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Properties

Label 36.36.8300243481...3125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{54}\cdot 5^{27}\cdot 7^{24}$
Root discriminant $63.58$
Ramified primes $3, 5, 7$
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![1, -24, -360, 2501, 21066, -47619, -351580, 423228, 2769012, -2131789, -12459300, 6642171, 35213992, -13456680, -65933622, 18269987, 84384807, -16934940, -75203780, 10844637, 47178711, -4829603, -20954832, 1493037, 6595406, -315834, -1463265, 44322, 225594, -3897, -23491, 192, 1566, -4, -60, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 60*x^34 - 4*x^33 + 1566*x^32 + 192*x^31 - 23491*x^30 - 3897*x^29 + 225594*x^28 + 44322*x^27 - 1463265*x^26 - 315834*x^25 + 6595406*x^24 + 1493037*x^23 - 20954832*x^22 - 4829603*x^21 + 47178711*x^20 + 10844637*x^19 - 75203780*x^18 - 16934940*x^17 + 84384807*x^16 + 18269987*x^15 - 65933622*x^14 - 13456680*x^13 + 35213992*x^12 + 6642171*x^11 - 12459300*x^10 - 2131789*x^9 + 2769012*x^8 + 423228*x^7 - 351580*x^6 - 47619*x^5 + 21066*x^4 + 2501*x^3 - 360*x^2 - 24*x + 1)
 
gp: K = bnfinit(x^36 - 60*x^34 - 4*x^33 + 1566*x^32 + 192*x^31 - 23491*x^30 - 3897*x^29 + 225594*x^28 + 44322*x^27 - 1463265*x^26 - 315834*x^25 + 6595406*x^24 + 1493037*x^23 - 20954832*x^22 - 4829603*x^21 + 47178711*x^20 + 10844637*x^19 - 75203780*x^18 - 16934940*x^17 + 84384807*x^16 + 18269987*x^15 - 65933622*x^14 - 13456680*x^13 + 35213992*x^12 + 6642171*x^11 - 12459300*x^10 - 2131789*x^9 + 2769012*x^8 + 423228*x^7 - 351580*x^6 - 47619*x^5 + 21066*x^4 + 2501*x^3 - 360*x^2 - 24*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 60 x^{34} - 4 x^{33} + 1566 x^{32} + 192 x^{31} - 23491 x^{30} - 3897 x^{29} + 225594 x^{28} + 44322 x^{27} - 1463265 x^{26} - 315834 x^{25} + 6595406 x^{24} + 1493037 x^{23} - 20954832 x^{22} - 4829603 x^{21} + 47178711 x^{20} + 10844637 x^{19} - 75203780 x^{18} - 16934940 x^{17} + 84384807 x^{16} + 18269987 x^{15} - 65933622 x^{14} - 13456680 x^{13} + 35213992 x^{12} + 6642171 x^{11} - 12459300 x^{10} - 2131789 x^{9} + 2769012 x^{8} + 423228 x^{7} - 351580 x^{6} - 47619 x^{5} + 21066 x^{4} + 2501 x^{3} - 360 x^{2} - 24 x + 1 \)

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:  \(83002434816200303192485381664744316138553796477615833282470703125=3^{54}\cdot 5^{27}\cdot 7^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $63.58$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(256,·)$, $\chi_{315}(1,·)$, $\chi_{315}(2,·)$, $\chi_{315}(4,·)$, $\chi_{315}(263,·)$, $\chi_{315}(8,·)$, $\chi_{315}(137,·)$, $\chi_{315}(128,·)$, $\chi_{315}(23,·)$, $\chi_{315}(16,·)$, $\chi_{315}(274,·)$, $\chi_{315}(46,·)$, $\chi_{315}(151,·)$, $\chi_{315}(158,·)$, $\chi_{315}(32,·)$, $\chi_{315}(289,·)$, $\chi_{315}(169,·)$, $\chi_{315}(302,·)$, $\chi_{315}(53,·)$, $\chi_{315}(184,·)$, $\chi_{315}(64,·)$, $\chi_{315}(197,·)$, $\chi_{315}(79,·)$, $\chi_{315}(211,·)$, $\chi_{315}(212,·)$, $\chi_{315}(214,·)$, $\chi_{315}(218,·)$, $\chi_{315}(92,·)$, $\chi_{315}(226,·)$, $\chi_{315}(233,·)$, $\chi_{315}(106,·)$, $\chi_{315}(107,·)$, $\chi_{315}(109,·)$, $\chi_{315}(113,·)$, $\chi_{315}(242,·)$, $\chi_{315}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $\frac{1}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{35} - \frac{112343084766339435154879424149013719411692509848826997423228954668375498990002855}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{34} - \frac{130029799355701447404969041799768187242008465273676668551118767628086581451204297}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{33} + \frac{40684314360517983446400335309870269645742291179284979559811412081120959760674113}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{32} + \frac{341470941146110758430586288909670007619768470522915963326342215001321709722995258}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{31} + \frac{27913237760667798850396725193792424105980170351375881897913912822025493300043925}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{30} + \frac{65470932008252850174613795729753009901342316123358751235963213406557692595494418}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{29} - \frac{6191790610093435693690936005614911979410431019914123397783991105148322051560553}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{28} - \frac{240891568900262645912014071227613523589643212677005779585225424560583476042349082}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{27} + \frac{349851801935976103430479120733807379130866028828103909922572468646698309099625805}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{26} - \frac{277707983042902521336319937894930515007152839221510273167304395026245172712464110}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{25} + \frac{273141008587516490654746492553900138464195052151487255875729232880060767014232282}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{24} - \frac{203841038705377355297410171408256662912293971887226010732420426640635727503404390}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{23} + \frac{336107052111840827121225039079598263412113622071219291850751426622904068689777921}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{22} + \frac{211799898460722910334219635131035146577530406524608634474874005517272837355574978}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{21} + \frac{346739037885554233485587237070267462168492706921613617908133833677736316163513940}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{20} + \frac{150849128698512504007870627353740412585133774398122661717415578101068766505247440}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{19} - \frac{104256743752216848182486483741913011509877278534968763873864607111480011433271773}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{18} - \frac{788157314278894144387306562178116079530218366388879492160857317697354388655055}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{17} - \frac{265789611033066123162008637345937514913343407834754410367226494460140939285537712}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{16} - \frac{250343119874610915333814334667846970400725951738504400359757700772869116638425148}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{15} - \frac{81506866815774924125329335289333617233557155100089653714885786280495288813543183}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{14} + \frac{281700086977414700672358850385855155908796529874032755631731910851574106520767051}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{13} + \frac{310988550943431476459927165897635484501848201430408101792078496448655708796705147}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{12} + \frac{208503305099733866014758966610831865530104948495564577215457315047778623329198242}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{11} + \frac{258127854240919426391896924765537669193176587281058273807803091173249074017636242}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{10} + \frac{284438955801052567288907820091005390873605634777995261219216550839811369301230863}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{9} - \frac{305328689339842170556596694593551733820017497913484601966508094242460530936033887}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{8} + \frac{243618201357030392594318184466624449452522561959464202709799503735002525633179668}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{7} + \frac{177202740718304081258979033354195027798679162057598183817966985113910278834264669}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{6} - \frac{82368270980980140313218013529563320720179372285248569433220293415460405496657147}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{5} - \frac{30379868247053015591731978645775418697681496629203418437204900067527760701325509}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{4} + \frac{14591911670992234687590915576200628787206265070437782945192047197117262953302390}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{3} + \frac{79053756948870756994522066628350994589785903467372090937320191283610048012149081}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a^{2} + \frac{270884137223942958939188112682200917188581896404115138266826902908960896347049848}{716301498946229402047582046491260714440498460249578640570052123759949582517175469} a - \frac{41457655494271782079154726947277958479505151716065697648612064645632590007327377}{716301498946229402047582046491260714440498460249578640570052123759949582517175469}$

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:  \( 982275678999294500000 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{15})^+\), 6.6.820125.1, 6.6.1969120125.2, 6.6.1969120125.1, 6.6.300125.1, 9.9.62523502209.1, \(\Q(\zeta_{45})^+\), 12.12.4362113325015017578125.2, 12.12.4362113325015017578125.1, 12.12.8208085798828125.1, 18.18.7635133454060210702501953125.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{12}$

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}$
7Data not computed