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Properties

Label 36.0.61466712532...2768.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{99}\cdot 3^{88}$
Root discriminant $98.66$
Ramified primes $2, 3$
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![512, 0, 82944, 0, 2509056, 0, 25943040, 0, 131300352, 0, 385913088, 0, 726219648, 0, 928015488, 0, 834837696, 0, 540076864, 0, 253983600, 0, 87049728, 0, 21644784, 0, 3859488, 0, 483804, 0, 41280, 0, 2268, 0, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 72*x^34 + 2268*x^32 + 41280*x^30 + 483804*x^28 + 3859488*x^26 + 21644784*x^24 + 87049728*x^22 + 253983600*x^20 + 540076864*x^18 + 834837696*x^16 + 928015488*x^14 + 726219648*x^12 + 385913088*x^10 + 131300352*x^8 + 25943040*x^6 + 2509056*x^4 + 82944*x^2 + 512)
 
gp: K = bnfinit(x^36 + 72*x^34 + 2268*x^32 + 41280*x^30 + 483804*x^28 + 3859488*x^26 + 21644784*x^24 + 87049728*x^22 + 253983600*x^20 + 540076864*x^18 + 834837696*x^16 + 928015488*x^14 + 726219648*x^12 + 385913088*x^10 + 131300352*x^8 + 25943040*x^6 + 2509056*x^4 + 82944*x^2 + 512, 1)
 

Normalized defining polynomial

\( x^{36} + 72 x^{34} + 2268 x^{32} + 41280 x^{30} + 483804 x^{28} + 3859488 x^{26} + 21644784 x^{24} + 87049728 x^{22} + 253983600 x^{20} + 540076864 x^{18} + 834837696 x^{16} + 928015488 x^{14} + 726219648 x^{12} + 385913088 x^{10} + 131300352 x^{8} + 25943040 x^{6} + 2509056 x^{4} + 82944 x^{2} + 512 \)

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:  \(614667125325361522818798575155151578949632894783197825857500612833312768=2^{99}\cdot 3^{88}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $98.66$
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:  $2, 3$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(432=2^{4}\cdot 3^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{432}(1,·)$, $\chi_{432}(259,·)$, $\chi_{432}(385,·)$, $\chi_{432}(265,·)$, $\chi_{432}(139,·)$, $\chi_{432}(115,·)$, $\chi_{432}(145,·)$, $\chi_{432}(19,·)$, $\chi_{432}(25,·)$, $\chi_{432}(409,·)$, $\chi_{432}(283,·)$, $\chi_{432}(289,·)$, $\chi_{432}(163,·)$, $\chi_{432}(169,·)$, $\chi_{432}(427,·)$, $\chi_{432}(49,·)$, $\chi_{432}(307,·)$, $\chi_{432}(313,·)$, $\chi_{432}(187,·)$, $\chi_{432}(193,·)$, $\chi_{432}(67,·)$, $\chi_{432}(73,·)$, $\chi_{432}(331,·)$, $\chi_{432}(337,·)$, $\chi_{432}(211,·)$, $\chi_{432}(43,·)$, $\chi_{432}(217,·)$, $\chi_{432}(91,·)$, $\chi_{432}(97,·)$, $\chi_{432}(355,·)$, $\chi_{432}(361,·)$, $\chi_{432}(235,·)$, $\chi_{432}(241,·)$, $\chi_{432}(403,·)$, $\chi_{432}(121,·)$, $\chi_{432}(379,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{16} a^{18}$, $\frac{1}{16} a^{19}$, $\frac{1}{32} a^{20}$, $\frac{1}{32} a^{21}$, $\frac{1}{32} a^{22}$, $\frac{1}{32} a^{23}$, $\frac{1}{64} a^{24}$, $\frac{1}{64} a^{25}$, $\frac{1}{64} a^{26}$, $\frac{1}{64} a^{27}$, $\frac{1}{128} a^{28}$, $\frac{1}{128} a^{29}$, $\frac{1}{128} a^{30}$, $\frac{1}{128} a^{31}$, $\frac{1}{256} a^{32}$, $\frac{1}{256} a^{33}$, $\frac{1}{297759589389948674706983682304} a^{34} + \frac{752806421719491764250795}{74439897347487168676745920576} a^{32} - \frac{267462755570500681269572343}{148879794694974337353491841152} a^{30} + \frac{388130429856080003531770233}{148879794694974337353491841152} a^{28} - \frac{13902309635611554231779925}{18609974336871792169186480144} a^{26} - \frac{204977696037520416636662025}{74439897347487168676745920576} a^{24} - \frac{274975411712143804261463313}{18609974336871792169186480144} a^{22} + \frac{74810138934737676241498651}{9304987168435896084593240072} a^{20} + \frac{69689843263160089159454337}{18609974336871792169186480144} a^{18} - \frac{4710875171771951318711127}{2326246792108974021148310018} a^{16} + \frac{280008149313563757264140565}{4652493584217948042296620036} a^{14} - \frac{312620464290862636635226}{1163123396054487010574155009} a^{12} + \frac{286028307548987129144555097}{4652493584217948042296620036} a^{10} + \frac{194010913546221209037158579}{2326246792108974021148310018} a^{8} + \frac{514433235935538473497130381}{2326246792108974021148310018} a^{6} - \frac{263381583102536673919037585}{2326246792108974021148310018} a^{4} + \frac{330697418327063979740389904}{1163123396054487010574155009} a^{2} - \frac{504622935428449945356808045}{1163123396054487010574155009}$, $\frac{1}{297759589389948674706983682304} a^{35} + \frac{752806421719491764250795}{74439897347487168676745920576} a^{33} - \frac{267462755570500681269572343}{148879794694974337353491841152} a^{31} + \frac{388130429856080003531770233}{148879794694974337353491841152} a^{29} - \frac{13902309635611554231779925}{18609974336871792169186480144} a^{27} - \frac{204977696037520416636662025}{74439897347487168676745920576} a^{25} - \frac{274975411712143804261463313}{18609974336871792169186480144} a^{23} + \frac{74810138934737676241498651}{9304987168435896084593240072} a^{21} + \frac{69689843263160089159454337}{18609974336871792169186480144} a^{19} - \frac{4710875171771951318711127}{2326246792108974021148310018} a^{17} + \frac{280008149313563757264140565}{4652493584217948042296620036} a^{15} - \frac{312620464290862636635226}{1163123396054487010574155009} a^{13} + \frac{286028307548987129144555097}{4652493584217948042296620036} a^{11} + \frac{194010913546221209037158579}{2326246792108974021148310018} a^{9} + \frac{514433235935538473497130381}{2326246792108974021148310018} a^{7} - \frac{263381583102536673919037585}{2326246792108974021148310018} a^{5} + \frac{330697418327063979740389904}{1163123396054487010574155009} a^{3} - \frac{504622935428449945356808045}{1163123396054487010574155009} a$

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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.2048.2, 6.6.3359232.1, \(\Q(\zeta_{27})^+\), 12.0.369768517790072832.1, 18.18.132173713091594538512566714368.1

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 $36$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{4}$ $36$ $36$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $36$ $18^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $18^{2}$ $36$ $18^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ $36$

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