Properties

Label 22.0.3273714539904703691187619.1
Degree 22
Signature $[0, 11]$
Discriminant $-\,61\cdot 1279\cdot 1609^{2}\cdot 4025911^{2}$
Ramified primes $61, 1279, 1609, 4025911$
Class number 1 (GRH)
Class group Trivial (GRH)
Galois Group 22T53

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -3, -17, -13, 24, 57, 32, -27, -49, 10, 92, 68, -4, -35, -26, 17, 7, 8, -7, 0, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1)
gp: K = bnfinit(x^22 - x^21 - 7*x^19 + 8*x^18 + 7*x^17 + 17*x^16 - 26*x^15 - 35*x^14 - 4*x^13 + 68*x^12 + 92*x^11 + 10*x^10 - 49*x^9 - 27*x^8 + 32*x^7 + 57*x^6 + 24*x^5 - 13*x^4 - 17*x^3 - 3*x^2 + 2*x + 1, 1)

Normalized defining polynomial

\(x^{22} \) \(\mathstrut -\mathstrut x^{21} \) \(\mathstrut -\mathstrut 7 x^{19} \) \(\mathstrut +\mathstrut 8 x^{18} \) \(\mathstrut +\mathstrut 7 x^{17} \) \(\mathstrut +\mathstrut 17 x^{16} \) \(\mathstrut -\mathstrut 26 x^{15} \) \(\mathstrut -\mathstrut 35 x^{14} \) \(\mathstrut -\mathstrut 4 x^{13} \) \(\mathstrut +\mathstrut 68 x^{12} \) \(\mathstrut +\mathstrut 92 x^{11} \) \(\mathstrut +\mathstrut 10 x^{10} \) \(\mathstrut -\mathstrut 49 x^{9} \) \(\mathstrut -\mathstrut 27 x^{8} \) \(\mathstrut +\mathstrut 32 x^{7} \) \(\mathstrut +\mathstrut 57 x^{6} \) \(\mathstrut +\mathstrut 24 x^{5} \) \(\mathstrut -\mathstrut 13 x^{4} \) \(\mathstrut -\mathstrut 17 x^{3} \) \(\mathstrut -\mathstrut 3 x^{2} \) \(\mathstrut +\mathstrut 2 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $22$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[0, 11]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-3273714539904703691187619=-\,61\cdot 1279\cdot 1609^{2}\cdot 4025911^{2}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $61, 1279, 1609, 4025911$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{11521304158449797} a^{21} - \frac{739608894551581}{11521304158449797} a^{20} - \frac{469895265641984}{11521304158449797} a^{19} + \frac{435265577257926}{11521304158449797} a^{18} - \frac{5710698697039737}{11521304158449797} a^{17} - \frac{168388016981139}{11521304158449797} a^{16} - \frac{1127221923624567}{11521304158449797} a^{15} + \frac{3033538536688846}{11521304158449797} a^{14} - \frac{2404752630075298}{11521304158449797} a^{13} - \frac{4108420174725029}{11521304158449797} a^{12} + \frac{215875868487319}{11521304158449797} a^{11} - \frac{3442447303482306}{11521304158449797} a^{10} + \frac{2519039132613763}{11521304158449797} a^{9} + \frac{2387586649481750}{11521304158449797} a^{8} + \frac{1815069984512517}{11521304158449797} a^{7} + \frac{848460951775526}{11521304158449797} a^{6} - \frac{2370387454248410}{11521304158449797} a^{5} - \frac{3135308891305600}{11521304158449797} a^{4} + \frac{5537270364796856}{11521304158449797} a^{3} + \frac{3619053001549274}{11521304158449797} a^{2} + \frac{1564141785378248}{11521304158449797} a - \frac{1076049322310070}{11521304158449797}$

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:  $10$
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:  \( \frac{4619541048721315}{11521304158449797} a^{21} - \frac{9677519636344999}{11521304158449797} a^{20} + \frac{5686387178429833}{11521304158449797} a^{19} - \frac{33891011683249298}{11521304158449797} a^{18} + \frac{73376834734817349}{11521304158449797} a^{17} - \frac{12477210118500305}{11521304158449797} a^{16} + \frac{53791349462704647}{11521304158449797} a^{15} - \frac{208369035729368442}{11521304158449797} a^{14} - \frac{27097027233046322}{11521304158449797} a^{13} + \frac{133333046992129340}{11521304158449797} a^{12} + \frac{328523940301592571}{11521304158449797} a^{11} + \frac{114125336509566031}{11521304158449797} a^{10} - \frac{390806048597857052}{11521304158449797} a^{9} - \frac{257109201603790417}{11521304158449797} a^{8} + \frac{47615687380798940}{11521304158449797} a^{7} + \frac{257635099726074094}{11521304158449797} a^{6} + \frac{96697895405238293}{11521304158449797} a^{5} - \frac{150741517844718994}{11521304158449797} a^{4} - \frac{159649267449191560}{11521304158449797} a^{3} - \frac{31387086069372839}{11521304158449797} a^{2} + \frac{49128373990480054}{11521304158449797} a + \frac{13283797833514413}{11521304158449797} \),  \( \frac{2490909900337597}{11521304158449797} a^{21} - \frac{14422555111507104}{11521304158449797} a^{20} + \frac{14940205665338105}{11521304158449797} a^{19} - \frac{22221876257386859}{11521304158449797} a^{18} + \frac{103794086120650793}{11521304158449797} a^{17} - \frac{96527381815817445}{11521304158449797} a^{16} - \frac{6687170437450252}{11521304158449797} a^{15} - \frac{249791407802716218}{11521304158449797} a^{14} + \frac{242703094765690728}{11521304158449797} a^{13} + \frac{303685978344706514}{11521304158449797} a^{12} + \frac{135087840780904239}{11521304158449797} a^{11} - \frac{484617222784094757}{11521304158449797} a^{10} - \frac{849963082139743963}{11521304158449797} a^{9} - \frac{98923127560411639}{11521304158449797} a^{8} + \frac{319827960104982918}{11521304158449797} a^{7} + \frac{170773761739164084}{11521304158449797} a^{6} - \frac{202642238615208329}{11521304158449797} a^{5} - \frac{477931947089346301}{11521304158449797} a^{4} - \frac{207368804032245845}{11521304158449797} a^{3} + \frac{58779877076202589}{11521304158449797} a^{2} + \frac{80743061335962174}{11521304158449797} a + \frac{5426386600000580}{11521304158449797} \),  \( \frac{1011675330575242}{11521304158449797} a^{21} - \frac{9459688237079684}{11521304158449797} a^{20} + \frac{10087618659599095}{11521304158449797} a^{19} - \frac{10532595607329229}{11521304158449797} a^{18} + \frac{68754551345993479}{11521304158449797} a^{17} - \frac{71903469486000150}{11521304158449797} a^{16} - \frac{15723174253775657}{11521304158449797} a^{15} - \frac{171794744857498938}{11521304158449797} a^{14} + \frac{199435295329566479}{11521304158449797} a^{13} + \frac{213196275082503941}{11521304158449797} a^{12} + \frac{91824820763489403}{11521304158449797} a^{11} - \frac{426608132956038946}{11521304158449797} a^{10} - \frac{631612985533485780}{11521304158449797} a^{9} - \frac{112342393567617064}{11521304158449797} a^{8} + \frac{235283301492854478}{11521304158449797} a^{7} + \frac{129555727605429441}{11521304158449797} a^{6} - \frac{183267662904341511}{11521304158449797} a^{5} - \frac{368117811107146410}{11521304158449797} a^{4} - \frac{189877124442060375}{11521304158449797} a^{3} + \frac{31738963939265555}{11521304158449797} a^{2} + \frac{58125332300198145}{11521304158449797} a + \frac{8271504010828231}{11521304158449797} \),  \( \frac{2499397437394739}{11521304158449797} a^{21} - \frac{11264628193970461}{11521304158449797} a^{20} + \frac{16792592454575659}{11521304158449797} a^{19} - \frac{29888511061437340}{11521304158449797} a^{18} + \frac{86562344923277985}{11521304158449797} a^{17} - \frac{108672902385012760}{11521304158449797} a^{16} + \frac{73689042011371509}{11521304158449797} a^{15} - \frac{196884176768757211}{11521304158449797} a^{14} + \frac{245541645970131183}{11521304158449797} a^{13} + \frac{36514891812948339}{11521304158449797} a^{12} + \frac{44618801108381294}{11521304158449797} a^{11} - \frac{253436417459763728}{11521304158449797} a^{10} - \frac{291908948203017252}{11521304158449797} a^{9} + \frac{236992031515382943}{11521304158449797} a^{8} + \frac{117173592389561235}{11521304158449797} a^{7} + \frac{12337884071354495}{11521304158449797} a^{6} - \frac{124837467243575601}{11521304158449797} a^{5} - \frac{163087734046162989}{11521304158449797} a^{4} + \frac{39566755599611475}{11521304158449797} a^{3} + \frac{61151763161006236}{11521304158449797} a^{2} + \frac{5379550067979228}{11521304158449797} a - \frac{18907342130229249}{11521304158449797} \),  \( \frac{11785426760979273}{11521304158449797} a^{21} - \frac{5344147848693466}{11521304158449797} a^{20} - \frac{9583709678115277}{11521304158449797} a^{19} - \frac{76412314129027756}{11521304158449797} a^{18} + \frac{46466679384520013}{11521304158449797} a^{17} + \frac{154574658030412244}{11521304158449797} a^{16} + \frac{202596307768314823}{11521304158449797} a^{15} - \frac{199766487774597216}{11521304158449797} a^{14} - \frac{600343680710026071}{11521304158449797} a^{13} - \frac{161835050965423116}{11521304158449797} a^{12} + \frac{825831831574299945}{11521304158449797} a^{11} + \frac{1401699834506103004}{11521304158449797} a^{10} + \frac{537081948558353842}{11521304158449797} a^{9} - \frac{618121663257924031}{11521304158449797} a^{8} - \frac{391726400263666253}{11521304158449797} a^{7} + \frac{334283663971562360}{11521304158449797} a^{6} + \frac{802425800200783146}{11521304158449797} a^{5} + \frac{525598596209503789}{11521304158449797} a^{4} - \frac{69340531578619775}{11521304158449797} a^{3} - \frac{196099161026080940}{11521304158449797} a^{2} - \frac{52438615261693529}{11521304158449797} a + \frac{8531664283695960}{11521304158449797} \),  \( \frac{8284714219235766}{11521304158449797} a^{21} - \frac{12808923047709059}{11521304158449797} a^{20} + \frac{2096537436527226}{11521304158449797} a^{19} - \frac{54756526724241087}{11521304158449797} a^{18} + \frac{96337755600882229}{11521304158449797} a^{17} + \frac{39628845391252825}{11521304158449797} a^{16} + \frac{83795398674664414}{11521304158449797} a^{15} - \frac{296750487978826501}{11521304158449797} a^{14} - \frac{214580093574049268}{11521304158449797} a^{13} + \frac{197795000854005563}{11521304158449797} a^{12} + \frac{629063199019027378}{11521304158449797} a^{11} + \frac{456089109601423913}{11521304158449797} a^{10} - \frac{473081299026599856}{11521304158449797} a^{9} - \frac{610332383878040560}{11521304158449797} a^{8} - \frac{3341158750800475}{11521304158449797} a^{7} + \frac{428811807288873398}{11521304158449797} a^{6} + \frac{338867744562203160}{11521304158449797} a^{5} - \frac{121450807548752349}{11521304158449797} a^{4} - \frac{287637211543574213}{11521304158449797} a^{3} - \frac{116237215552972610}{11521304158449797} a^{2} + \frac{48245488188197149}{11521304158449797} a + \frac{39996686271630050}{11521304158449797} \),  \( \frac{26690183654645599}{11521304158449797} a^{21} - \frac{37022357966571995}{11521304158449797} a^{20} + \frac{13142390929198077}{11521304158449797} a^{19} - \frac{187492944135153579}{11521304158449797} a^{18} + \frac{281143664326227384}{11521304158449797} a^{17} + \frac{89170196077053711}{11521304158449797} a^{16} + \frac{385897094576105386}{11521304158449797} a^{15} - \frac{813663599430369571}{11521304158449797} a^{14} - \frac{638640164743390220}{11521304158449797} a^{13} + \frac{224223147285870729}{11521304158449797} a^{12} + \frac{1663678209168036233}{11521304158449797} a^{11} + \frac{1766736179843060539}{11521304158449797} a^{10} - \frac{482531569541103421}{11521304158449797} a^{9} - \frac{1041520286612635725}{11521304158449797} a^{8} - \frac{147228271776613030}{11521304158449797} a^{7} + \frac{921563421113748399}{11521304158449797} a^{6} + \frac{1117914180747119731}{11521304158449797} a^{5} + \frac{164573233400392977}{11521304158449797} a^{4} - \frac{382225025759390410}{11521304158449797} a^{3} - \frac{218162358797269903}{11521304158449797} a^{2} + \frac{46714991477693143}{11521304158449797} a + \frac{36051234952779583}{11521304158449797} \),  \( \frac{3341168700959298}{11521304158449797} a^{21} + \frac{6736119740892870}{11521304158449797} a^{20} - \frac{14505210996202946}{11521304158449797} a^{19} - \frac{17178185560281946}{11521304158449797} a^{18} - \frac{44366085807170189}{11521304158449797} a^{17} + \frac{132474328304319177}{11521304158449797} a^{16} + \frac{81890619359154014}{11521304158449797} a^{15} + \frac{57502918461726884}{11521304158449797} a^{14} - \frac{422433471647741935}{11521304158449797} a^{13} - \frac{229861488100346929}{11521304158449797} a^{12} + \frac{318930801155096338}{11521304158449797} a^{11} + \frac{898337541198269047}{11521304158449797} a^{10} + \frac{656466763243663538}{11521304158449797} a^{9} - \frac{340947650879599134}{11521304158449797} a^{8} - \frac{401267400407592079}{11521304158449797} a^{7} + \frac{107375425695433704}{11521304158449797} a^{6} + \frac{503453644464918789}{11521304158449797} a^{5} + \frac{460383669842435094}{11521304158449797} a^{4} + \frac{13448100232175646}{11521304158449797} a^{3} - \frac{155310771803336718}{11521304158449797} a^{2} - \frac{59721086200290391}{11521304158449797} a + \frac{20078705617327690}{11521304158449797} \),  \( \frac{5919661291745849}{11521304158449797} a^{21} - \frac{6356401308907265}{11521304158449797} a^{20} - \frac{6896519644152276}{11521304158449797} a^{19} - \frac{30739671367943545}{11521304158449797} a^{18} + \frac{44669437789221921}{11521304158449797} a^{17} + \frac{91861671956790204}{11521304158449797} a^{16} + \frac{15225655410072116}{11521304158449797} a^{15} - \frac{169234873312278694}{11521304158449797} a^{14} - \frac{317656484334171815}{11521304158449797} a^{13} + \frac{224371003665006658}{11521304158449797} a^{12} + \frac{535504062051344686}{11521304158449797} a^{11} + \frac{493306370352024823}{11521304158449797} a^{10} - \frac{416963789968785175}{11521304158449797} a^{9} - \frac{742699578245988885}{11521304158449797} a^{8} - \frac{67948555071444095}{11521304158449797} a^{7} + \frac{400123240240389573}{11521304158449797} a^{6} + \frac{359106775441433535}{11521304158449797} a^{5} - \frac{89628404965054092}{11521304158449797} a^{4} - \frac{369341933221950068}{11521304158449797} a^{3} - \frac{170870986664117794}{11521304158449797} a^{2} + \frac{39210122383445856}{11521304158449797} a + \frac{53581238079471570}{11521304158449797} \),  \( \frac{11417506936906238}{11521304158449797} a^{21} - \frac{11277307732122226}{11521304158449797} a^{20} + \frac{5279103977124701}{11521304158449797} a^{19} - \frac{85322025576585783}{11521304158449797} a^{18} + \frac{91031771629469296}{11521304158449797} a^{17} + \frac{42431203034259684}{11521304158449797} a^{16} + \frac{238779036770809405}{11521304158449797} a^{15} - \frac{262190865133753234}{11521304158449797} a^{14} - \frac{304586810068383433}{11521304158449797} a^{13} - \frac{190649246717418916}{11521304158449797} a^{12} + \frac{605800351465568681}{11521304158449797} a^{11} + \frac{1014853846227369665}{11521304158449797} a^{10} + \frac{485110630410384769}{11521304158449797} a^{9} - \frac{87581558784394248}{11521304158449797} a^{8} - \frac{203873129193034585}{11521304158449797} a^{7} + \frac{148722848191446894}{11521304158449797} a^{6} + \frac{559030711120391809}{11521304158449797} a^{5} + \frac{432424892279609831}{11521304158449797} a^{4} + \frac{142679916957473675}{11521304158449797} a^{3} - \frac{61624770921891223}{11521304158449797} a^{2} - \frac{68858819327537042}{11521304158449797} a - \frac{24423500776831785}{11521304158449797} \) (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 1054.33143917 \) (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 $22$ $16{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.14.0.1}{14} }{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ $22$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.12.0.1}{12} }{,}\,{\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ ${\href{/LocalNumberField/17.10.0.1}{10} }{,}\,{\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ $22$ $16{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.11.0.1}{11} }^{2}$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$61$61.2.1.1$x^{2} - 4 x - 57$$2$$1$$1$$C_2$$[\ ]_{2}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
61.6.0.1$x^{6} - 4 x + 10$$1$$6$$0$$C_6$$[\ ]^{6}$
61.8.0.1$x^{8} - x + 17$$1$$8$$0$$C_8$$[\ ]^{8}$
1279Data not computed
1609Data not computed
4025911Data not computed