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Properties

Label 21.21.7747469504...7344.1
Degree $21$
Signature $[21, 0]$
Discriminant $2^{18}\cdot 7^{36}\cdot 13^{6}\cdot 229^{7}\cdot 9421^{6}$
Root discriminant $8853.25$
Ramified primes $2, 7, 13, 229, 9421$
Class number Not computed
Class group Not computed
Galois group 21T46

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-148311093030591252973592576, 13821648271223598394767360, 19321372641008808426900480, 2109712653074731793670400, -267242834707638039181568, -53750525437959992879328, 24955225147594433328, 475341663746724551300, 18990798567294353702, -1885199336157315059, -136940737048345762, 2813759104373693, 434805185770494, 2604295338050, -697989570102, -14002943339, 540787226, 16745526, -159446, -7707, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^21 - 7707*x^19 - 159446*x^18 + 16745526*x^17 + 540787226*x^16 - 14002943339*x^15 - 697989570102*x^14 + 2604295338050*x^13 + 434805185770494*x^12 + 2813759104373693*x^11 - 136940737048345762*x^10 - 1885199336157315059*x^9 + 18990798567294353702*x^8 + 475341663746724551300*x^7 + 24955225147594433328*x^6 - 53750525437959992879328*x^5 - 267242834707638039181568*x^4 + 2109712653074731793670400*x^3 + 19321372641008808426900480*x^2 + 13821648271223598394767360*x - 148311093030591252973592576)
 
gp: K = bnfinit(x^21 - 7707*x^19 - 159446*x^18 + 16745526*x^17 + 540787226*x^16 - 14002943339*x^15 - 697989570102*x^14 + 2604295338050*x^13 + 434805185770494*x^12 + 2813759104373693*x^11 - 136940737048345762*x^10 - 1885199336157315059*x^9 + 18990798567294353702*x^8 + 475341663746724551300*x^7 + 24955225147594433328*x^6 - 53750525437959992879328*x^5 - 267242834707638039181568*x^4 + 2109712653074731793670400*x^3 + 19321372641008808426900480*x^2 + 13821648271223598394767360*x - 148311093030591252973592576, 1)
 

Normalized defining polynomial

\( x^{21} - 7707 x^{19} - 159446 x^{18} + 16745526 x^{17} + 540787226 x^{16} - 14002943339 x^{15} - 697989570102 x^{14} + 2604295338050 x^{13} + 434805185770494 x^{12} + 2813759104373693 x^{11} - 136940737048345762 x^{10} - 1885199336157315059 x^{9} + 18990798567294353702 x^{8} + 475341663746724551300 x^{7} + 24955225147594433328 x^{6} - 53750525437959992879328 x^{5} - 267242834707638039181568 x^{4} + 2109712653074731793670400 x^{3} + 19321372641008808426900480 x^{2} + 13821648271223598394767360 x - 148311093030591252973592576 \)

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

Invariants

Degree:  $21$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[21, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(77474695045030405305071775338889924369299860224383774504674641727894441069982777344=2^{18}\cdot 7^{36}\cdot 13^{6}\cdot 229^{7}\cdot 9421^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8853.25$
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:  $2, 7, 13, 229, 9421$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{8} a^{8} - \frac{1}{8} a^{6} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} + \frac{1}{4} a^{3}$, $\frac{1}{8} a^{9} - \frac{1}{4} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{4} a^{5} - \frac{3}{8} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} + \frac{1}{8} a^{5} + \frac{1}{4} a^{4} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{16} a^{10} - \frac{1}{32} a^{8} - \frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{11}{32} a^{4} + \frac{7}{16} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{128} a^{13} - \frac{1}{64} a^{12} - \frac{1}{64} a^{11} - \frac{5}{128} a^{9} - \frac{3}{64} a^{8} + \frac{1}{64} a^{7} + \frac{1}{32} a^{6} + \frac{25}{128} a^{5} + \frac{17}{64} a^{4} + \frac{3}{32} a^{3} + \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{128} a^{14} - \frac{1}{64} a^{12} + \frac{1}{32} a^{11} + \frac{3}{128} a^{10} + \frac{1}{64} a^{8} + \frac{9}{128} a^{6} + \frac{1}{32} a^{5} + \frac{1}{32} a^{4} - \frac{1}{4} a^{3} - \frac{3}{8} a^{2}$, $\frac{1}{128} a^{15} - \frac{1}{128} a^{11} - \frac{1}{16} a^{9} + \frac{1}{32} a^{8} - \frac{3}{128} a^{7} - \frac{1}{32} a^{6} - \frac{13}{64} a^{5} - \frac{11}{32} a^{4} + \frac{7}{16} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{128} a^{16} - \frac{1}{128} a^{12} - \frac{1}{16} a^{10} + \frac{1}{32} a^{9} - \frac{3}{128} a^{8} - \frac{1}{32} a^{7} + \frac{3}{64} a^{6} + \frac{5}{32} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{256} a^{17} - \frac{1}{256} a^{15} + \frac{1}{128} a^{12} + \frac{15}{256} a^{11} - \frac{1}{64} a^{10} + \frac{7}{128} a^{8} - \frac{13}{256} a^{7} - \frac{3}{64} a^{6} + \frac{27}{256} a^{5} + \frac{33}{128} a^{4} + \frac{23}{64} a^{3} + \frac{7}{16} a^{2} + \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{1536} a^{18} + \frac{1}{768} a^{17} + \frac{1}{1536} a^{16} + \frac{1}{384} a^{15} - \frac{1}{768} a^{14} + \frac{1}{768} a^{13} - \frac{1}{512} a^{12} + \frac{7}{384} a^{11} - \frac{13}{256} a^{10} + \frac{1}{256} a^{9} - \frac{25}{512} a^{8} + \frac{5}{96} a^{7} - \frac{43}{1536} a^{6} + \frac{21}{128} a^{5} - \frac{91}{192} a^{4} + \frac{5}{192} a^{3} + \frac{5}{24} a^{2} - \frac{3}{8} a - \frac{1}{6}$, $\frac{1}{83509248} a^{19} + \frac{1849}{20877312} a^{18} + \frac{112781}{83509248} a^{17} + \frac{28101}{13918208} a^{16} - \frac{43599}{13918208} a^{15} + \frac{32225}{41754624} a^{14} + \frac{129517}{83509248} a^{13} - \frac{480313}{41754624} a^{12} + \frac{1409017}{41754624} a^{11} + \frac{494841}{13918208} a^{10} - \frac{1023657}{27836416} a^{9} - \frac{2076239}{41754624} a^{8} - \frac{976913}{27836416} a^{7} - \frac{2797963}{41754624} a^{6} - \frac{1935191}{20877312} a^{5} - \frac{644895}{1739776} a^{4} - \frac{247523}{1304832} a^{3} + \frac{212129}{652416} a^{2} + \frac{13217}{326208} a - \frac{32609}{81552}$, $\frac{1}{16699028354238412692013785898741080019363002857676226218953238533229503489975838277396664703876694228551965191382302130747800131850473821689672004732501092728791140872171465226837021119381504} a^{20} - \frac{14570172112204193070264524087805934578413712744038413303928149921751576154039006456742595404327574796185158846586991155957024806177320482524685473826048748878526123833130991087836211}{4174757088559603173003446474685270004840750714419056554738309633307375872493959569349166175969173557137991297845575532686950032962618455422418001183125273182197785218042866306709255279845376} a^{19} - \frac{43085795262115663877314550626941814742869886773737078730578142215476627921696561010699248001984761406659106502005709399077179064136605706944491318347093351514921271199629682061945804549}{726044711053844030087555908640916522581000124246792444302314718836065369129384272930289769733769314284868051799230527423817397036977122682159652379673960553425701777050933270732044396494848} a^{18} + \frac{1278423606052651583902021216069839325012019757968797570695252835660419100269461901029081265179646388798142582261699227144506387459463606749928552052240541451608417382489790717969550032349}{2783171392373068782002297649790180003227167142946037703158873088871583914995973046232777450646115704758660865230383688457966688641745636948278667455416848788131856812028577537806170186563584} a^{17} + \frac{26104646173087136609686284850612713455365946350332355805695549492663379694770375489643475723517503003448620761121352437248759809064869425792528947293716199911537210085984471295104304225379}{8349514177119206346006892949370540009681501428838113109476619266614751744987919138698332351938347114275982595691151065373900065925236910844836002366250546364395570436085732613418510559690752} a^{16} + \frac{8709916967794455187080610331401873092149007119013539346752434931415468131223510037734480862732860717389807728231459513955304183406773068222336540319329650684625414110291599404630202813531}{2783171392373068782002297649790180003227167142946037703158873088871583914995973046232777450646115704758660865230383688457966688641745636948278667455416848788131856812028577537806170186563584} a^{15} + \frac{25317507277069803534056883437953351700807478660794137864557234102784241646432235544198325514052247064369431037071565183401428393953875155073161687966720334911115656853235681901767571463181}{16699028354238412692013785898741080019363002857676226218953238533229503489975838277396664703876694228551965191382302130747800131850473821689672004732501092728791140872171465226837021119381504} a^{14} - \frac{4349526184545286276083859584419784967820440716643834626774725261862776637928811663463225238117338108047087197676158944228232773036425403075066659831994037560163512446470812727164085476411}{2783171392373068782002297649790180003227167142946037703158873088871583914995973046232777450646115704758660865230383688457966688641745636948278667455416848788131856812028577537806170186563584} a^{13} + \frac{43083108327171657314123202502703027860159049409551168992576550413042486314535354538198918075185872826240380024021868052315365279776815727429201802623697432453915482628215734479596120490179}{2783171392373068782002297649790180003227167142946037703158873088871583914995973046232777450646115704758660865230383688457966688641745636948278667455416848788131856812028577537806170186563584} a^{12} - \frac{394964726900438289165370679060731675911820202774044853148669840597463392472233277056515089464022371374039667033106191491553047733650079960369732562320042110274474658620287097606957845470981}{8349514177119206346006892949370540009681501428838113109476619266614751744987919138698332351938347114275982595691151065373900065925236910844836002366250546364395570436085732613418510559690752} a^{11} + \frac{190222317257719031001908660285231069935540924685844415784801150655040853775967484329877648727427271859720031904761265959976982020450036560896123432884213259353168357940793402279741992285303}{5566342784746137564004595299580360006454334285892075406317746177743167829991946092465554901292231409517321730460767376915933377283491273896557334910833697576263713624057155075612340373127168} a^{10} - \frac{165053426656445118882523489109761681247554027389173264845507136470626285990810080837642933362212863818283361503434637483782974360644707008904289911446683312550447406063414228195068704401}{363022355526922015043777954320458261290500062123396222151157359418032684564692136465144884866884657142434025899615263711908698518488561341079826189836980276712850888525466635366022198247424} a^{9} + \frac{734076496175619775019885858116495576478152999550537579010646096204516305229802922137398353540336702755467086362637611448992967774833399465867656155877450960660533553463638268151774915066797}{16699028354238412692013785898741080019363002857676226218953238533229503489975838277396664703876694228551965191382302130747800131850473821689672004732501092728791140872171465226837021119381504} a^{8} - \frac{192165146804455182767440091941380048841353483279564124545182100037933131372572550771144534313108988373606255099224296862323135465362075991681769430125887320357358837018935355239793612026179}{8349514177119206346006892949370540009681501428838113109476619266614751744987919138698332351938347114275982595691151065373900065925236910844836002366250546364395570436085732613418510559690752} a^{7} + \frac{144133617128677361922941425699665004451460220476565990095536235131954682843023967697436570467577533872956292379244413767870661724332210636399442893849275119544915314681358176538486414411595}{1391585696186534391001148824895090001613583571473018851579436544435791957497986523116388725323057852379330432615191844228983344320872818474139333727708424394065928406014288768903085093281792} a^{6} - \frac{220785353702505535419237030743340456862183928495762026850084679532573135075678437845504685056900065122760660976807664624786970894870058596099147006169307317940772915365307074349399858940993}{1043689272139900793250861618671317501210187678604764138684577408326843968123489892337291543992293389284497824461393883171737508240654613855604500295781318295549446304510716576677313819961344} a^{5} - \frac{1746719788981280971547707337835278836192309484770873182010957189969305994944950852595793049164823015316708991410021470947597278481537115305994073986494790249011469429798287830602279022681}{86974106011658399437571801555943125100848973217063678223714784027236997343624157694774295332691115773708152038449490264311459020054551154633708357981776524629120525375893048056442818330112} a^{4} + \frac{50390086377134149822902801058480031299280840184636946421971588230462534762859719774258423875611578797307757996453018402482170161487542062658637546666559659872669513133447619315260987044673}{130461159017487599156357702333914687651273459825595517335572176040855496015436236542161442999036673660562228057674235396467188530081826731950562536972664786943680788063839572084664227495168} a^{3} - \frac{3835652330917507286001154699872754030116020949354130148356327520213410820690568145445923192160886691293769184830576455813901655134838140458844832106770195746286481960363919122818465268431}{65230579508743799578178851166957343825636729912797758667786088020427748007718118271080721499518336830281114028837117698233594265040913365975281268486332393471840394031919786042332113747584} a^{2} - \frac{1901715034420612742571707062475786664422261131195542474694774388998330146760527307787522010898876487018460969321613067116314846018977352646462559131159008550841582538770915892435660417047}{5435881625728649964848237597246445318803060826066479888982174001702312333976509855923393458293194735856759502403093141519466188753409447164606772373861032789320032835993315503527676145632} a + \frac{233101660991972935355367822300245970067857216251907386364914292617225616473533833583882210063836279828360822572483376925496368620354103869288451546224553813701188484418158654670611005939}{1019227804824121868409044549483708497275573904887464979184157625319183562620595597985636273429974012973142406700579964034899910391264271343363769820098943647997506156748746656911439277306}$

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:  $20$
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

21T46:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 8232
The 70 conjugacy class representatives for t21n46 are not computed
Character table for t21n46 is not computed

Intermediate fields

3.3.229.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 28 sibling: data not computed
Degree 42 siblings: 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 ${\href{/LocalNumberField/3.3.0.1}{3} }^{7}$ $21$ R $21$ R $21$ $21$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.7.0.1}{7} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.14.0.1}{14} }{,}\,{\href{/LocalNumberField/41.7.0.1}{7} }$ $21$ ${\href{/LocalNumberField/47.14.0.1}{14} }{,}\,{\href{/LocalNumberField/47.7.0.1}{7} }$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
2.4.4.3$x^{4} + 2 x^{2} + 4 x + 4$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
7Data not computed
$13$13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.4.0.1$x^{4} + x^{2} - x + 2$$1$$4$$0$$C_4$$[\ ]^{4}$
13.7.6.1$x^{7} - 13$$7$$1$$6$$D_{7}$$[\ ]_{7}^{2}$
229Data not computed
9421Data not computed