Properties

Label 22.2.289863980721044983135295513.1
Degree 22
Signature $[2, 10]$
Discriminant $359753\cdot 28385393161^{2}$
Ramified primes $359753, 28385393161$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T53

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, -15, 15, 53, -41, -101, 75, 100, -81, -45, 32, 16, 11, -18, -4, 5, 1, 3, -3, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 + x^20 - 3*x^19 + 3*x^18 + x^17 + 5*x^16 - 4*x^15 - 18*x^14 + 11*x^13 + 16*x^12 + 32*x^11 - 45*x^10 - 81*x^9 + 100*x^8 + 75*x^7 - 101*x^6 - 41*x^5 + 53*x^4 + 15*x^3 - 15*x^2 - 4*x + 1)
gp: K = bnfinit(x^22 - x^21 + x^20 - 3*x^19 + 3*x^18 + x^17 + 5*x^16 - 4*x^15 - 18*x^14 + 11*x^13 + 16*x^12 + 32*x^11 - 45*x^10 - 81*x^9 + 100*x^8 + 75*x^7 - 101*x^6 - 41*x^5 + 53*x^4 + 15*x^3 - 15*x^2 - 4*x + 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut -\mathstrut x^{21} \) \(\mathstrut +\mathstrut x^{20} \) \(\mathstrut -\mathstrut 3 x^{19} \) \(\mathstrut +\mathstrut 3 x^{18} \) \(\mathstrut +\mathstrut x^{17} \) \(\mathstrut +\mathstrut 5 x^{16} \) \(\mathstrut -\mathstrut 4 x^{15} \) \(\mathstrut -\mathstrut 18 x^{14} \) \(\mathstrut +\mathstrut 11 x^{13} \) \(\mathstrut +\mathstrut 16 x^{12} \) \(\mathstrut +\mathstrut 32 x^{11} \) \(\mathstrut -\mathstrut 45 x^{10} \) \(\mathstrut -\mathstrut 81 x^{9} \) \(\mathstrut +\mathstrut 100 x^{8} \) \(\mathstrut +\mathstrut 75 x^{7} \) \(\mathstrut -\mathstrut 101 x^{6} \) \(\mathstrut -\mathstrut 41 x^{5} \) \(\mathstrut +\mathstrut 53 x^{4} \) \(\mathstrut +\mathstrut 15 x^{3} \) \(\mathstrut -\mathstrut 15 x^{2} \) \(\mathstrut -\mathstrut 4 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 10]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(289863980721044983135295513=359753\cdot 28385393161^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $359753, 28385393161$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{1905641144321} a^{21} - \frac{174163645302}{1905641144321} a^{20} + \frac{200880859921}{1905641144321} a^{19} - \frac{625037984800}{1905641144321} a^{18} - \frac{282703934710}{1905641144321} a^{17} + \frac{455664307019}{1905641144321} a^{16} + \frac{618485916718}{1905641144321} a^{15} - \frac{793283426287}{1905641144321} a^{14} - \frac{200498667426}{1905641144321} a^{13} + \frac{37721721529}{1905641144321} a^{12} + \frac{805073550203}{1905641144321} a^{11} + \frac{139691662963}{1905641144321} a^{10} + \frac{327071997433}{1905641144321} a^{9} - \frac{484157436258}{1905641144321} a^{8} - \frac{509817804171}{1905641144321} a^{7} + \frac{558788108214}{1905641144321} a^{6} - \frac{363903847855}{1905641144321} a^{5} + \frac{714674888219}{1905641144321} a^{4} - \frac{885861087370}{1905641144321} a^{3} - \frac{441615939835}{1905641144321} a^{2} + \frac{636477542155}{1905641144321} a + \frac{888963507629}{1905641144321}$

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

Class group and class number

Trivial Abelian group, 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:  $11$
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:  \( \frac{3768728854142}{1905641144321} a^{21} - \frac{6167589910592}{1905641144321} a^{20} + \frac{7519723568573}{1905641144321} a^{19} - \frac{15853623460267}{1905641144321} a^{18} + \frac{20764568361318}{1905641144321} a^{17} - \frac{8342847494564}{1905641144321} a^{16} + \frac{22528363768782}{1905641144321} a^{15} - \frac{27425551748298}{1905641144321} a^{14} - \frac{53682052462685}{1905641144321} a^{13} + \frac{78719359542282}{1905641144321} a^{12} + \frac{8411459498969}{1905641144321} a^{11} + \frac{116536800016243}{1905641144321} a^{10} - \frac{243581091462724}{1905641144321} a^{9} - \frac{156189312652512}{1905641144321} a^{8} + \frac{483988318079050}{1905641144321} a^{7} - \frac{28995558804042}{1905641144321} a^{6} - \frac{365367224828456}{1905641144321} a^{5} + \frac{84292659131978}{1905641144321} a^{4} + \frac{145118905435797}{1905641144321} a^{3} - \frac{40150306614134}{1905641144321} a^{2} - \frac{29162737047133}{1905641144321} a + \frac{7551444814109}{1905641144321} \),  \( \frac{406625973316}{1905641144321} a^{21} - \frac{1639035342013}{1905641144321} a^{20} + \frac{3372365619074}{1905641144321} a^{19} - \frac{6638292849980}{1905641144321} a^{18} + \frac{11387600589727}{1905641144321} a^{17} - \frac{15786576426026}{1905641144321} a^{16} + \frac{20534792714157}{1905641144321} a^{15} - \frac{27416374215739}{1905641144321} a^{14} + \frac{25059381252713}{1905641144321} a^{13} - \frac{11255157773246}{1905641144321} a^{12} - \frac{99457537426}{1905641144321} a^{11} + \frac{18378451547820}{1905641144321} a^{10} - \frac{61703637503141}{1905641144321} a^{9} + \frac{76077166843427}{1905641144321} a^{8} - \frac{17862540717404}{1905641144321} a^{7} - \frac{53177234679716}{1905641144321} a^{6} + \frac{53896994246448}{1905641144321} a^{5} - \frac{5216908155023}{1905641144321} a^{4} - \frac{26500061837285}{1905641144321} a^{3} + \frac{12422527740978}{1905641144321} a^{2} + \frac{6355370944415}{1905641144321} a - \frac{4329194007635}{1905641144321} \),  \( \frac{6284591426259}{1905641144321} a^{21} - \frac{11164413916710}{1905641144321} a^{20} + \frac{15131418430283}{1905641144321} a^{19} - \frac{31260652963459}{1905641144321} a^{18} + \frac{43820563416380}{1905641144321} a^{17} - \frac{29300497204341}{1905641144321} a^{16} + \frac{56672121927949}{1905641144321} a^{15} - \frac{71411078729053}{1905641144321} a^{14} - \frac{55200948864055}{1905641144321} a^{13} + \frac{106784658211370}{1905641144321} a^{12} + \frac{17556930652995}{1905641144321} a^{11} + \frac{193530156420163}{1905641144321} a^{10} - \frac{433522799321522}{1905641144321} a^{9} - \frac{168342906579420}{1905641144321} a^{8} + \frac{733785619199345}{1905641144321} a^{7} - \frac{93207241106888}{1905641144321} a^{6} - \frac{526335478335158}{1905641144321} a^{5} + \frac{132121161584644}{1905641144321} a^{4} + \frac{207175755463998}{1905641144321} a^{3} - \frac{54884744322167}{1905641144321} a^{2} - \frac{42990108545067}{1905641144321} a + \frac{6006552618379}{1905641144321} \),  \( \frac{5891020371580}{1905641144321} a^{21} - \frac{10825794494890}{1905641144321} a^{20} + \frac{14846969712485}{1905641144321} a^{19} - \frac{29400556230104}{1905641144321} a^{18} + \frac{41591726554795}{1905641144321} a^{17} - \frac{27483369870222}{1905641144321} a^{16} + \frac{49849032895764}{1905641144321} a^{15} - \frac{62673799259405}{1905641144321} a^{14} - \frac{55404879890189}{1905641144321} a^{13} + \frac{116896237052082}{1905641144321} a^{12} - \frac{4970905377666}{1905641144321} a^{11} + \frac{184854050894373}{1905641144321} a^{10} - \frac{417769972685268}{1905641144321} a^{9} - \frac{127479977275503}{1905641144321} a^{8} + \frac{723942153091989}{1905641144321} a^{7} - \frac{179482537630422}{1905641144321} a^{6} - \frac{484915096609374}{1905641144321} a^{5} + \frac{195533996809617}{1905641144321} a^{4} + \frac{178091181674148}{1905641144321} a^{3} - \frac{78577543144891}{1905641144321} a^{2} - \frac{35995250209310}{1905641144321} a + \frac{11469328502926}{1905641144321} \),  \( \frac{9526465526330}{1905641144321} a^{21} - \frac{17546459795690}{1905641144321} a^{20} + \frac{24003905182041}{1905641144321} a^{19} - \frac{48686618355714}{1905641144321} a^{18} + \frac{68981182535746}{1905641144321} a^{17} - \frac{47569713524065}{1905641144321} a^{16} + \frac{86887127083780}{1905641144321} a^{15} - \frac{110528576331401}{1905641144321} a^{14} - \frac{81025658356265}{1905641144321} a^{13} + \frac{173082622263375}{1905641144321} a^{12} + \frac{9375539485445}{1905641144321} a^{11} + \frac{297882896553063}{1905641144321} a^{10} - \frac{678088344925179}{1905641144321} a^{9} - \frac{213331738620451}{1905641144321} a^{8} + \frac{1134500744133811}{1905641144321} a^{7} - \frac{229203896715090}{1905641144321} a^{6} - \frac{772502394331918}{1905641144321} a^{5} + \frac{257178079553635}{1905641144321} a^{4} + \frac{285082848094511}{1905641144321} a^{3} - \frac{99000283342910}{1905641144321} a^{2} - \frac{54904207061339}{1905641144321} a + \frac{11783776983131}{1905641144321} \),  \( \frac{3936676684659}{1905641144321} a^{21} - \frac{7262891328540}{1905641144321} a^{20} + \frac{9942012189116}{1905641144321} a^{19} - \frac{19997709028491}{1905641144321} a^{18} + \frac{28360363508386}{1905641144321} a^{17} - \frac{19214324946537}{1905641144321} a^{16} + \frac{34909115694617}{1905641144321} a^{15} - \frac{44210962511161}{1905641144321} a^{14} - \frac{35273417612984}{1905641144321} a^{13} + \frac{75243980284280}{1905641144321} a^{12} + \frac{184101158424}{1905641144321} a^{11} + \frac{125663422689401}{1905641144321} a^{10} - \frac{283398496497223}{1905641144321} a^{9} - \frac{85003096390380}{1905641144321} a^{8} + \frac{474052120025688}{1905641144321} a^{7} - \frac{101915066650664}{1905641144321} a^{6} - \frac{317408811047291}{1905641144321} a^{5} + \frac{107062390409884}{1905641144321} a^{4} + \frac{118473443937935}{1905641144321} a^{3} - \frac{39167871776301}{1905641144321} a^{2} - \frac{24025656045513}{1905641144321} a + \frac{5204348086216}{1905641144321} \),  \( \frac{984427042237}{1905641144321} a^{21} - \frac{1274051499886}{1905641144321} a^{20} + \frac{1377633092563}{1905641144321} a^{19} - \frac{3261388781916}{1905641144321} a^{18} + \frac{3642926272391}{1905641144321} a^{17} + \frac{541245158866}{1905641144321} a^{16} + \frac{3435781425506}{1905641144321} a^{15} - \frac{2679379055668}{1905641144321} a^{14} - \frac{20133772613411}{1905641144321} a^{13} + \frac{21513761957244}{1905641144321} a^{12} + \frac{3183086950823}{1905641144321} a^{11} + \frac{36684070726735}{1905641144321} a^{10} - \frac{59672605070899}{1905641144321} a^{9} - \frac{58898936243864}{1905641144321} a^{8} + \frac{117327376118215}{1905641144321} a^{7} + \frac{29841647140782}{1905641144321} a^{6} - \frac{95187189449181}{1905641144321} a^{5} - \frac{16416214333067}{1905641144321} a^{4} + \frac{50596320012521}{1905641144321} a^{3} + \frac{4736436928456}{1905641144321} a^{2} - \frac{14555175317569}{1905641144321} a - \frac{1494090380620}{1905641144321} \),  \( \frac{7766446366187}{1905641144321} a^{21} - \frac{12414489410708}{1905641144321} a^{20} + \frac{15973453255490}{1905641144321} a^{19} - \frac{33572570958261}{1905641144321} a^{18} + \frac{44065453703040}{1905641144321} a^{17} - \frac{20873117508260}{1905641144321} a^{16} + \frac{53177976149911}{1905641144321} a^{15} - \frac{61454346359489}{1905641144321} a^{14} - \frac{99518425399502}{1905641144321} a^{13} + \frac{143470909864584}{1905641144321} a^{12} + \frac{22364036802146}{1905641144321} a^{11} + \frac{243280318807470}{1905641144321} a^{10} - \frac{484332929906417}{1905641144321} a^{9} - \frac{310199063499038}{1905641144321} a^{8} + \frac{931909533473963}{1905641144321} a^{7} - \frac{43415449537389}{1905641144321} a^{6} - \frac{689110842623022}{1905641144321} a^{5} + \frac{150764708120052}{1905641144321} a^{4} + \frac{260735712524480}{1905641144321} a^{3} - \frac{66458238366602}{1905641144321} a^{2} - \frac{52237029824272}{1905641144321} a + \frac{6793646422111}{1905641144321} \),  \( \frac{6579428896915}{1905641144321} a^{21} - \frac{12169217738586}{1905641144321} a^{20} + \frac{16862997364065}{1905641144321} a^{19} - \frac{33800179683670}{1905641144321} a^{18} + \frac{48427196017968}{1905641144321} a^{17} - \frac{34041390130445}{1905641144321} a^{16} + \frac{61252533062103}{1905641144321} a^{15} - \frac{78295726976823}{1905641144321} a^{14} - \frac{52112106324230}{1905641144321} a^{13} + \frac{118125958609346}{1905641144321} a^{12} + \frac{7432220371545}{1905641144321} a^{11} + \frac{201350286374259}{1905641144321} a^{10} - \frac{468293774224837}{1905641144321} a^{9} - \frac{138243892222159}{1905641144321} a^{8} + \frac{786271531921571}{1905641144321} a^{7} - \frac{166991456839498}{1905641144321} a^{6} - \frac{537233488523989}{1905641144321} a^{5} + \frac{185336998511112}{1905641144321} a^{4} + \frac{198594042392744}{1905641144321} a^{3} - \frac{67917970702851}{1905641144321} a^{2} - \frac{38859021887116}{1905641144321} a + \frac{6466476572487}{1905641144321} \),  \( \frac{9135135203093}{1905641144321} a^{21} - \frac{15830166387517}{1905641144321} a^{20} + \frac{21787241316325}{1905641144321} a^{19} - \frac{44422740002040}{1905641144321} a^{18} + \frac{61932812366723}{1905641144321} a^{17} - \frac{40415638549028}{1905641144321} a^{16} + \frac{79863384234985}{1905641144321} a^{15} - \frac{97820105192479}{1905641144321} a^{14} - \frac{83687189164091}{1905641144321} a^{13} + \frac{156011704957735}{1905641144321} a^{12} + \frac{20797950699008}{1905641144321} a^{11} + \frac{283286113259978}{1905641144321} a^{10} - \frac{614081899842696}{1905641144321} a^{9} - \frac{250894837660172}{1905641144321} a^{8} + \frac{1058103356642796}{1905641144321} a^{7} - \frac{138545049894604}{1905641144321} a^{6} - \frac{753784430649425}{1905641144321} a^{5} + \frac{200935950358378}{1905641144321} a^{4} + \frac{291215636398257}{1905641144321} a^{3} - \frac{83325330630170}{1905641144321} a^{2} - \frac{61212365991980}{1905641144321} a + \frac{8093815508820}{1905641144321} \),  \( \frac{17792269672330}{1905641144321} a^{21} - \frac{29142162458155}{1905641144321} a^{20} + \frac{36219174303942}{1905641144321} a^{19} - \frac{75365607276549}{1905641144321} a^{18} + \frac{99944138452140}{1905641144321} a^{17} - \frac{42875910306232}{1905641144321} a^{16} + \frac{110820360579157}{1905641144321} a^{15} - \frac{135145351315999}{1905641144321} a^{14} - \frac{241087876788255}{1905641144321} a^{13} + \frac{362259065530917}{1905641144321} a^{12} + \frac{45092380847369}{1905641144321} a^{11} + \frac{536158914675151}{1905641144321} a^{10} - \frac{1140158233467242}{1905641144321} a^{9} - \frac{717772160459828}{1905641144321} a^{8} + \frac{2279364486074958}{1905641144321} a^{7} - \frac{156681348498175}{1905641144321} a^{6} - \frac{1727800163206219}{1905641144321} a^{5} + \frac{420528365857497}{1905641144321} a^{4} + \frac{682913474934971}{1905641144321} a^{3} - \frac{189110116179019}{1905641144321} a^{2} - \frac{146152594630065}{1905641144321} a + \frac{25905183581521}{1905641144321} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 15832.2441917 \) (assuming GRH)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

22T53:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A non-solvable group of order 81749606400
Conjugacy class representatives for 22T53
Character table for 22T53

Intermediate fields

Deg 11

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.5.0.1}{5} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.10.0.1}{10} }{,}\,{\href{/LocalNumberField/11.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/19.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ $16{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.7.0.1}{7} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ $16{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.11.0.1}{11} }^{2}$ $22$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.14.0.1}{14} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}$

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
359753Data not computed
28385393161Data not computed