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Properties

Label 36.0.55320041279...4912.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{99}\cdot 3^{90}$
Root discriminant $104.87$
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![4608, 0, 248832, 0, 4167936, 0, 33039360, 0, 146810880, 0, 405022464, 0, 740057472, 0, 933856128, 0, 836167104, 0, 540202944, 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 + 540202944*x^18 + 836167104*x^16 + 933856128*x^14 + 740057472*x^12 + 405022464*x^10 + 146810880*x^8 + 33039360*x^6 + 4167936*x^4 + 248832*x^2 + 4608)
 
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 + 540202944*x^18 + 836167104*x^16 + 933856128*x^14 + 740057472*x^12 + 405022464*x^10 + 146810880*x^8 + 33039360*x^6 + 4167936*x^4 + 248832*x^2 + 4608, 1)
 

Normalized defining polynomial

\( x^{36} + 72 x^{34} + 2268 x^{32} + 41280 x^{30} + 483804 x^{28} + 3859488 x^{26} + 21644784 x^{24} + 87049728 x^{22} + 253983600 x^{20} + 540202944 x^{18} + 836167104 x^{16} + 933856128 x^{14} + 740057472 x^{12} + 405022464 x^{10} + 146810880 x^{8} + 33039360 x^{6} + 4167936 x^{4} + 248832 x^{2} + 4608 \)

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:  \(5532004127928253705369187176396364210546696053048780432717505515499814912=2^{99}\cdot 3^{90}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $104.87$
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$$\chi_{432}(1,·)$, $\chi_{432}(389,·)$, $\chi_{432}(385,·)$, $\chi_{432}(265,·)$, $\chi_{432}(269,·)$, $\chi_{432}(145,·)$, $\chi_{432}(149,·)$, $\chi_{432}(25,·)$, $\chi_{432}(409,·)$, $\chi_{432}(29,·)$, $\chi_{432}(5,·)$, $\chi_{432}(289,·)$, $\chi_{432}(293,·)$, $\chi_{432}(169,·)$, $\chi_{432}(173,·)$, $\chi_{432}(413,·)$, $\chi_{432}(49,·)$, $\chi_{432}(53,·)$, $\chi_{432}(313,·)$, $\chi_{432}(317,·)$, $\chi_{432}(193,·)$, $\chi_{432}(197,·)$, $\chi_{432}(73,·)$, $\chi_{432}(77,·)$, $\chi_{432}(337,·)$, $\chi_{432}(341,·)$, $\chi_{432}(217,·)$, $\chi_{432}(221,·)$, $\chi_{432}(97,·)$, $\chi_{432}(101,·)$, $\chi_{432}(361,·)$, $\chi_{432}(365,·)$, $\chi_{432}(241,·)$, $\chi_{432}(245,·)$, $\chi_{432}(121,·)$, $\chi_{432}(125,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{2} a^{6}$, $\frac{1}{2} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{4} a^{9}$, $\frac{1}{4} a^{10}$, $\frac{1}{4} a^{11}$, $\frac{1}{8} a^{12}$, $\frac{1}{8} a^{13}$, $\frac{1}{8} a^{14}$, $\frac{1}{8} a^{15}$, $\frac{1}{16} a^{16}$, $\frac{1}{16} a^{17}$, $\frac{1}{48} a^{18}$, $\frac{1}{48} a^{19}$, $\frac{1}{96} a^{20}$, $\frac{1}{96} a^{21}$, $\frac{1}{96} a^{22}$, $\frac{1}{96} a^{23}$, $\frac{1}{192} a^{24}$, $\frac{1}{192} a^{25}$, $\frac{1}{192} a^{26}$, $\frac{1}{192} a^{27}$, $\frac{1}{384} a^{28}$, $\frac{1}{384} a^{29}$, $\frac{1}{384} a^{30}$, $\frac{1}{384} a^{31}$, $\frac{1}{662784} a^{32} - \frac{109}{165696} a^{30} - \frac{337}{331392} a^{28} + \frac{397}{165696} a^{26} + \frac{1}{863} a^{24} + \frac{5}{41424} a^{22} - \frac{49}{13808} a^{20} - \frac{133}{13808} a^{18} + \frac{383}{13808} a^{16} - \frac{399}{6904} a^{14} - \frac{95}{1726} a^{12} + \frac{78}{863} a^{10} - \frac{63}{863} a^{8} + \frac{273}{1726} a^{6} - \frac{167}{863} a^{4} + \frac{211}{863} a^{2} - \frac{207}{863}$, $\frac{1}{662784} a^{33} - \frac{109}{165696} a^{31} - \frac{337}{331392} a^{29} + \frac{397}{165696} a^{27} + \frac{1}{863} a^{25} + \frac{5}{41424} a^{23} - \frac{49}{13808} a^{21} - \frac{133}{13808} a^{19} + \frac{383}{13808} a^{17} - \frac{399}{6904} a^{15} - \frac{95}{1726} a^{13} + \frac{78}{863} a^{11} - \frac{63}{863} a^{9} + \frac{273}{1726} a^{7} - \frac{167}{863} a^{5} + \frac{211}{863} a^{3} - \frac{207}{863} a$, $\frac{1}{6162346530732423844306176} a^{34} - \frac{763030662079597181}{1027057755122070640717696} a^{32} + \frac{2923435859564969221403}{3081173265366211922153088} a^{30} - \frac{833560083227853660361}{770293316341552980538272} a^{28} - \frac{848886422231959117779}{513528877561035320358848} a^{26} + \frac{432682614938729579941}{513528877561035320358848} a^{24} + \frac{109991477070310272175}{256764438780517660179424} a^{22} + \frac{1062998553771274779363}{256764438780517660179424} a^{20} - \frac{1914077898989161566647}{192573329085388245134568} a^{18} - \frac{142665337353871119991}{32095554847564707522428} a^{16} + \frac{1698203216085092503575}{32095554847564707522428} a^{14} + \frac{1715912971386473175929}{64191109695129415044856} a^{12} - \frac{918472319959002528785}{16047777423782353761214} a^{10} + \frac{565629521922957946665}{16047777423782353761214} a^{8} + \frac{2001319721478938007714}{8023888711891176880607} a^{6} - \frac{1817892719642209002927}{8023888711891176880607} a^{4} - \frac{3455054067858305387238}{8023888711891176880607} a^{2} - \frac{2446810146890980221194}{8023888711891176880607}$, $\frac{1}{6162346530732423844306176} a^{35} - \frac{763030662079597181}{1027057755122070640717696} a^{33} + \frac{2923435859564969221403}{3081173265366211922153088} a^{31} - \frac{833560083227853660361}{770293316341552980538272} a^{29} - \frac{848886422231959117779}{513528877561035320358848} a^{27} + \frac{432682614938729579941}{513528877561035320358848} a^{25} + \frac{109991477070310272175}{256764438780517660179424} a^{23} + \frac{1062998553771274779363}{256764438780517660179424} a^{21} - \frac{1914077898989161566647}{192573329085388245134568} a^{19} - \frac{142665337353871119991}{32095554847564707522428} a^{17} + \frac{1698203216085092503575}{32095554847564707522428} a^{15} + \frac{1715912971386473175929}{64191109695129415044856} a^{13} - \frac{918472319959002528785}{16047777423782353761214} a^{11} + \frac{565629521922957946665}{16047777423782353761214} a^{9} + \frac{2001319721478938007714}{8023888711891176880607} a^{7} - \frac{1817892719642209002927}{8023888711891176880607} a^{5} - \frac{3455054067858305387238}{8023888711891176880607} a^{3} - \frac{2446810146890980221194}{8023888711891176880607} a$

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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{9})^+\), 4.0.18432.2, 6.6.3359232.1, \(\Q(\zeta_{27})^+\), 12.0.3327916660110655488.1, 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 $36$ $18^{2}$ $36$ $36$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $36$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ $36$ $18^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ $36$

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