↑

Properties

Label 36.36.4965241496...0000.1
Degree $36$
Signature $[36, 0]$
Discriminant $2^{36}\cdot 3^{88}\cdot 5^{27}$
Root discriminant $98.07$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 54, -369, -18696, 162621, 445752, -5288694, -2685978, 57031200, 6111114, -299222487, -5409414, 878655639, 585108, -1551012606, 1973988, 1720942785, -1460934, -1241952110, 488160, 599479200, -84042, -197837730, 6804, 45285660, -162, -7227216, -4, 799020, 0, -59832, 0, 2889, 0, -81, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 81*x^34 + 2889*x^32 - 59832*x^30 + 799020*x^28 - 4*x^27 - 7227216*x^26 - 162*x^25 + 45285660*x^24 + 6804*x^23 - 197837730*x^22 - 84042*x^21 + 599479200*x^20 + 488160*x^19 - 1241952110*x^18 - 1460934*x^17 + 1720942785*x^16 + 1973988*x^15 - 1551012606*x^14 + 585108*x^13 + 878655639*x^12 - 5409414*x^11 - 299222487*x^10 + 6111114*x^9 + 57031200*x^8 - 2685978*x^7 - 5288694*x^6 + 445752*x^5 + 162621*x^4 - 18696*x^3 - 369*x^2 + 54*x + 1)
 
gp: K = bnfinit(x^36 - 81*x^34 + 2889*x^32 - 59832*x^30 + 799020*x^28 - 4*x^27 - 7227216*x^26 - 162*x^25 + 45285660*x^24 + 6804*x^23 - 197837730*x^22 - 84042*x^21 + 599479200*x^20 + 488160*x^19 - 1241952110*x^18 - 1460934*x^17 + 1720942785*x^16 + 1973988*x^15 - 1551012606*x^14 + 585108*x^13 + 878655639*x^12 - 5409414*x^11 - 299222487*x^10 + 6111114*x^9 + 57031200*x^8 - 2685978*x^7 - 5288694*x^6 + 445752*x^5 + 162621*x^4 - 18696*x^3 - 369*x^2 + 54*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 81 x^{34} + 2889 x^{32} - 59832 x^{30} + 799020 x^{28} - 4 x^{27} - 7227216 x^{26} - 162 x^{25} + 45285660 x^{24} + 6804 x^{23} - 197837730 x^{22} - 84042 x^{21} + 599479200 x^{20} + 488160 x^{19} - 1241952110 x^{18} - 1460934 x^{17} + 1720942785 x^{16} + 1973988 x^{15} - 1551012606 x^{14} + 585108 x^{13} + 878655639 x^{12} - 5409414 x^{11} - 299222487 x^{10} + 6111114 x^{9} + 57031200 x^{8} - 2685978 x^{7} - 5288694 x^{6} + 445752 x^{5} + 162621 x^{4} - 18696 x^{3} - 369 x^{2} + 54 x + 1 \)

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:  \(496524149651212084672932834388009377566802432000000000000000000000000000=2^{36}\cdot 3^{88}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.07$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(540=2^{2}\cdot 3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{540}(1,·)$, $\chi_{540}(43,·)$, $\chi_{540}(7,·)$, $\chi_{540}(523,·)$, $\chi_{540}(529,·)$, $\chi_{540}(403,·)$, $\chi_{540}(409,·)$, $\chi_{540}(283,·)$, $\chi_{540}(289,·)$, $\chi_{540}(163,·)$, $\chi_{540}(421,·)$, $\chi_{540}(169,·)$, $\chi_{540}(427,·)$, $\chi_{540}(301,·)$, $\chi_{540}(49,·)$, $\chi_{540}(307,·)$, $\chi_{540}(181,·)$, $\chi_{540}(187,·)$, $\chi_{540}(61,·)$, $\chi_{540}(67,·)$, $\chi_{540}(463,·)$, $\chi_{540}(469,·)$, $\chi_{540}(343,·)$, $\chi_{540}(349,·)$, $\chi_{540}(223,·)$, $\chi_{540}(481,·)$, $\chi_{540}(229,·)$, $\chi_{540}(103,·)$, $\chi_{540}(361,·)$, $\chi_{540}(487,·)$, $\chi_{540}(109,·)$, $\chi_{540}(367,·)$, $\chi_{540}(241,·)$, $\chi_{540}(247,·)$, $\chi_{540}(121,·)$, $\chi_{540}(127,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{2554566011968592384631575221925161} a^{34} + \frac{418699151222525923299225189801157}{2554566011968592384631575221925161} a^{33} + \frac{594227639895242336302355844338645}{2554566011968592384631575221925161} a^{32} - \frac{1020342406788394935324525825960823}{2554566011968592384631575221925161} a^{31} - \frac{120026083753932293278556582952042}{2554566011968592384631575221925161} a^{30} + \frac{49082700021851853672238678671783}{2554566011968592384631575221925161} a^{29} - \frac{784792332635008016409355483729425}{2554566011968592384631575221925161} a^{28} + \frac{227906753931108736551116070501070}{2554566011968592384631575221925161} a^{27} - \frac{1268294928900941728936347917046707}{2554566011968592384631575221925161} a^{26} + \frac{890174542286928054211931486799352}{2554566011968592384631575221925161} a^{25} + \frac{1114157411822034610469254516581878}{2554566011968592384631575221925161} a^{24} - \frac{118543485495564850697299220264551}{2554566011968592384631575221925161} a^{23} + \frac{727757771854073705217780508169946}{2554566011968592384631575221925161} a^{22} + \frac{1239240447604663655800109092130483}{2554566011968592384631575221925161} a^{21} + \frac{1023581657189419544475901673765227}{2554566011968592384631575221925161} a^{20} - \frac{15439765294130184885808159708717}{2554566011968592384631575221925161} a^{19} - \frac{230801002414608788085240942215888}{2554566011968592384631575221925161} a^{18} + \frac{691178571573774473166843376131727}{2554566011968592384631575221925161} a^{17} - \frac{499209111754132617198220632726480}{2554566011968592384631575221925161} a^{16} + \frac{7209754533691312456013437021097}{2554566011968592384631575221925161} a^{15} + \frac{446082319621982464535904791695277}{2554566011968592384631575221925161} a^{14} + \frac{517350130297071349750489429672851}{2554566011968592384631575221925161} a^{13} - \frac{356092235234519399994987870361258}{2554566011968592384631575221925161} a^{12} - \frac{220458261080134352699525529454263}{2554566011968592384631575221925161} a^{11} + \frac{161359619557599278516141019079991}{2554566011968592384631575221925161} a^{10} - \frac{284257507273096674811415903079297}{2554566011968592384631575221925161} a^{9} + \frac{68700677719129852162722918218518}{2554566011968592384631575221925161} a^{8} + \frac{1034469391742864311587226743308010}{2554566011968592384631575221925161} a^{7} + \frac{375843956758678443406975081373541}{2554566011968592384631575221925161} a^{6} + \frac{876694776260231415700603273534917}{2554566011968592384631575221925161} a^{5} + \frac{790378332703578647910656148252122}{2554566011968592384631575221925161} a^{4} - \frac{113660876281103158226703367895147}{2554566011968592384631575221925161} a^{3} + \frac{1206282934615697771255551301243735}{2554566011968592384631575221925161} a^{2} - \frac{395477751327894741460287234908950}{2554566011968592384631575221925161} a + \frac{729462989208320359200527104348346}{2554566011968592384631575221925161}$, $\frac{1}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{35} + \frac{21137927792738529620070707679690976623083413278184}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{34} - \frac{182771850258329270249185883403952322699662827338916711835641846339691827980923645365}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{33} + \frac{70029921058221354717743749837948981568083068958471191216119525699995891415072073203}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{32} + \frac{63110468066888327067791915786846795258954011472342223039195128112458569650297816878}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{31} + \frac{117372895308880687317623986885643571394346250617996521381051969258733568923413162900}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{30} - \frac{133129845408005912763510024865191086324020948632092740639274101781894749886842040051}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{29} + \frac{119819351981291019051262010796107608468232243382207497513205486472474065180798682970}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{28} - \frac{157636655654646600812412383698490339289481144302865375934976223288896366647098704217}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{27} + \frac{42060465883687248257137214247605149589485629250520758355978566293142974289249871381}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{26} + \frac{182346221817586799083948304700954947512941763157662449400654429089939478778742360189}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{25} - \frac{98377679558793706326895799216239343155360332642717801977977021450632226586875000893}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{24} + \frac{87518699321998302162402911488010770206190912698712288701853151824384097087605973672}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{23} - \frac{114803463297761037449757026910185470059866387854442999034962663470316778125457565161}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{22} - \frac{63978436177501231800916999721698743513712751180254055771811514767337714363832473292}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{21} - \frac{59944709137136092818164341579490266309108353534439591319701536488856667359293633196}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{20} - \frac{119649237417405855431257900469154868571116510226519208821604870208758392646913061967}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{19} - \frac{94736787029992975225999410473826416804711304157518437436218094849170467389405676146}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{18} - \frac{162544305608433268281045306003141454658351989213546087132066887121414246293872484262}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{17} + \frac{21475705270659838599614774951987491372478085645883570398551269054167605834635667351}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{16} - \frac{2607818102938535964521438932156047828135304716335127965496836734999049998271939179}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{15} + \frac{50787764299050995048712218345768475318677068914020950966591748803190569664015245120}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{14} - \frac{76544429930667935055943025431450213501292064341004337907061657196345938671843396153}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{13} + \frac{144251289915146527423475736482437594106445505762226176248645608774251740599334700374}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{12} + \frac{164221204048004492564778668254849616997368856416206005386944522597827469112740890548}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{11} - \frac{77238426579166278170995400450826027311409968420372224548063569872428123916606759294}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{10} + \frac{181199456054877540034896472981530293475463009994978931149071609395921262140143252389}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{9} - \frac{106707508733810191182052334182899350895076076555710763505627308303950687260193382225}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{8} - \frac{28395261046701456040125520333379718101451648995128618644113812056777040118461629364}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{7} + \frac{174007030531808195191194932641859849613558734412966592219920779005724805568610256861}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{6} - \frac{158538688914225392115727978230322703091712197617326733716787406312039845019138981809}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{5} - \frac{183980088922405790167073929926291241648096545300850738681813838539992497813833934277}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{4} + \frac{124680618233698409450711218125450188160343944111968174499083318365241659731201444217}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{3} + \frac{85338386910336164922709935019737986295566594868822727273143117341504502270432951614}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a^{2} + \frac{165296363837080612548358570011461003575668317715829207850344640295213407567959894138}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921} a + \frac{162418580186917804555995554800013436365745728425008482613827435558467286730022163060}{370009211682965298610228821483659846090106962366846658692126345590645564203321181921}$

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:  \( 2134359545650035000000000 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{20})^+\), 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.12.344373768000000000.1, 18.18.1923380668327365689220703125.1

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 R R R $36$ $18^{2}$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $36$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $36$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
2Data not computed
3Data not computed
5Data not computed