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Properties

Label 22.10.2541719265...9184.1
Degree $22$
Signature $[10, 6]$
Discriminant $2^{30}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}$
Root discriminant $939.64$
Ramified primes $2, 3, 7, 23, 137$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T49

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2634468582334464, -13115399746724928, 30990808995877088, -24165278101734848, 8751031478070272, -647257611403968, -813912422758992, 261815058820176, 9822827363220, -13494358927986, 682357346031, 256034067942, -16539946488, -406653528, -19963545, -36397800, 3327324, 208542, -23031, 978, -92, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464)
 
gp: K = bnfinit(x^22 - 4*x^21 - 92*x^20 + 978*x^19 - 23031*x^18 + 208542*x^17 + 3327324*x^16 - 36397800*x^15 - 19963545*x^14 - 406653528*x^13 - 16539946488*x^12 + 256034067942*x^11 + 682357346031*x^10 - 13494358927986*x^9 + 9822827363220*x^8 + 261815058820176*x^7 - 813912422758992*x^6 - 647257611403968*x^5 + 8751031478070272*x^4 - 24165278101734848*x^3 + 30990808995877088*x^2 - 13115399746724928*x - 2634468582334464, 1)
 

Normalized defining polynomial

\( x^{22} - 4 x^{21} - 92 x^{20} + 978 x^{19} - 23031 x^{18} + 208542 x^{17} + 3327324 x^{16} - 36397800 x^{15} - 19963545 x^{14} - 406653528 x^{13} - 16539946488 x^{12} + 256034067942 x^{11} + 682357346031 x^{10} - 13494358927986 x^{9} + 9822827363220 x^{8} + 261815058820176 x^{7} - 813912422758992 x^{6} - 647257611403968 x^{5} + 8751031478070272 x^{4} - 24165278101734848 x^{3} + 30990808995877088 x^{2} - 13115399746724928 x - 2634468582334464 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(254171926510889578368436897006059571070510600002237220914603229184=2^{30}\cdot 3^{28}\cdot 7^{4}\cdot 23^{4}\cdot 137^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $939.64$
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:  $2, 3, 7, 23, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{9} + \frac{1}{6} a^{7} - \frac{1}{6} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{12} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{5}{12} a^{4} + \frac{1}{6} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{6} a^{7} + \frac{1}{4} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{12} a^{14} - \frac{1}{6} a^{8} + \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{15} - \frac{1}{6} a^{9} + \frac{1}{4} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{12} a^{16} - \frac{1}{6} a^{10} - \frac{1}{4} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{24} a^{17} - \frac{1}{24} a^{13} - \frac{1}{24} a^{9} + \frac{1}{6} a^{7} - \frac{1}{2} a^{6} - \frac{7}{24} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{24} a^{18} - \frac{1}{24} a^{14} - \frac{1}{24} a^{10} + \frac{1}{6} a^{8} - \frac{1}{2} a^{7} - \frac{7}{24} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{48} a^{19} - \frac{1}{24} a^{16} + \frac{1}{48} a^{15} - \frac{1}{24} a^{14} - \frac{1}{48} a^{11} + \frac{1}{12} a^{10} - \frac{1}{4} a^{9} + \frac{5}{24} a^{8} + \frac{23}{48} a^{7} - \frac{3}{8} a^{6} - \frac{1}{6} a^{5} - \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{2016} a^{20} - \frac{1}{168} a^{18} + \frac{1}{336} a^{17} + \frac{19}{672} a^{16} - \frac{13}{336} a^{15} - \frac{1}{56} a^{14} - \frac{1}{168} a^{13} - \frac{1}{224} a^{12} - \frac{3}{56} a^{11} + \frac{19}{84} a^{10} - \frac{53}{336} a^{9} - \frac{41}{224} a^{8} + \frac{3}{16} a^{7} - \frac{5}{168} a^{6} - \frac{17}{168} a^{5} + \frac{1}{7} a^{4} + \frac{3}{7} a^{3} - \frac{7}{18} a^{2} - \frac{4}{21} a - \frac{2}{7}$, $\frac{1}{261198760289684633926635798859925789675282211354036004174053527014837231135804501121741769488162777624905863208415929059666133858781361658211794449621610799558537280} a^{21} + \frac{25462983319371853391616789037551224728389846591150534568470908084929281245552688402051622893727219283739813320902529269564676662198055846370339851075524913485871}{130599380144842316963317899429962894837641105677018002087026763507418615567902250560870884744081388812452931604207964529833066929390680829105897224810805399779268640} a^{20} + \frac{24766572358476514964600879735003420627635485979721495759161780660606649657467479448394333750683667319646557963489172685361111106394101572501429539700522658920329}{4353312671494743898777263314332096494588036855900600069567558783580620518930075018695696158136046293748431053473598817661102230979689360970196574160360179992642288} a^{19} + \frac{259473222580038029536968285372659341001838982816987725676479606508698164536759310070220478584108098720202727682472409120481973457767565465555526280465283786096363}{43533126714947438987772633143320964945880368559006000695675587835806205189300750186956961581360462937484310534735988176611022309796893609701965741603601799926422880} a^{18} + \frac{44688247692403274356668904098867588018335758459054554630537628092598685623623709923322008636811679134546586543544483460410838133466623746694962388284687820242919}{87066253429894877975545266286641929891760737118012001391351175671612410378601500373913923162720925874968621069471976353222044619593787219403931483207203599852845760} a^{17} + \frac{103057640173641157299093840482673816697414318027127716063766342363203302030715228349695633388332120984241870059269211462620841367218774914089177358615939313876013}{5441640839368429873471579142915120618235046069875750086959448479475775648662593773369620197670057867185538816841998522076377788724611701212745717700450224990802860} a^{16} - \frac{56564169235979195202040973281867061513072419800378485657242948116603383434211031678245686017486517339415630605873656468797059560375080851253824569622675161594161}{3627760559578953248981052761943413745490030713250500057972965652983850432441729182246413465113371911457025877894665681384251859149741134141830478466966816660535240} a^{15} + \frac{8478264581371845903711167167340146995277519914174966340141516098016614017560639792998287679362005148883718846108863114990601347627281049687530277217986912740119}{3627760559578953248981052761943413745490030713250500057972965652983850432441729182246413465113371911457025877894665681384251859149741134141830478466966816660535240} a^{14} - \frac{3309659619033869555964138509642963615610875268185078216740303531758330869771362533054993669304809151934575280088263626319275281912341575246280955140208038066728379}{87066253429894877975545266286641929891760737118012001391351175671612410378601500373913923162720925874968621069471976353222044619593787219403931483207203599852845760} a^{13} + \frac{272697209522443441228130904381907290783637436962581789113655148659772067076996627756271745111200356891589312150014048236696598996844097368883833110685724471647165}{8706625342989487797554526628664192989176073711801200139135117567161241037860150037391392316272092587496862106947197635322204461959378721940393148320720359985284576} a^{12} - \frac{100853791810202196780227012341478896803493754753889481973473090598936661592270348954383666359130317664872279812932213330964956626760548303956029865424982512468861}{1554754525533837106706165469404320176638584591393071453416985279850221613903598220962748627905730819195868233383428149164679368207031914632213062200128635711657960} a^{11} + \frac{3021632724639199831282949980797720637165008340552110096000871137677713902046976007118067139478735891573935711692418837424215002143403235019413206269111407200495389}{43533126714947438987772633143320964945880368559006000695675587835806205189300750186956961581360462937484310534735988176611022309796893609701965741603601799926422880} a^{10} + \frac{3545600491163467974323169807748414197751736598797841607957197271814704437380566322134663588856405160567014943916806966676555594404636046690184256798229469212238173}{17413250685978975595109053257328385978352147423602400278270235134322482075720300074782784632544185174993724213894395270644408923918757443880786296641440719970569152} a^{9} + \frac{362188929247651926405173971871557591517171048643757139401610927827980918718049440176220020838991620640554910449282164250815695796514774726063893046494536697360377}{21766563357473719493886316571660482472940184279503000347837793917903102594650375093478480790680231468742155267367994088305511154898446804850982870801800899963211440} a^{8} + \frac{734741559238693665003783499686521990395758785217147989914077701158988672393304629602998516540838819730019647452727667388500734891855747020258411424345554445247433}{5441640839368429873471579142915120618235046069875750086959448479475775648662593773369620197670057867185538816841998522076377788724611701212745717700450224990802860} a^{7} + \frac{1734424618046256752737312737430690258826785678286087380539336430253062495675657553817016596338297171361752012309510695249066298693116934487631881290249290684877}{4202039258199559747854501268660324801725904301062355279505365621216815172712427624223644940285758970799643874009265268012646941100086255762738005946293610031508} a^{6} - \frac{1062338134897199924892943277008963721102033474824707262486472365414117700154687093776963642778499062821847639081871011065622339676769634598316550910306883034747563}{10883281678736859746943158285830241236470092139751500173918896958951551297325187546739240395340115734371077633683997044152755577449223402425491435400900449981605720} a^{5} + \frac{140807440375647867832733256205805450650246681198150696296196262738885633582845182879281165277619692082702355919011120125366183171474042456333499334829059064254115}{1088328167873685974694315828583024123647009213975150017391889695895155129732518754673924039534011573437107763368399704415275557744922340242549143540090044998160572} a^{4} - \frac{1687812706767681934773228214556234473973901193449346198002707778588771772457303201618381696133985434156874076144254594159511040325101629666491696702321671471461227}{4081230629526322405103684357186340463676284552406812565219586359606831736496945330027215148252543400389154112631498891557283341543458775909559288275337668743102145} a^{3} - \frac{1482923307793682740319957861396497343863953453369823401638727679339457299766734792325109463916686674237007595280040132028054291166168047183373743281218678457265464}{4081230629526322405103684357186340463676284552406812565219586359606831736496945330027215148252543400389154112631498891557283341543458775909559288275337668743102145} a^{2} - \frac{41951143362742047932008463408802996460385316403992885263615249519158046536551519679215678785968448648739663879748569831882827306775151671919456608285253940550581}{906940139894738312245263190485853436372507678312625014493241413245962608110432295561603366278342977864256469473666420346062964787435283535457619616741704165133810} a + \frac{11526255296485123139650615550253849031581688427081855532081729324729966849317885658331063473932720915858562949945317839095171378970397598869807190687689778854658}{151156689982456385374210531747642239395417946385437502415540235540993768018405382593600561046390496310709411578944403391010494131239213922576269936123617360855635}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  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:  \( 739930020231000000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T49:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 20437401600
The 200 conjugacy class representatives for t22n49 are not computed
Character table for t22n49 is not computed

Intermediate fields

11.7.63019333158425674204677255696384.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/11.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.16.69$x^{8} + 12 x^{6} + 28 x^{4} + 52$$8$$1$$16$$C_4\wr C_2$$[2, 2, 5/2]^{4}$
2.12.14.2$x^{12} + 2 x^{4} + 2 x^{3} + 2$$12$$1$$14$12T27$[4/3, 4/3]_{3}^{4}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.9.13.2$x^{9} + 6 x^{5} + 3 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6} $$[3/2, 3/2, 5/3]_{2}^{2}$
3.9.13.2$x^{9} + 6 x^{5} + 3 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6} $$[3/2, 3/2, 5/3]_{2}^{2}$
$7$7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
7.7.0.1$x^{7} - x + 2$$1$$7$$0$$C_7$$[\ ]^{7}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
23.7.0.1$x^{7} - x + 8$$1$$7$$0$$C_7$$[\ ]^{7}$
137Data not computed