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Properties

Label 36.0.44687173468...0000.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{90}\cdot 5^{27}$
Root discriminant $104.25$
Ramified primes $2, 3, 5$
Class number Not computed
Class group Not computed
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![17578125, 0, 474609375, 0, 4461328125, 0, 20334375000, 0, 52017187500, 0, 82533515625, 0, 86985140625, 0, 63846984375, 0, 33719118750, 0, 13095318750, 0, 3789888750, 0, 822251250, 0, 133603875, 0, 16119000, 0, 1417950, 0, 88050, 0, 3645, 0, 90, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 90*x^34 + 3645*x^32 + 88050*x^30 + 1417950*x^28 + 16119000*x^26 + 133603875*x^24 + 822251250*x^22 + 3789888750*x^20 + 13095318750*x^18 + 33719118750*x^16 + 63846984375*x^14 + 86985140625*x^12 + 82533515625*x^10 + 52017187500*x^8 + 20334375000*x^6 + 4461328125*x^4 + 474609375*x^2 + 17578125)
 
gp: K = bnfinit(x^36 + 90*x^34 + 3645*x^32 + 88050*x^30 + 1417950*x^28 + 16119000*x^26 + 133603875*x^24 + 822251250*x^22 + 3789888750*x^20 + 13095318750*x^18 + 33719118750*x^16 + 63846984375*x^14 + 86985140625*x^12 + 82533515625*x^10 + 52017187500*x^8 + 20334375000*x^6 + 4461328125*x^4 + 474609375*x^2 + 17578125, 1)
 

Normalized defining polynomial

\( x^{36} + 90 x^{34} + 3645 x^{32} + 88050 x^{30} + 1417950 x^{28} + 16119000 x^{26} + 133603875 x^{24} + 822251250 x^{22} + 3789888750 x^{20} + 13095318750 x^{18} + 33719118750 x^{16} + 63846984375 x^{14} + 86985140625 x^{12} + 82533515625 x^{10} + 52017187500 x^{8} + 20334375000 x^{6} + 4461328125 x^{4} + 474609375 x^{2} + 17578125 \)

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:  \(4468717346860908762056395509492084398101221888000000000000000000000000000=2^{36}\cdot 3^{90}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.25$
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:  $2, 3, 5$
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$$\chi_{540}(1,·)$, $\chi_{540}(263,·)$, $\chi_{540}(407,·)$, $\chi_{540}(143,·)$, $\chi_{540}(529,·)$, $\chi_{540}(23,·)$, $\chi_{540}(409,·)$, $\chi_{540}(287,·)$, $\chi_{540}(289,·)$, $\chi_{540}(421,·)$, $\chi_{540}(167,·)$, $\chi_{540}(169,·)$, $\chi_{540}(301,·)$, $\chi_{540}(47,·)$, $\chi_{540}(49,·)$, $\chi_{540}(181,·)$, $\chi_{540}(443,·)$, $\chi_{540}(61,·)$, $\chi_{540}(323,·)$, $\chi_{540}(203,·)$, $\chi_{540}(83,·)$, $\chi_{540}(469,·)$, $\chi_{540}(527,·)$, $\chi_{540}(347,·)$, $\chi_{540}(349,·)$, $\chi_{540}(481,·)$, $\chi_{540}(227,·)$, $\chi_{540}(229,·)$, $\chi_{540}(361,·)$, $\chi_{540}(107,·)$, $\chi_{540}(109,·)$, $\chi_{540}(241,·)$, $\chi_{540}(467,·)$, $\chi_{540}(503,·)$, $\chi_{540}(121,·)$, $\chi_{540}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{125} a^{14}$, $\frac{1}{125} a^{15}$, $\frac{1}{625} a^{16}$, $\frac{1}{625} a^{17}$, $\frac{1}{1875} a^{18}$, $\frac{1}{1875} a^{19}$, $\frac{1}{9375} a^{20}$, $\frac{1}{9375} a^{21}$, $\frac{1}{9375} a^{22}$, $\frac{1}{9375} a^{23}$, $\frac{1}{46875} a^{24}$, $\frac{1}{46875} a^{25}$, $\frac{1}{46875} a^{26}$, $\frac{1}{46875} a^{27}$, $\frac{1}{234375} a^{28}$, $\frac{1}{234375} a^{29}$, $\frac{1}{234375} a^{30}$, $\frac{1}{234375} a^{31}$, $\frac{1}{505078125} a^{32} - \frac{199}{101015625} a^{30} + \frac{172}{101015625} a^{28} + \frac{52}{6734375} a^{26} - \frac{7}{20203125} a^{24} - \frac{14}{1346875} a^{22} + \frac{58}{1346875} a^{20} - \frac{64}{269375} a^{18} + \frac{18}{269375} a^{16} + \frac{16}{10775} a^{14} + \frac{17}{53875} a^{12} + \frac{119}{10775} a^{10} - \frac{164}{10775} a^{8} + \frac{102}{2155} a^{6} + \frac{2}{431} a^{4} + \frac{83}{431} a^{2} + \frac{145}{431}$, $\frac{1}{505078125} a^{33} - \frac{199}{101015625} a^{31} + \frac{172}{101015625} a^{29} + \frac{52}{6734375} a^{27} - \frac{7}{20203125} a^{25} - \frac{14}{1346875} a^{23} + \frac{58}{1346875} a^{21} - \frac{64}{269375} a^{19} + \frac{18}{269375} a^{17} + \frac{16}{10775} a^{15} + \frac{17}{53875} a^{13} + \frac{119}{10775} a^{11} - \frac{164}{10775} a^{9} + \frac{102}{2155} a^{7} + \frac{2}{431} a^{5} + \frac{83}{431} a^{3} + \frac{145}{431} a$, $\frac{1}{5612760971484375} a^{34} - \frac{752728}{1122552194296875} a^{32} + \frac{637092272}{1122552194296875} a^{30} - \frac{3656342}{374184064765625} a^{28} + \frac{457973209}{74836812953125} a^{26} - \frac{47824727}{224510438859375} a^{24} - \frac{197007572}{44902087771875} a^{22} + \frac{748437637}{14967362590625} a^{20} + \frac{671251816}{8980417554375} a^{18} + \frac{935640359}{2993472518125} a^{16} + \frac{1012807872}{598694503625} a^{14} - \frac{49281216}{23947780145} a^{12} + \frac{2356512101}{119738900725} a^{10} + \frac{2329935526}{119738900725} a^{8} - \frac{2152770664}{23947780145} a^{6} + \frac{169919429}{23947780145} a^{4} + \frac{78972754}{4789556029} a^{2} - \frac{1788688379}{4789556029}$, $\frac{1}{5612760971484375} a^{35} - \frac{752728}{1122552194296875} a^{33} + \frac{637092272}{1122552194296875} a^{31} - \frac{3656342}{374184064765625} a^{29} + \frac{457973209}{74836812953125} a^{27} - \frac{47824727}{224510438859375} a^{25} - \frac{197007572}{44902087771875} a^{23} + \frac{748437637}{14967362590625} a^{21} + \frac{671251816}{8980417554375} a^{19} + \frac{935640359}{2993472518125} a^{17} + \frac{1012807872}{598694503625} a^{15} - \frac{49281216}{23947780145} a^{13} + \frac{2356512101}{119738900725} a^{11} + \frac{2329935526}{119738900725} a^{9} - \frac{2152770664}{23947780145} a^{7} + \frac{169919429}{23947780145} a^{5} + \frac{78972754}{4789556029} a^{3} - \frac{1788688379}{4789556029} a$

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:  $17$
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:  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

$C_{36}$ (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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 4.0.18000.1, 6.6.820125.1, \(\Q(\zeta_{27})^+\), 12.0.3099363912000000000.1, 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 $36$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $36$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $18^{2}$ $36$ $36$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ $18^{2}$

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