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Properties

Label 22.22.2437540593...0928.1
Degree $22$
Signature $[22, 0]$
Discriminant $2^{54}\cdot 3^{23}\cdot 337^{8}\cdot 310501^{8}$
Root discriminant $14{,}254.51$
Ramified primes $2, 3, 337, 310501$
Class number Not computed
Class group Not computed
Galois group 22T27

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-9655790329507504656039075, 0, 20751898885985795263838019, 0, -2110663170149138275578021, 0, 43869578581905971661981, 0, -411884307283617185094, 0, 2145714219228954438, 0, -6763660792069842, 0, 13399040743938, 0, -16760085111, 0, 12830103, 0, -5481, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - 5481*x^20 + 12830103*x^18 - 16760085111*x^16 + 13399040743938*x^14 - 6763660792069842*x^12 + 2145714219228954438*x^10 - 411884307283617185094*x^8 + 43869578581905971661981*x^6 - 2110663170149138275578021*x^4 + 20751898885985795263838019*x^2 - 9655790329507504656039075)
 
gp: K = bnfinit(x^22 - 5481*x^20 + 12830103*x^18 - 16760085111*x^16 + 13399040743938*x^14 - 6763660792069842*x^12 + 2145714219228954438*x^10 - 411884307283617185094*x^8 + 43869578581905971661981*x^6 - 2110663170149138275578021*x^4 + 20751898885985795263838019*x^2 - 9655790329507504656039075, 1)
 

Normalized defining polynomial

\( x^{22} - 5481 x^{20} + 12830103 x^{18} - 16760085111 x^{16} + 13399040743938 x^{14} - 6763660792069842 x^{12} + 2145714219228954438 x^{10} - 411884307283617185094 x^{8} + 43869578581905971661981 x^{6} - 2110663170149138275578021 x^{4} + 20751898885985795263838019 x^{2} - 9655790329507504656039075 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[22, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(24375405939112817739172491615990528404119519459053029993296057131489380667213319381532540928=2^{54}\cdot 3^{23}\cdot 337^{8}\cdot 310501^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14{,}254.51$
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:  $2, 3, 337, 310501$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{5} - \frac{1}{2} a$, $\frac{1}{72} a^{6} + \frac{1}{24} a^{4} - \frac{3}{8} a^{2} + \frac{3}{8}$, $\frac{1}{72} a^{7} + \frac{1}{24} a^{5} - \frac{1}{24} a^{3} + \frac{3}{8} a$, $\frac{1}{216} a^{8} - \frac{1}{2} a^{2} + \frac{1}{8}$, $\frac{1}{432} a^{9} - \frac{1}{432} a^{8} - \frac{1}{12} a^{5} - \frac{1}{12} a^{4} + \frac{1}{12} a^{3} + \frac{1}{4} a^{2} + \frac{5}{16} a + \frac{3}{16}$, $\frac{1}{864} a^{10} + \frac{1}{864} a^{8} - \frac{1}{144} a^{6} + \frac{1}{48} a^{4} + \frac{3}{32} a^{2} + \frac{11}{32}$, $\frac{1}{2592} a^{11} - \frac{1}{864} a^{9} - \frac{1}{144} a^{7} - \frac{1}{16} a^{5} - \frac{1}{96} a^{3} - \frac{7}{32} a$, $\frac{1}{8843904} a^{12} - \frac{487}{1473984} a^{10} - \frac{3959}{2947968} a^{8} - \frac{911}{245664} a^{6} + \frac{4859}{109184} a^{4} - \frac{1}{6} a^{3} - \frac{3545}{54592} a^{2} - \frac{1}{2} a - \frac{17647}{109184}$, $\frac{1}{26531712} a^{13} - \frac{487}{4421952} a^{11} + \frac{955}{2947968} a^{9} - \frac{1}{432} a^{8} - \frac{1441}{245664} a^{7} - \frac{8789}{327552} a^{5} - \frac{1}{12} a^{4} + \frac{1093}{54592} a^{3} - \frac{1}{4} a^{2} - \frac{8157}{109184} a - \frac{5}{16}$, $\frac{1}{53063424} a^{14} - \frac{1}{17687808} a^{12} + \frac{1789}{5895936} a^{10} + \frac{737}{655104} a^{8} - \frac{9851}{1965312} a^{6} - \frac{1}{12} a^{5} - \frac{44885}{655104} a^{4} - \frac{1}{6} a^{3} - \frac{48607}{218368} a^{2} + \frac{1}{4} a - \frac{14995}{218368}$, $\frac{1}{53063424} a^{15} - \frac{1}{53063424} a^{13} - \frac{1135}{5895936} a^{11} + \frac{191}{655104} a^{9} - \frac{859}{218368} a^{7} + \frac{33073}{655104} a^{5} - \frac{1}{12} a^{4} + \frac{37895}{655104} a^{3} - \frac{1}{2} a^{2} - \frac{51781}{218368} a + \frac{1}{4}$, $\frac{1}{5415334672896} a^{16} + \frac{293}{75212981568} a^{14} - \frac{14977}{300851926272} a^{12} - \frac{7856123}{16713995904} a^{10} + \frac{4462357}{2089249488} a^{8} - \frac{15122071}{4178498976} a^{6} - \frac{849938665}{11142663936} a^{4} - \frac{1}{6} a^{3} - \frac{372676663}{1857110656} a^{2} - \frac{1}{2} a - \frac{1163353489}{7428442624}$, $\frac{1}{5415334672896} a^{17} + \frac{293}{75212981568} a^{15} - \frac{10913}{902555778816} a^{13} + \frac{3199727}{16713995904} a^{11} + \frac{7270747}{50141987712} a^{9} - \frac{1}{432} a^{8} + \frac{4600667}{1044624744} a^{7} + \frac{81303375}{3714221312} a^{5} - \frac{1}{12} a^{4} + \frac{270278609}{5571331968} a^{3} + \frac{1}{4} a^{2} - \frac{3111156133}{7428442624} a + \frac{3}{16}$, $\frac{1}{32492008037376} a^{18} - \frac{1}{10830669345792} a^{16} - \frac{3973}{601703852544} a^{14} - \frac{2765}{200567950848} a^{12} + \frac{1377485}{2785665984} a^{10} + \frac{81074203}{100283975424} a^{8} - \frac{121112395}{66855983616} a^{6} + \frac{269533501}{22285327872} a^{4} - \frac{1}{6} a^{3} + \frac{504618461}{14856885248} a^{2} - \frac{1}{2} a + \frac{429067509}{14856885248}$, $\frac{1}{25408750285228032} a^{19} - \frac{1}{64984016074752} a^{18} - \frac{167}{2823194476136448} a^{17} + \frac{1}{21661338691584} a^{16} - \frac{87367}{17427126395904} a^{15} + \frac{3973}{1203407705088} a^{14} - \frac{16680085}{1411597238068224} a^{13} - \frac{59741}{1203407705088} a^{12} - \frac{1397136227}{9802758597696} a^{11} + \frac{2075281}{4178498976} a^{10} + \frac{2874000523}{26140689593856} a^{9} + \frac{169672475}{200567950848} a^{8} + \frac{45943947821}{52281379187712} a^{7} - \frac{95242085}{133711967232} a^{6} - \frac{1449513592835}{17427126395904} a^{5} - \frac{265672203}{14856885248} a^{4} - \frac{318956489129}{2050250164224} a^{3} + \frac{1852965011}{29713770496} a^{2} - \frac{1120932831615}{11618084263936} a - \frac{7777635869}{29713770496}$, $\frac{1}{2289080061020400131049369650529139436180077853106595020190476702554112} a^{20} - \frac{1885091781733974440774562454718347719550598075356037341}{381513343503400021841561608421523239363346308851099170031746117092352} a^{18} - \frac{12406693279498483214042690278226741884645933594814720717}{254342229002266681227707738947682159575564205900732780021164078061568} a^{16} + \frac{40504831731218378971803655824394862059331532712873577827949}{10597592875094445051154489122820089982315175245863865834215169919232} a^{14} - \frac{704768827404508721833862677818802334914639399669020712670557}{14130123833459260068205985497093453309753566994485154445620226558976} a^{12} + \frac{165818078145350501462343249913982438221183306778234469767984965}{785006879636625559344776972060747406097420388582508580312234808832} a^{10} - \frac{380566638648211428865503585332392773127521721675860278912128285}{523337919757750372896517981373831604064946925721672386874823205888} a^{8} - \frac{1}{144} a^{7} - \frac{50786121607133346632610134872107230308861315175824317998949943}{43611493313145864408043165114485967005412243810139365572901933824} a^{6} + \frac{1}{16} a^{5} - \frac{1171533678659911905559700572725169867344683144915099913992525527}{20523055676774524427314430642111043296664585322418524975483262976} a^{4} + \frac{1}{48} a^{3} - \frac{66197938856956720387357500002721668848798959038903832306095141129}{523337919757750372896517981373831604064946925721672386874823205888} a^{2} - \frac{7}{16} a - \frac{26027555401623499657130016123045752174294276977078644278197435}{52488633444436123855024119289286555745945231003627941113767936}$, $\frac{1}{129667904415179586841461252787920538597990363190042847959240699483507481815040} a^{21} - \frac{51686047269452479511953978179577055805479193010954131446101}{21611317402529931140243542131320089766331727198340474659873449913917913635840} a^{19} + \frac{1154327837221745183399074920861961724178086588957012253491722147}{14407544935019954093495694754213393177554484798893649773248966609278609090560} a^{17} - \frac{724869969868039692853008125027233862959637930370110025092653773413}{300157186146249043614493640712779024532385099976951036942686804359971022720} a^{15} + \frac{4353162219011452931588718302828554779699950716300146123000079432199}{800419163056664116305316375234077398753026933271869431847164811626589393920} a^{13} - \frac{7597319823275627206766659464636491447801578609500067954778907401880077}{44467731280925784239184243068559855486279274070659412880398045090366077440} a^{11} + \frac{9187107171706963002747364876091279406700128663738512291733736225811327}{29645154187283856159456162045706570324186182713772941920265363393577384960} a^{9} - \frac{1274797174668101937242898537198006637473530601831763947722749954020389}{1235214757803494006644006751904440430174424279740539246677723474732391040} a^{7} - \frac{1}{144} a^{6} + \frac{19339511462847069147688751703106774489782118034803836249560717074345313}{1162555066167994359194359295910061581340634616226389879226092682101073920} a^{5} - \frac{1}{48} a^{4} + \frac{3803412868399888263783937501921838318003512621760028774045128000900835451}{29645154187283856159456162045706570324186182713772941920265363393577384960} a^{3} + \frac{3}{16} a^{2} - \frac{27290279718445799128926379581115531787241712515240049227263208150773669}{68385592127529079952609370347650681255331448013317051719181922476533760} a - \frac{3}{16}$

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:  $21$
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

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

\(\Q(\sqrt{3}) \), 11.11.118769262421915560193703211428553337469927424.1

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 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 ${\href{/LocalNumberField/5.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $22$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/13.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/37.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ $22$ ${\href{/LocalNumberField/59.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.4.1$x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
2.8.24.66$x^{8} + 20 x^{4} + 52$$8$$1$$24$$QD_{16}$$[2, 3, 4]^{2}$
2.8.24.66$x^{8} + 20 x^{4} + 52$$8$$1$$24$$QD_{16}$$[2, 3, 4]^{2}$
3Data not computed
337Data not computed
310501Data not computed