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Properties

Label 36.0.20413967918...9349.1
Degree $36$
Signature $[0, 18]$
Discriminant $109^{35}$
Root discriminant $95.68$
Ramified prime $109$
Class number $17153$ (GRH)
Class group $[17153]$ (GRH)
Galois group $C_{36}$ (as 36T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![103177, -52981, 873358, -565945, -330453, 675039, -2756219, -897617, 1926836, 132851, 220271, 1177166, 2721446, 987382, -49640, 3146481, 1198241, -784527, 2238405, 640151, -454935, 887126, 182604, -121616, 199142, 29347, -17945, 25554, 2590, -1467, 1837, 115, -61, 68, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - x^35 + 2*x^34 + 68*x^33 - 61*x^32 + 115*x^31 + 1837*x^30 - 1467*x^29 + 2590*x^28 + 25554*x^27 - 17945*x^26 + 29347*x^25 + 199142*x^24 - 121616*x^23 + 182604*x^22 + 887126*x^21 - 454935*x^20 + 640151*x^19 + 2238405*x^18 - 784527*x^17 + 1198241*x^16 + 3146481*x^15 - 49640*x^14 + 987382*x^13 + 2721446*x^12 + 1177166*x^11 + 220271*x^10 + 132851*x^9 + 1926836*x^8 - 897617*x^7 - 2756219*x^6 + 675039*x^5 - 330453*x^4 - 565945*x^3 + 873358*x^2 - 52981*x + 103177)
 
gp: K = bnfinit(x^36 - x^35 + 2*x^34 + 68*x^33 - 61*x^32 + 115*x^31 + 1837*x^30 - 1467*x^29 + 2590*x^28 + 25554*x^27 - 17945*x^26 + 29347*x^25 + 199142*x^24 - 121616*x^23 + 182604*x^22 + 887126*x^21 - 454935*x^20 + 640151*x^19 + 2238405*x^18 - 784527*x^17 + 1198241*x^16 + 3146481*x^15 - 49640*x^14 + 987382*x^13 + 2721446*x^12 + 1177166*x^11 + 220271*x^10 + 132851*x^9 + 1926836*x^8 - 897617*x^7 - 2756219*x^6 + 675039*x^5 - 330453*x^4 - 565945*x^3 + 873358*x^2 - 52981*x + 103177, 1)
 

Normalized defining polynomial

\( x^{36} - x^{35} + 2 x^{34} + 68 x^{33} - 61 x^{32} + 115 x^{31} + 1837 x^{30} - 1467 x^{29} + 2590 x^{28} + 25554 x^{27} - 17945 x^{26} + 29347 x^{25} + 199142 x^{24} - 121616 x^{23} + 182604 x^{22} + 887126 x^{21} - 454935 x^{20} + 640151 x^{19} + 2238405 x^{18} - 784527 x^{17} + 1198241 x^{16} + 3146481 x^{15} - 49640 x^{14} + 987382 x^{13} + 2721446 x^{12} + 1177166 x^{11} + 220271 x^{10} + 132851 x^{9} + 1926836 x^{8} - 897617 x^{7} - 2756219 x^{6} + 675039 x^{5} - 330453 x^{4} - 565945 x^{3} + 873358 x^{2} - 52981 x + 103177 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 18]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(204139679185162956270051790668558119706910323589259444440398107939799349=109^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.68$
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:  $109$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(109\)
Dirichlet character group:    $\lbrace$$\chi_{109}(1,·)$, $\chi_{109}(2,·)$, $\chi_{109}(4,·)$, $\chi_{109}(8,·)$, $\chi_{109}(16,·)$, $\chi_{109}(17,·)$, $\chi_{109}(19,·)$, $\chi_{109}(23,·)$, $\chi_{109}(27,·)$, $\chi_{109}(32,·)$, $\chi_{109}(33,·)$, $\chi_{109}(34,·)$, $\chi_{109}(38,·)$, $\chi_{109}(41,·)$, $\chi_{109}(43,·)$, $\chi_{109}(45,·)$, $\chi_{109}(46,·)$, $\chi_{109}(54,·)$, $\chi_{109}(55,·)$, $\chi_{109}(63,·)$, $\chi_{109}(64,·)$, $\chi_{109}(66,·)$, $\chi_{109}(68,·)$, $\chi_{109}(71,·)$, $\chi_{109}(75,·)$, $\chi_{109}(76,·)$, $\chi_{109}(77,·)$, $\chi_{109}(82,·)$, $\chi_{109}(86,·)$, $\chi_{109}(90,·)$, $\chi_{109}(92,·)$, $\chi_{109}(93,·)$, $\chi_{109}(101,·)$, $\chi_{109}(105,·)$, $\chi_{109}(107,·)$, $\chi_{109}(108,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{281} a^{34} - \frac{81}{281} a^{33} + \frac{110}{281} a^{32} - \frac{86}{281} a^{31} - \frac{31}{281} a^{30} + \frac{108}{281} a^{29} - \frac{70}{281} a^{28} - \frac{89}{281} a^{27} - \frac{32}{281} a^{26} + \frac{64}{281} a^{25} - \frac{125}{281} a^{24} - \frac{70}{281} a^{23} + \frac{39}{281} a^{22} + \frac{121}{281} a^{21} + \frac{5}{281} a^{20} - \frac{58}{281} a^{19} + \frac{41}{281} a^{18} - \frac{95}{281} a^{17} + \frac{50}{281} a^{16} + \frac{25}{281} a^{15} + \frac{78}{281} a^{14} + \frac{93}{281} a^{13} + \frac{36}{281} a^{12} - \frac{89}{281} a^{11} - \frac{40}{281} a^{10} - \frac{65}{281} a^{9} + \frac{122}{281} a^{8} - \frac{1}{281} a^{7} - \frac{39}{281} a^{6} + \frac{116}{281} a^{5} - \frac{74}{281} a^{4} - \frac{26}{281} a^{3} + \frac{126}{281} a^{2} - \frac{93}{281} a + \frac{89}{281}$, $\frac{1}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{35} + \frac{1477130455613953515823611818227812849663785985895128415508234023613255601641025020514682282103076146972839136310009580746062805}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{34} + \frac{88991451314302324632709563893552735835467373195377830691094619070792313120773908647428126799808968282855046825020687205073024077}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{33} + \frac{19076173411887626298187788038767127263404278398608653529803624696741256247606604070824668464024763483555910483372640953517590880}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{32} + \frac{263487904529809002222916908286662379321699438447449889267384511162058990842767003477497660003479836191133817284643746038420614639}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{31} - \frac{50354566809340800103257711371427333879913331894296107024339266180628445422016260312257862015815887132030108738581761034148438084}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{30} - \frac{386158178247185205368409134442490803674678605399987893451915153328598900272090409322037520284430930835017811210809571869102261861}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{29} - \frac{86687554202567568697141401007609558559542814997571130129919790192671276371670552890470466100894620880314603760403982405967170345}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{28} - \frac{359605837272247657521823791792468412940488363565418705751311270943281181378567896681609387413540615068249030730677089076607642693}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{27} - \frac{24854498037407209326944988958040456694912807977850715472834569309867320441518653708572109065842790498685191147774217369887165882}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{26} + \frac{181063682853894331486511295926660020207055651065882440105572727962561333095475707677423637092830566366338521161442742539340389168}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{25} + \frac{254972667548780400926626711119426942187896315642080575290059115399678876801588191657716700087171802228915633348918923060605120466}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{24} + \frac{319345698625471411405411082732368324092804082035424343690668707417621898240498265602464486061355763836228958599074179352653042179}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{23} + \frac{236519262642456929479276763642611351273690364424269742489775727629762835725556998113523866506183729703655806171782386347280993799}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{22} + \frac{273316091212947404269837469916599578506017078818852045510628094658661514271319717584911810774258812018445909158519216292268362786}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{21} + \frac{50455521708337005053005143886345891581428038861166692732022161716242328104816614024308612918236291986481525442567179160843384802}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{20} - \frac{412926452687532805309553097192964457581651375097002028173798453985368293557400935083919762886992988363202024927514372236263939725}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{19} + \frac{355589724451865821020985027422654298414122350919950895540186609986135291690812898294118338528949301158106196050921979528229409028}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{18} - \frac{279659887254951519168079484517737670139918394603288397859385064059548092590844654536787774302202347123454563740604963042972230719}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{17} - \frac{200072425327281103689754638804914714649059986756129682128291492641030398604687745056006258713947749108544318483051904286334107974}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{16} - \frac{116154893943832095134433696566532496295151210980780564206558715065863240809505266671169159510099404420015080668625912236516406519}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{15} - \frac{114878293171587468018859139382935249446053172841544203289553130806995561534258396073987013202969813334298068205366380965522022760}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{14} - \frac{29877862109678441589209398880346731402864078544311338392147806245140710535737733612971155907321120472481488957706728969829597664}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{13} + \frac{30067794112578701461861100521680640416233765825766193345967814404977265434790782634136300525714134190019335917373497988157986611}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{12} - \frac{361113657777363437755354027645916260315019475014353810067521405880135568217418104919300654355748273747110962651010109696739344401}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{11} - \frac{129722764382615124259323931602674878705764502659744394307253756148357686463209580877945189845591989655077681067776403245942571420}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{10} + \frac{61107049735203240276720423274069039389918368540765169981314057762341192590715439122297425899200760219000318092892671874766436540}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{9} + \frac{79432717654076919911314545103277826872548018530461723382319553748119749985294136430024511760698513681770459969717052506663129244}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{8} - \frac{350108607588579790447384314012676860107794783461014646693011440445912068320725767580662017820951029316016020129364117599208106109}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{7} + \frac{311708320748419464741231048521033470420873721563012757645415529406890598327222662730794419899413224757750014629121390451180800844}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{6} - \frac{147900091584406178889274662979908174048201029841850375713427454925190073446537636777929482853181838017493289732238452255955833523}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{5} - \frac{408988472188869447456144114705096204864585519024964608940855480504380189917546037747988355105225693311362183711703928549932033337}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{4} + \frac{368238623685157708289635066147381422869356571032402465495314560277514938299178305553219019119867394192019974728172549710645632000}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{3} - \frac{159955091972377469830003033001197524719982089021303863447112827921387243516312123706319988349756978082126062343140975271464754874}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a^{2} + \frac{144563772034920178954511185981095499933175523989784788218539271365613150719728793079541994914098384808752853047659779061549071712}{841464930511750699234509677869675465304221396386307077195870004864593512016155181942670563592756120040211900928715470936708375253} a - \frac{2507168996325369677156288332273740558620128566412203877962179295142488383515143516151527789503565097301035490460092660527725}{8155547559162901608250963663119449734962456714057465105555210995324476501702464521576228845505840643168650968032754111252589}$

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

Class group and class number

$C_{17153}$, which has order $17153$ (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:  $17$
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:  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:  \( 26453170924052510 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{36}$ (as 36T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 36
The 36 conjugacy class representatives for $C_{36}$
Character table for $C_{36}$ is not computed

Intermediate fields

\(\Q(\sqrt{109}) \), 3.3.11881.1, 4.0.1295029.1, 6.6.15386239549.1, 9.9.19925626416901921.1, 12.0.25804264053054077850709.1, 18.18.43276334103547425867991106950436269.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$ $36$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ $18^{2}$ $18^{2}$ $36$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{6}$ $36$ $36$ $36$

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
109Data not computed