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Properties

Label 36.36.2351680550...3125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{54}\cdot 5^{27}\cdot 13^{24}$
Root discriminant $96.06$
Ramified primes $3, 5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_{12}$ (as 36T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![288991, 71286, -10886424, 10150922, 139324140, -255756294, -646510065, 1795533942, 964975548, -5631020628, 944334231, 9570453009, -5143540984, -9724914267, 7929872535, 6167942709, -6918173706, -2406058791, 3915786614, 487842723, -1517917803, 15579817, 412140267, -41557041, -78595284, 13221999, 10357989, -2337941, -903138, 258882, 46936, -17841, -1014, 701, -18, -12, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 12*x^35 - 18*x^34 + 701*x^33 - 1014*x^32 - 17841*x^31 + 46936*x^30 + 258882*x^29 - 903138*x^28 - 2337941*x^27 + 10357989*x^26 + 13221999*x^25 - 78595284*x^24 - 41557041*x^23 + 412140267*x^22 + 15579817*x^21 - 1517917803*x^20 + 487842723*x^19 + 3915786614*x^18 - 2406058791*x^17 - 6918173706*x^16 + 6167942709*x^15 + 7929872535*x^14 - 9724914267*x^13 - 5143540984*x^12 + 9570453009*x^11 + 944334231*x^10 - 5631020628*x^9 + 964975548*x^8 + 1795533942*x^7 - 646510065*x^6 - 255756294*x^5 + 139324140*x^4 + 10150922*x^3 - 10886424*x^2 + 71286*x + 288991)
 
gp: K = bnfinit(x^36 - 12*x^35 - 18*x^34 + 701*x^33 - 1014*x^32 - 17841*x^31 + 46936*x^30 + 258882*x^29 - 903138*x^28 - 2337941*x^27 + 10357989*x^26 + 13221999*x^25 - 78595284*x^24 - 41557041*x^23 + 412140267*x^22 + 15579817*x^21 - 1517917803*x^20 + 487842723*x^19 + 3915786614*x^18 - 2406058791*x^17 - 6918173706*x^16 + 6167942709*x^15 + 7929872535*x^14 - 9724914267*x^13 - 5143540984*x^12 + 9570453009*x^11 + 944334231*x^10 - 5631020628*x^9 + 964975548*x^8 + 1795533942*x^7 - 646510065*x^6 - 255756294*x^5 + 139324140*x^4 + 10150922*x^3 - 10886424*x^2 + 71286*x + 288991, 1)
 

Normalized defining polynomial

\( x^{36} - 12 x^{35} - 18 x^{34} + 701 x^{33} - 1014 x^{32} - 17841 x^{31} + 46936 x^{30} + 258882 x^{29} - 903138 x^{28} - 2337941 x^{27} + 10357989 x^{26} + 13221999 x^{25} - 78595284 x^{24} - 41557041 x^{23} + 412140267 x^{22} + 15579817 x^{21} - 1517917803 x^{20} + 487842723 x^{19} + 3915786614 x^{18} - 2406058791 x^{17} - 6918173706 x^{16} + 6167942709 x^{15} + 7929872535 x^{14} - 9724914267 x^{13} - 5143540984 x^{12} + 9570453009 x^{11} + 944334231 x^{10} - 5631020628 x^{9} + 964975548 x^{8} + 1795533942 x^{7} - 646510065 x^{6} - 255756294 x^{5} + 139324140 x^{4} + 10150922 x^{3} - 10886424 x^{2} + 71286 x + 288991 \)

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

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(235168055015185353161031494672207623739277412596389718353748321533203125=3^{54}\cdot 5^{27}\cdot 13^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.06$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(585=3^{2}\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{585}(256,·)$, $\chi_{585}(1,·)$, $\chi_{585}(263,·)$, $\chi_{585}(139,·)$, $\chi_{585}(16,·)$, $\chi_{585}(529,·)$, $\chi_{585}(274,·)$, $\chi_{585}(406,·)$, $\chi_{585}(152,·)$, $\chi_{585}(68,·)$, $\chi_{585}(542,·)$, $\chi_{585}(287,·)$, $\chi_{585}(289,·)$, $\chi_{585}(113,·)$, $\chi_{585}(391,·)$, $\chi_{585}(302,·)$, $\chi_{585}(308,·)$, $\chi_{585}(53,·)$, $\chi_{585}(443,·)$, $\chi_{585}(61,·)$, $\chi_{585}(451,·)$, $\chi_{585}(196,·)$, $\chi_{585}(458,·)$, $\chi_{585}(334,·)$, $\chi_{585}(79,·)$, $\chi_{585}(211,·)$, $\chi_{585}(469,·)$, $\chi_{585}(347,·)$, $\chi_{585}(92,·)$, $\chi_{585}(94,·)$, $\chi_{585}(482,·)$, $\chi_{585}(484,·)$, $\chi_{585}(107,·)$, $\chi_{585}(497,·)$, $\chi_{585}(503,·)$, $\chi_{585}(248,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $\frac{1}{181} a^{33} + \frac{79}{181} a^{32} - \frac{30}{181} a^{31} - \frac{4}{181} a^{30} + \frac{3}{181} a^{29} + \frac{19}{181} a^{28} - \frac{27}{181} a^{27} + \frac{55}{181} a^{26} + \frac{49}{181} a^{25} + \frac{38}{181} a^{24} + \frac{43}{181} a^{23} - \frac{64}{181} a^{22} + \frac{54}{181} a^{21} - \frac{77}{181} a^{20} + \frac{66}{181} a^{19} - \frac{22}{181} a^{18} + \frac{35}{181} a^{17} + \frac{83}{181} a^{16} - \frac{34}{181} a^{15} + \frac{18}{181} a^{14} + \frac{55}{181} a^{13} + \frac{79}{181} a^{12} + \frac{4}{181} a^{11} + \frac{8}{181} a^{10} + \frac{86}{181} a^{9} - \frac{70}{181} a^{8} - \frac{26}{181} a^{7} - \frac{19}{181} a^{6} - \frac{25}{181} a^{5} + \frac{10}{181} a^{4} - \frac{58}{181} a^{3} - \frac{63}{181} a^{2} + \frac{21}{181} a - \frac{34}{181}$, $\frac{1}{181} a^{34} + \frac{64}{181} a^{32} + \frac{13}{181} a^{31} - \frac{43}{181} a^{30} - \frac{37}{181} a^{29} - \frac{80}{181} a^{28} + \frac{16}{181} a^{27} + \frac{48}{181} a^{26} - \frac{32}{181} a^{25} - \frac{63}{181} a^{24} - \frac{22}{181} a^{23} + \frac{42}{181} a^{22} + \frac{1}{181} a^{21} - \frac{5}{181} a^{20} + \frac{13}{181} a^{19} - \frac{37}{181} a^{18} + \frac{33}{181} a^{17} - \frac{75}{181} a^{16} - \frac{11}{181} a^{15} + \frac{81}{181} a^{14} + \frac{78}{181} a^{13} - \frac{83}{181} a^{12} + \frac{54}{181} a^{11} - \frac{3}{181} a^{10} + \frac{14}{181} a^{9} + \frac{74}{181} a^{8} + \frac{44}{181} a^{7} + \frac{28}{181} a^{6} - \frac{6}{181} a^{5} + \frac{57}{181} a^{4} - \frac{6}{181} a^{3} - \frac{70}{181} a^{2} - \frac{64}{181} a - \frac{29}{181}$, $\frac{1}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{35} - \frac{305687143300650726688623183213265198374755235608313112624790208057167953216074689487501013671520943519793427758645661}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{34} - \frac{231204445430854032095583622278842133668393242663429453333752929312253935359803745991286785043037587958972195051161836}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{33} + \frac{60765059220375541344083369331233684351007994959892397295122240882543099001241268414443571281462238865233633391513758564}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{32} - \frac{84168459293493590428452332204888623317024264341083210036442980202041480031930651639528536142085725648811580542554550387}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{31} - \frac{72790861249288205243772487895061167680478824577777744140186252528268316062040159635367374459826600811001418476434815832}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{30} - \frac{61531883292649621834201695965594647983161202818697429724970138255622179131964686814348501895741027381042615892685516196}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{29} + \frac{91407265749067743963625346722589237754338543732004183840778528894667371474342320642218552509541753386775639492757326165}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{28} + \frac{67352840044333189616485080444618290182996566484631788918874434930100501914510267594514949295582481640930454690155346810}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{27} - \frac{29110570519192888268444440587162250354894388442822929328277632793477446371200304604558309629387287295712754644129830608}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{26} - \frac{24686989943640011612720107952226747137548457436363008789726682971937851398339094536585788354447396139729786423746589890}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{25} - \frac{42793224619587120025059227050630508628303272107535730682226242070609604878689425817276610819841526009188752158724940913}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{24} + \frac{88148996027128729405687850792120772351380952126374592709244048142076229091758406134024702899994279315958633803693357567}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{23} + \frac{52187236016121941953039470326656770097859140685038618680334461907349113135827898694251119501418721629357874329064410136}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{22} + \frac{73493905601079721026677598657709504472345151753109573383674152026515987871878752807440996717601885046555547192380563546}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{21} + \frac{16494802678901275400740036593788920025690157278452377029619090166648823239777083908039272110036259166468910534480582267}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{20} - \frac{33520586763428699653923970537826816741281813247081441450567350423741752330444553504190064485743481924640323301077639236}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{19} - \frac{41154603823836559738831048240773863836195719061637689567294509619157508538111167680226323488852393235327398362762869049}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{18} - \frac{78546095870025976999694140642448990823794029001664125105068261784462984810344600938703668971348335218009887417572763073}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{17} - \frac{37933523161090396864213598559596524010641241016645672245296266669107913052845529078766514771449275126277100318596122210}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{16} - \frac{10364207347928872049226130163744126838604650148336702400905640129711228881064266801013593217596301084629739671514248072}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{15} + \frac{5919780317915723839082995899355031516996068422540257027842910546541241105705087579529580897969555829403191573642328544}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{14} + \frac{73091163069299091314534631416205178599867153877355627021688003705817164000486253836035975417854518265948157851816639763}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{13} + \frac{11218675448777884838195782611473358279191250827306772141252306506937590784250004025365898986377168840264440073936148551}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{12} + \frac{57434096053506252006654414974340221921786900440877579133394302287994597865838847343055433725725367164919827822975362299}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{11} - \frac{12906415647489723680191926410707460831398735486392038435903078514553421437601034138536375959834074491395661823091756671}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{10} + \frac{4472367385197096864681948105375390221684669208346824699675137780316919811582482113157024828199956459358689227295396933}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{9} - \frac{26401569860360229577969776346225759879091576276851971197493407419540871672266524886244670368502018427999603484340157369}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{8} + \frac{22002295533786997764249114356835843746497740539543285187004253736708482641089302395139424388260135181452028458471828554}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{7} - \frac{22945364279480469247135948880771298890493804881628510269755107246203443118227048024211728596121924629708451720445958977}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{6} - \frac{27294373523922364507553351074384471363101578366980891722957619022720164811471763371680983273855740269171079234379221664}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{5} - \frac{71840830049199531371218858131250807511108701985238633350142761984465722180704384945853954274125150836336074719219411224}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{4} - \frac{25393350776149787938682199967598317633924987713905928976115552747230218862462176144324130559995287534240415735294130980}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{3} + \frac{77653177090615605349052086198369391040135504722804054728238396443542181310260545002590892399654200888409047394208722385}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a^{2} + \frac{62677906303466270201246715168004386669386495208540296361781658820096155216112205947849892060593263950083396303314820746}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801} a - \frac{33228740018857389584500487412980934566213236552106785254557150678746352904363787026653907610990758589077726181620969038}{186618767852465449099168681937523436820734760394692188823875965847458278743578641171100574038054641725080411675785281801}$

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

Class group and class number

Trivial group, which has 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:  $35$
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:  \( 1731579138067769200000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_{12}$ (as 36T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 36
The 36 conjugacy class representatives for $C_3\times C_{12}$
Character table for $C_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.169.1, 3.3.13689.1, 3.3.13689.2, \(\Q(\zeta_{15})^+\), 6.6.820125.1, 6.6.3570125.1, 6.6.23423590125.1, 6.6.23423590125.2, 9.9.2565164201769.1, \(\Q(\zeta_{45})^+\), 12.12.1161460342986328125.1, 12.12.617247646136997205078125.2, 12.12.617247646136997205078125.1, 18.18.12851694105541388560018283203125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{3}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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
3Data not computed
$5$5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
13Data not computed