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Properties

Label 22.14.2683021359...0784.1
Degree $22$
Signature $[14, 4]$
Discriminant $2^{72}\cdot 3^{20}\cdot 337^{8}\cdot 947^{10}\cdot 310501^{8}\cdot 53591959^{10}$
Root discriminant $1{,}589{,}648{,}471.61$
Ramified primes $2, 3, 337, 947, 310501, 53591959$
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![-387525510081628637297743170535017129158275865687798305019045138474571366001139953640250564464680282754821621623700592281054857750134784, 0, 465315646285731804566448193725361044248288059478367061599274499469392698960388352711250887836367396465243767319244482010112, 0, 5343558891203926971053678141516280096820104179732811762443202895791628733302502206317811912655111486804391018496, 0, -6274369569526916448536833950155270998368078365193513543624088680948481195760518524041106824138496768, 0, -3747833231991359307927309897387471198885933309684020485859542287332808873765923846934528, 0, 6224971581116867291176830299635009030608381084723679035467660012998285012992, 0, -711369580334732818426577644309882828779774928375610743409128704, 0, -1100067537115825097282039898823065852802526737569408, 0, 282033335825689492595606193310007218368, 0, 29455972774636519059028044, 0, -13804431167056, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 13804431167056*x^20 + 29455972774636519059028044*x^18 + 282033335825689492595606193310007218368*x^16 - 1100067537115825097282039898823065852802526737569408*x^14 - 711369580334732818426577644309882828779774928375610743409128704*x^12 + 6224971581116867291176830299635009030608381084723679035467660012998285012992*x^10 - 3747833231991359307927309897387471198885933309684020485859542287332808873765923846934528*x^8 - 6274369569526916448536833950155270998368078365193513543624088680948481195760518524041106824138496768*x^6 + 5343558891203926971053678141516280096820104179732811762443202895791628733302502206317811912655111486804391018496*x^4 + 465315646285731804566448193725361044248288059478367061599274499469392698960388352711250887836367396465243767319244482010112*x^2 - 387525510081628637297743170535017129158275865687798305019045138474571366001139953640250564464680282754821621623700592281054857750134784)
 
gp: K = bnfinit(x^22 - 13804431167056*x^20 + 29455972774636519059028044*x^18 + 282033335825689492595606193310007218368*x^16 - 1100067537115825097282039898823065852802526737569408*x^14 - 711369580334732818426577644309882828779774928375610743409128704*x^12 + 6224971581116867291176830299635009030608381084723679035467660012998285012992*x^10 - 3747833231991359307927309897387471198885933309684020485859542287332808873765923846934528*x^8 - 6274369569526916448536833950155270998368078365193513543624088680948481195760518524041106824138496768*x^6 + 5343558891203926971053678141516280096820104179732811762443202895791628733302502206317811912655111486804391018496*x^4 + 465315646285731804566448193725361044248288059478367061599274499469392698960388352711250887836367396465243767319244482010112*x^2 - 387525510081628637297743170535017129158275865687798305019045138474571366001139953640250564464680282754821621623700592281054857750134784, 1)
 

Normalized defining polynomial

\( x^{22} - 13804431167056 x^{20} + 29455972774636519059028044 x^{18} + 282033335825689492595606193310007218368 x^{16} - 1100067537115825097282039898823065852802526737569408 x^{14} - 711369580334732818426577644309882828779774928375610743409128704 x^{12} + 6224971581116867291176830299635009030608381084723679035467660012998285012992 x^{10} - 3747833231991359307927309897387471198885933309684020485859542287332808873765923846934528 x^{8} - 6274369569526916448536833950155270998368078365193513543624088680948481195760518524041106824138496768 x^{6} + 5343558891203926971053678141516280096820104179732811762443202895791628733302502206317811912655111486804391018496 x^{4} + 465315646285731804566448193725361044248288059478367061599274499469392698960388352711250887836367396465243767319244482010112 x^{2} - 387525510081628637297743170535017129158275865687798305019045138474571366001139953640250564464680282754821621623700592281054857750134784 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(26830213597708412349017401539020628303767583956076604194625199878276562431883058860650728011650318953900945540680338981382014874223474436016018325969989526051132498414439997937223230143956258052357750784=2^{72}\cdot 3^{20}\cdot 337^{8}\cdot 947^{10}\cdot 310501^{8}\cdot 53591959^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1{,}589{,}648{,}471.61$
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:  $2, 3, 337, 947, 310501, 53591959$
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 \(a\))

$1$, $a$, $\frac{1}{101503170346} a^{2}$, $\frac{1}{101503170346} a^{3}$, $\frac{1}{10302893590289093759716} a^{4}$, $\frac{1}{10302893590289093759716} a^{5}$, $\frac{1}{1045776363151825415278418740581736} a^{6}$, $\frac{1}{1045776363151825415278418740581736} a^{7}$, $\frac{1}{106149616332820092587857528442786732344400656} a^{8}$, $\frac{1}{212299232665640185175715056885573464688801312} a^{9} - \frac{1}{2} a$, $\frac{1}{10774522588792781688516794680706721808102407725242146976} a^{10}$, $\frac{1}{21549045177585563377033589361413443616204815450484293952} a^{11} - \frac{1}{203006340692} a^{3}$, $\frac{1}{1093648201727058630224709722557681063355051814348000008603696773696} a^{12}$, $\frac{1}{2187296403454117260449419445115362126710103628696000017207393547392} a^{13} - \frac{1}{20605787180578187519432} a^{5}$, $\frac{1}{111008759718498203541228135227174699784466242591421411797710499225795700018816} a^{14}$, $\frac{1}{222017519436996407082456270454349399568932485182842823595420998451591400037632} a^{15} - \frac{1}{2091552726303650830556837481163472} a^{7}$, $\frac{1}{11267741047604906161340259844011836960524086506443608039974822895708642056404195853230336} a^{16}$, $\frac{1}{22535482095209812322680519688023673921048173012887216079949645791417284112808391706460672} a^{17} - \frac{1}{2} a$, $\frac{1}{2287422877939314570799730709229273750088910460199171147048839251868590574071069663839171218765632512} a^{18} - \frac{1}{203006340692} a^{2}$, $\frac{1}{4574845755878629141599461418458547500177820920398342294097678503737181148142139327678342437531265024} a^{19} - \frac{1}{45070964190419624645361039376047347842096346025774432159899291582834568225616783412921344} a^{17} + \frac{1}{406012681384} a^{3} - \frac{1}{4} a$, $\frac{1}{6720158558262927413160213668794990174322676877986733256967749804990513757101368312102765410660463856264916826622537943011731423309317018551204027302957293049599333257216} a^{20} - \frac{2950442336895763005424408194123522436283571976756718491405}{66206390749722557568952858811575057463897009280383736251340495691185824521057549511181308719636352630865999715505457035941274433829367737443498465589861124096} a^{18} - \frac{2737194544524833140993105902792583037168669141457279278769}{163064834635314410460475556414338801897645431797948917858868824920717675198676768016734506381612997403710670584893774614251678956689342956937817344} a^{16} + \frac{448010863468600608733251996452712734757586789849524172045}{146045446909746257653360436280551615386509141122386197170059685186070824084957786897766273478511746385936843907094586937396483705841024} a^{14} + \frac{3640387740344206143639524182262773313786356565007975397991}{15827091021206877980750601362956050512203777233004065254821597777473415203965885945268827280966840454645687279231108325950784} a^{12} + \frac{3318259023159591003027624581862711037207471778757466352799}{77963530435828333830152271857571927983943767806044766547875398011429789729997947118844898512658130203649113511952} a^{10} + \frac{2101450659797321063465309600911427871138970121122844413023}{768089609123235550904595110128895677635938496738673744635702336857822087499252534123845804084813134312} a^{8} + \frac{877980835922178739125904347982375677210722872205491182553}{15134297904292092817634425657408335895406939247971728463979860187407078519376842127623247944} a^{6} + \frac{7624824584200078298315306400558358002958294580349176864833}{596406904442605904164333584741973312369505217541857151716905651868596404732709456} a^{4} - \frac{26237684833586530354514463222437439180316890934918410021673}{5875746564462938377788170616023778165994992787984746161860594894374536} a^{2} - \frac{3510112041698694215882322931925786995678446606960950735414}{14471830151792120008931436633118625422631665599858903351629}$, $\frac{1}{3069528295011712591835854305960549804970519078426512794895274049479171794605024947378740709211755682264252603802851475025656967214549881723138670280299890796894353277948484067725312} a^{21} - \frac{1959783172471210244868658607986596572226174110421941450506584227992915}{30240713512183173359220961509577455784452045950940299656354874122457311906956157404462444347972087353192125719801420743552791404891926583480418122863307818723196999657472} a^{19} + \frac{181810418681324941268807866897263957835720945695523101818345136370705}{49654793062291918176714644108681293097922756960287802188505371768389368390793162133385981539727264473149499786629092776955955825372025803082623849192395843072} a^{17} - \frac{35495023511405413583051665984741678597312227202048073327751141523951}{22236113813906510517337575874682563895133467972447579708027567034643319345274104729554705415674499645960545988849151083761592585003092221400611456} a^{15} - \frac{380189116389071757205463897234269962476614988951511273014116889990983}{2409749874010813251280447198629101653877400828519372253305984805570168597401803483813143512395443815367957924467060684467041981146376896} a^{13} - \frac{451281702145464245225294087685876128357879057940443700274227740901103}{47481273063620633942251804088868151536611331699012195764464793332420245611897657835806481842900521363937061837693324977852352} a^{11} + \frac{29871581481954186747169233787318785093973348457622682966632944966389}{38981765217914166915076135928785963991971883903022383273937699005714894864998973559422449256329065101824556755976} a^{9} - \frac{893756303849359279721646967174407347518597553880040497981430950208019}{2304268827369706652713785330386687032907815490216021233907107010573466262497757602371537412254439402936} a^{7} + \frac{1258890062488282723142009994797484351267800913175168971121456112651389}{30268595808584185635268851314816671790813878495943456927959720374814157038753684255246495888} a^{5} + \frac{11219686829171603718331468734585896435955541035190487646803534074316755}{2683831069991726568739501131338879905662773478938357182726075433408683821297192552} a^{3} - \frac{1810919264659928904655863004144458431084212393730128920030213716925851}{13220429770041611350023383886053500873488733772965678864186338512342706} 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:  $17$
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 $16{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ $16{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/37.10.0.1}{10} }{,}\,{\href{/LocalNumberField/37.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/59.10.0.1}{10} }{,}\,{\href{/LocalNumberField/59.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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
337Data not computed
947Data not computed
310501Data not computed
53591959Data not computed