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Properties

Label 22.0.12914016300...0000.1
Degree $22$
Signature $[0, 11]$
Discriminant $-\,2^{36}\cdot 3^{17}\cdot 5^{36}$
Root discriminant $101.17$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 22T27

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8100, 252450, 8853525, -29801250, 134263875, 63496890, 8234605, 25194460, 8937300, 334550, 2422871, 700482, -45851, 134890, 31155, -4172, 4906, 762, -145, 110, 9, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 2*x^21 + 9*x^20 + 110*x^19 - 145*x^18 + 762*x^17 + 4906*x^16 - 4172*x^15 + 31155*x^14 + 134890*x^13 - 45851*x^12 + 700482*x^11 + 2422871*x^10 + 334550*x^9 + 8937300*x^8 + 25194460*x^7 + 8234605*x^6 + 63496890*x^5 + 134263875*x^4 - 29801250*x^3 + 8853525*x^2 + 252450*x + 8100)
 
gp: K = bnfinit(x^22 - 2*x^21 + 9*x^20 + 110*x^19 - 145*x^18 + 762*x^17 + 4906*x^16 - 4172*x^15 + 31155*x^14 + 134890*x^13 - 45851*x^12 + 700482*x^11 + 2422871*x^10 + 334550*x^9 + 8937300*x^8 + 25194460*x^7 + 8234605*x^6 + 63496890*x^5 + 134263875*x^4 - 29801250*x^3 + 8853525*x^2 + 252450*x + 8100, 1)
 

Normalized defining polynomial

\( x^{22} - 2 x^{21} + 9 x^{20} + 110 x^{19} - 145 x^{18} + 762 x^{17} + 4906 x^{16} - 4172 x^{15} + 31155 x^{14} + 134890 x^{13} - 45851 x^{12} + 700482 x^{11} + 2422871 x^{10} + 334550 x^{9} + 8937300 x^{8} + 25194460 x^{7} + 8234605 x^{6} + 63496890 x^{5} + 134263875 x^{4} - 29801250 x^{3} + 8853525 x^{2} + 252450 x + 8100 \)

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:  \(-129140163000000000000000000000000000000000000=-\,2^{36}\cdot 3^{17}\cdot 5^{36}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $101.17$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{3} a^{4} - \frac{1}{2} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{15} - \frac{1}{6} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{18} a^{16} + \frac{1}{18} a^{15} + \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{6} a^{10} - \frac{1}{6} a^{9} + \frac{1}{6} a^{8} - \frac{1}{9} a^{7} + \frac{7}{18} a^{6} + \frac{1}{6} a^{5} - \frac{1}{9} a^{4} + \frac{1}{18} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{90} a^{17} - \frac{1}{45} a^{16} - \frac{1}{15} a^{15} - \frac{1}{18} a^{14} - \frac{1}{18} a^{13} - \frac{1}{30} a^{12} - \frac{1}{10} a^{11} - \frac{2}{15} a^{10} - \frac{1}{6} a^{9} + \frac{5}{18} a^{8} - \frac{41}{90} a^{7} + \frac{7}{15} a^{6} - \frac{22}{45} a^{5} - \frac{1}{18} a^{4} + \frac{1}{6} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{180} a^{18} + \frac{2}{45} a^{15} + \frac{1}{15} a^{13} + \frac{1}{18} a^{12} + \frac{7}{60} a^{10} - \frac{4}{9} a^{9} - \frac{1}{30} a^{8} - \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{4}{15} a^{5} - \frac{4}{9} a^{3} + \frac{5}{12} a^{2} + \frac{1}{3} a$, $\frac{1}{180} a^{19} - \frac{1}{90} a^{16} - \frac{1}{18} a^{15} + \frac{1}{15} a^{14} - \frac{1}{18} a^{12} + \frac{7}{60} a^{11} + \frac{1}{18} a^{10} - \frac{1}{5} a^{9} - \frac{1}{6} a^{8} - \frac{5}{18} a^{7} - \frac{29}{90} a^{6} - \frac{1}{2} a^{5} + \frac{1}{36} a^{3} + \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{28410702929304120} a^{20} + \frac{23569802922049}{28410702929304120} a^{19} - \frac{5237733982677}{3156744769922680} a^{18} - \frac{1112295389677}{14205351464652060} a^{17} + \frac{115597381356311}{4735117154884020} a^{16} - \frac{69119744251105}{2841070292930412} a^{15} - \frac{253778347815367}{4735117154884020} a^{14} + \frac{123331685857843}{3551337866163015} a^{13} + \frac{4107197422359857}{28410702929304120} a^{12} - \frac{1614496569174023}{28410702929304120} a^{11} + \frac{153435338520635}{5682140585860824} a^{10} - \frac{2207694387710809}{4735117154884020} a^{9} + \frac{1702512696873839}{14205351464652060} a^{8} - \frac{589223674017001}{4735117154884020} a^{7} + \frac{1041885452569391}{7102675732326030} a^{6} + \frac{1999946941719787}{4735117154884020} a^{5} - \frac{1247843124679769}{5682140585860824} a^{4} + \frac{561237311031733}{5682140585860824} a^{3} - \frac{61853650535661}{631348953984536} a^{2} + \frac{468236732138773}{947023430976804} a - \frac{12910845677065}{157837238496134}$, $\frac{1}{71630116220391919673553806146722067681538882352910932840} a^{21} + \frac{384640105752918989780285551493725852147}{71630116220391919673553806146722067681538882352910932840} a^{20} + \frac{6515249160712368143252579827801089130858132286340047}{23876705406797306557851268715574022560512960784303644280} a^{19} + \frac{11920028965366934315214189259204001855107635379151659}{8953764527548989959194225768340258460192360294113866605} a^{18} - \frac{174376926407948464542714456074995493723076594317303927}{35815058110195959836776903073361033840769441176455466420} a^{17} - \frac{21692401143530898642337963416921239447667648994338325}{2387670540679730655785126871557402256051296078430364428} a^{16} - \frac{799567897020571648157620702473234841801663679928669157}{35815058110195959836776903073361033840769441176455466420} a^{15} + \frac{754054916011793952939963632571220306916168107747562803}{17907529055097979918388451536680516920384720588227733210} a^{14} + \frac{117008877913782778336791469176998027397870856699185407}{2652967267421922950872363190619335840056995642700404920} a^{13} - \frac{5006061581945653650682018155080537783801188541479410893}{71630116220391919673553806146722067681538882352910932840} a^{12} - \frac{401592641537894482923040194342268818660525406459761859}{14326023244078383934710761229344413536307776470582186568} a^{11} + \frac{147125821087191059233798733695824204388059228889927447}{1989725450566442213154272392964501880042746732025303690} a^{10} + \frac{1212491948569225473749413948099889675503276217458439197}{35815058110195959836776903073361033840769441176455466420} a^{9} - \frac{6871744336017505597968651573152515295032339969129779701}{35815058110195959836776903073361033840769441176455466420} a^{8} - \frac{1488457101863896668934282014039584117247151504627058389}{2984588175849663319731408589446752820064120098037955535} a^{7} + \frac{14364516783063029632587464467385599312520762325736179041}{35815058110195959836776903073361033840769441176455466420} a^{6} + \frac{27021307830358351627887552941096316570865029269189414443}{71630116220391919673553806146722067681538882352910932840} a^{5} - \frac{1827208887721016406639456286156979803306034140588924159}{4775341081359461311570253743114804512102592156860728856} a^{4} - \frac{1582178485928763550194014793307874022252884002158576125}{4775341081359461311570253743114804512102592156860728856} a^{3} + \frac{90228753500715156568098633493250669911305705893102161}{198972545056644221315427239296450188004274673202530369} a^{2} + \frac{48038292979662080489030129337989717808270987721502329}{132648363371096147543618159530966792002849782135020246} a - \frac{16793281019230836899649901512217408981815797058950043}{66324181685548073771809079765483396001424891067510123}$

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:  $10$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{809128533497725797853767253303669927}{226911332531158032445824627290324721753630} a^{21} - \frac{164363632262745262136222049131766530}{22691133253115803244582462729032472175363} a^{20} + \frac{54311862496210223172685898812604333}{1680824685415985425524626868817220161138} a^{19} + \frac{8876143673778331480052885605908986048}{22691133253115803244582462729032472175363} a^{18} - \frac{24009784941088248620349124694929202591}{45382266506231606489164925458064944350726} a^{17} + \frac{103299372255741751907458473438951395239}{37818555421859672074304104548387453625605} a^{16} + \frac{1974796497417083731046334320950359507581}{113455666265579016222912313645162360876815} a^{15} - \frac{349673448420161006191081693631355403448}{22691133253115803244582462729032472175363} a^{14} + \frac{1685799896304804064208970251339897284915}{15127422168743868829721641819354981450242} a^{13} + \frac{10834208500520269808214636586905497457346}{22691133253115803244582462729032472175363} a^{12} - \frac{40458134752903593872027630523955560101417}{226911332531158032445824627290324721753630} a^{11} + \frac{94518391652561790206071418593286847584834}{37818555421859672074304104548387453625605} a^{10} + \frac{388497209411678410501806995914057708403425}{45382266506231606489164925458064944350726} a^{9} + \frac{20958669735249481680065831274005141473508}{22691133253115803244582462729032472175363} a^{8} + \frac{240265707028358304486517213014129393832180}{7563711084371934414860820909677490725121} a^{7} + \frac{2015498963777528298612523363070018608837054}{22691133253115803244582462729032472175363} a^{6} + \frac{6017432972569181276962305578588702526557203}{226911332531158032445824627290324721753630} a^{5} + \frac{188948729990327565299130856967350544124200}{840412342707992712762313434408610080569} a^{4} + \frac{7131920249697432355797411314974336470059975}{15127422168743868829721641819354981450242} a^{3} - \frac{306623344884797051018962622799706389702590}{2521237028123978138286940303225830241707} a^{2} + \frac{156902113473377473580422136860893966314555}{5042474056247956276573880606451660483414} a + \frac{745662347061070087663202069540941371845}{840412342707992712762313434408610080569} \) (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:  \( 174561035136000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

22T27:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 15840
The 20 conjugacy class representatives for t22n27
Character table for t22n27

Intermediate fields

\(\Q(\sqrt{-3}) \), 11.3.6561000000000000000000.1

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 24 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 R ${\href{/LocalNumberField/7.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/13.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/23.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/37.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/41.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/59.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.8.16.9$x^{8} + 2 x^{4} + 8 x + 12$$4$$2$$16$$S_4$$[8/3, 8/3]_{3}^{2}$
2.12.20.34$x^{12} + 14 x^{10} + 16 x^{8} - 8 x^{6} - 8 x^{4} + 16 x^{2} + 16$$6$$2$$20$$S_4$$[8/3, 8/3]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
3.8.7.1$x^{8} + 3$$8$$1$$7$$QD_{16}$$[\ ]_{8}^{2}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.10.18.2$x^{10} + 10 x^{8} + 40 x^{6} + 60 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$
5.10.18.2$x^{10} + 10 x^{8} + 40 x^{6} + 60 x^{5} + 80 x^{4} - 75 x^{3} + 80 x^{2} + 75 x + 57$$5$$2$$18$$F_{5}\times C_2$$[9/4]_{4}^{2}$