972334057500000000} a^{10} + \frac{668944192819}{403763527434867097654516995327159223315692642675000000} a^{8} - \frac{44612707549}{573731564650872613694857278261940350000} a^{6} + \frac{669217431301}{183431078663274120042907500} a^{4} - \frac{669216079831}{3909721629725} a^{2}$, $\frac{1}{5003680615433895538490124850990245381591419170592877685173360049172065127237831823000000000000000} a^{17} - \frac{941090456393}{1667893538477965179496708283663415127197139723530959228391120016390688375745943941000000000000000} a^{15} + \frac{2430572857301}{35550134093887749539654946762886638289784193448388955736873513471477230000000000000} a^{13} - \frac{226756129657}{63144119860246533513223956479128446725455802283459723340575000000000} a^{11} - \frac{475722262817}{3028226455761503232408877464953694174867694820062500000} a^{9} - \frac{7089196902427}{1032716816371570704650743100871492630000000} a^{7} + \frac{5600036318311}{22011729439592894405148900000} a^{5} - \frac{221229975087}{19548608148625} a^{3}$, $\frac{1}{46951195992954242526602441755757531888143818896942384490608579329447710254899524774468066572820000000000000000} a^{18} + \frac{4899265354169}{5216799554772693614066937972861947987571535432993598276734286592160856694988836086052007396980000000000000000} a^{16} - \frac{1768675514749}{41697338461949129487417707091585378179928493088273980709778000409767209393648598525000000000000} a^{14} - \frac{1498829676109}{711002681877754990793098935257732765795683868967779114737470269429544600000000000} a^{12} + \frac{587671138891}{6314411986024653351322395647912844672545580228345972334057500000000} a^{10} + \frac{7493655539713}{1615054109739468390618067981308636893262770570700000000} a^{8} - \frac{44075000191}{215149336744077230135571479348227631250} a^{6} - \frac{7493652836773}{733724314653096480171630000} a^{4} + \frac{3526000020137}{7819443259450} a^{2} - \frac{1}{2}$, $\frac{1}{469511959929542425266024417557575318881438188969423844906085793294477102548995247744680665728200000000000000000} a^{19} + \frac{4899265354169}{52167995547726936140669379728619479875715354329935982767342865921608566949888360860520073969800000000000000000} a^{17} - \frac{1768675514749}{416973384619491294874177070915853781799284930882739807097780004097672093936485985250000000000000} a^{15} - \frac{1498829676109}{7110026818777549907930989352577327657956838689677791147374702694295446000000000000} a^{13} + \frac{587671138891}{63144119860246533513223956479128446725455802283459723340575000000000} a^{11} + \frac{6842080100239}{48451623292184051718542039439259106797883117121000000000} a^{9} - \frac{4576023116387}{1032716816371570704650743100871492630000000} a^{7} + \frac{1808383737929}{7337243146530964801716300000} a^{5} - \frac{2284102472193}{156388865189000} a^{3} - \frac{1}{4} a$, $\frac{1}{63756900362699197908125746771364242225948574601007678513979747159962897057030628632118341059782365866698265488815736775504544973836112452271101800272413518719456652000000000000000000} a^{20} + \frac{4426349508283418493522554419673483850055162830349557065417910857132341}{787122226699990097631182058905731385505537958037131833505922804443986383420131217680473346417066245267879820849576997228451172516495215460137059262622389119993292000000000000000000} a^{18} - \frac{9305481163255044358491925754756341629442412767625455730165094622262713}{16777030465025594378990918089915296075317141250575393685014978155118591126994177852323511271946442737435654762184749581597467979149683089486582651706760000000000000000} a^{16} - \frac{1497583740454328845400348303167772154273242923130943741732253496645963}{43886312727499394506117165768106458513488045199525627475723660886193800590991790593033396754501299440545349971011851844994303678049536417980000000000000} a^{14} - \frac{84202509587182560339346923578925768483719067039761146626819724462903}{6430943781002222139168487035290282919210632829208477800502388225318918255198580266724767596974846661989553525563745571759461345875000000000} a^{12} - \frac{53176815555252043006921766145675447571731428482013511353053929841738421}{548286586621134846466543560001498227399535591899509451249558102392727440774564846735971122964750377759884368648828005000000000} a^{10} - \frac{84449722207330262242891164163201454660859506360063598059766220216189933}{23372788762103418085889582879042296384237948117613259405871477198254963663985353725449542921880695104670300000000} a^{8} - \frac{27832655434496528617209693787091478524259386688257544340362971870939747}{166058915126987366628898148041043724937097667164566893494613561271849124237300098008077466603000000} a^{6} - \frac{350730820060603130030204478780832094945302085769982313238138351764261}{44243056916248710241399965396634185426704612800980313248156249375078690617333688625} a^{4} - \frac{146990684380761798458182576867337593307115050116861911661867593002661407}{2715879713576067915395145503473061716637918041318635424415145533451457200} a^{2} - \frac{25993217966688477012438465072113728408979759222961088211993}{57887320607168480035725746532474501690526662399435613406516}$, $\frac{1}{637569003626991979081257467713642422259485746010076785139797471599628970570306286321183410597823658666982654888157367755045449738361124522711018002724135187194566520000000000000000000} a^{21} + \frac{4426349508283418493522554419673483850055162830349557065417910857132341}{7871222266999900976311820589057313855055379580371318335059228044439863834201312176804733464170662452678798208495769972284511725164952154601370592626223891199932920000000000000000000} a^{19} - \frac{9305481163255044358491925754756341629442412767625455730165094622262713}{167770304650255943789909180899152960753171412505753936850149781551185911269941778523235112719464427374356547621847495815974679791496830894865826517067600000000000000000} a^{17} - \frac{1497583740454328845400348303167772154273242923130943741732253496645963}{438863127274993945061171657681064585134880451995256274757236608861938005909917905930333967545012994405453499710118518449943036780495364179800000000000000} a^{15} - \frac{84202509587182560339346923578925768483719067039761146626819724462903}{64309437810022221391684870352902829192106328292084778005023882253189182551985802667247675969748466619895535255637455717594613458750000000000} a^{13} - \frac{53176815555252043006921766145675447571731428482013511353053929841738421}{5482865866211348464665435600014982273995355918995094512495581023927274407745648467359711229647503777598843686488280050000000000} a^{11} - \frac{27025857157318460445744700533515887596085348970304508811369866194281699}{701183662863102542576687486371268891527138443528397782176144315947648909919560611763486287656420853140109000000000} a^{9} - \frac{629831511469807599300412751084026613828909279582276452782116089236397}{249088372690481049943347222061565587405646500746850340241920341907773686355950147012116199904500000} a^{7} - \frac{290459041183613396812498132890964864503137010442259298331657983943793}{19909375612311919608629984428485383442017075760441140961670312218785410777800159881250} a^{5} - \frac{218284196425810432370404446781647691935420846779105250227745663246885023}{27158797135760679153951455034730617166379180413186354244151455334514572000} a^{3} - \frac{28353571836205087416777991627412546358006616804366463005005}{115774641214336960071451493064949003381053324798871226813032} a$

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

Class group and class number

Not computed

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

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T43:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 8110080
The 52 conjugacy class representatives for t22n43 are not computed
Character table for t22n43 is not computed

Intermediate fields

11.11.118769262421915560193703211428553337469927424.1

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

Sibling fields

Degree 22 sibling: data not computed
Degree 44 sibling: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
2Data not computed
3Data not computed
$67$67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.1.2$x^{2} + 268$$2$$1$$1$$C_2$$[\ ]_{2}$
67.2.0.1$x^{2} - x + 12$$1$$2$$0$$C_2$$[\ ]^{2}$
67.8.4.1$x^{8} + 17956 x^{4} - 300763 x^{2} + 80604484$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
67.8.4.2$x^{8} - 300763 x^{2} + 40302242$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
337Data not computed
2851Data not computed
310501Data not computed
818717Data not computed