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Properties

Label 36.0.98150669491...3921.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 13^{18}$
Root discriminant $56.20$
Ramified primes $3, 13$
Class number Not computed
Class group Not computed
Galois group $C_2\times C_{18}$ (as 36T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![387420489, 0, 0, 0, 0, 0, 0, 0, 0, 35626230, 0, 0, 0, 0, 0, 0, 0, 0, 3295783, 0, 0, 0, 0, 0, 0, 0, 0, -1810, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 1810*x^27 + 3295783*x^18 + 35626230*x^9 + 387420489)
 
gp: K = bnfinit(x^36 - 1810*x^27 + 3295783*x^18 + 35626230*x^9 + 387420489, 1)
 

Normalized defining polynomial

\( x^{36} - 1810 x^{27} + 3295783 x^{18} + 35626230 x^{9} + 387420489 \)

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:  \(981506694911161790066526032030527389205087841773474128847803921=3^{90}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $56.20$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(351=3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{351}(1,·)$, $\chi_{351}(259,·)$, $\chi_{351}(14,·)$, $\chi_{351}(272,·)$, $\chi_{351}(274,·)$, $\chi_{351}(131,·)$, $\chi_{351}(235,·)$, $\chi_{351}(25,·)$, $\chi_{351}(155,·)$, $\chi_{351}(157,·)$, $\chi_{351}(287,·)$, $\chi_{351}(38,·)$, $\chi_{351}(40,·)$, $\chi_{351}(92,·)$, $\chi_{351}(298,·)$, $\chi_{351}(53,·)$, $\chi_{351}(311,·)$, $\chi_{351}(313,·)$, $\chi_{351}(181,·)$, $\chi_{351}(64,·)$, $\chi_{351}(194,·)$, $\chi_{351}(196,·)$, $\chi_{351}(326,·)$, $\chi_{351}(77,·)$, $\chi_{351}(79,·)$, $\chi_{351}(209,·)$, $\chi_{351}(142,·)$, $\chi_{351}(220,·)$, $\chi_{351}(350,·)$, $\chi_{351}(337,·)$, $\chi_{351}(233,·)$, $\chi_{351}(103,·)$, $\chi_{351}(116,·)$, $\chi_{351}(118,·)$, $\chi_{351}(248,·)$, $\chi_{351}(170,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{508} a^{18} - \frac{143}{508} a^{9} + \frac{129}{508}$, $\frac{1}{1524} a^{19} + \frac{365}{1524} a^{10} + \frac{637}{1524} a$, $\frac{1}{4572} a^{20} + \frac{1889}{4572} a^{11} + \frac{2161}{4572} a^{2}$, $\frac{1}{13716} a^{21} + \frac{1889}{13716} a^{12} + \frac{2161}{13716} a^{3}$, $\frac{1}{41148} a^{22} + \frac{15605}{41148} a^{13} + \frac{2161}{41148} a^{4}$, $\frac{1}{123444} a^{23} + \frac{56753}{123444} a^{14} + \frac{2161}{123444} a^{5}$, $\frac{1}{370332} a^{24} - \frac{66691}{370332} a^{15} + \frac{2161}{370332} a^{6}$, $\frac{1}{1110996} a^{25} - \frac{437023}{1110996} a^{16} + \frac{2161}{1110996} a^{7}$, $\frac{1}{3332988} a^{26} + \frac{673973}{3332988} a^{17} + \frac{2161}{3332988} a^{8}$, $\frac{1}{32954415568812} a^{27} - \frac{3295}{9998964} a^{18} + \frac{2361961}{9998964} a^{9} + \frac{184564753}{837128882}$, $\frac{1}{98863246706436} a^{28} - \frac{3295}{29996892} a^{19} + \frac{12360925}{29996892} a^{10} + \frac{340564545}{837128882} a$, $\frac{1}{296589740119308} a^{29} - \frac{3295}{89990676} a^{20} + \frac{12360925}{89990676} a^{11} + \frac{1177693427}{2511386646} a^{2}$, $\frac{1}{889769220357924} a^{30} - \frac{3295}{269972028} a^{21} + \frac{12360925}{269972028} a^{12} + \frac{3689080073}{7534159938} a^{3}$, $\frac{1}{2669307661073772} a^{31} - \frac{3295}{809916084} a^{22} + \frac{282332953}{809916084} a^{13} - \frac{3845079865}{22602479814} a^{4}$, $\frac{1}{8007922983221316} a^{32} - \frac{3295}{2429748252} a^{23} + \frac{1092249037}{2429748252} a^{14} + \frac{18757399949}{67807439442} a^{5}$, $\frac{1}{24023768949663948} a^{33} - \frac{3295}{7289244756} a^{24} + \frac{3521997289}{7289244756} a^{15} + \frac{86564839391}{203422318326} a^{6}$, $\frac{1}{72071306848991844} a^{34} - \frac{3295}{21867734268} a^{25} - \frac{3767247467}{21867734268} a^{16} - \frac{116857478935}{610266954978} a^{7}$, $\frac{1}{216213920546975532} a^{35} - \frac{3295}{65603202804} a^{26} - \frac{3767247467}{65603202804} a^{17} + \frac{493409476043}{1830800864934} a^{8}$

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:  \( -\frac{1048895}{49431623353218} a^{28} + \frac{1159}{29996892} a^{19} - \frac{2117473}{29996892} a^{10} + \frac{7604199}{1674257764} a \) (order $54$)
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

$C_2\times C_{18}$ (as 36T2):

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 $C_2\times C_{18}$
Character table for $C_2\times C_{18}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-39}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\zeta_{9})\), 6.6.14414517.1, 6.0.43243551.1, \(\Q(\zeta_{27})^+\), 12.0.1870004703089601.1, \(\Q(\zeta_{27})\), 18.18.10443002414754749649962321483613.1, 18.0.31329007244264248949886964450839.1

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 $18^{2}$ R $18^{2}$ $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
3Data not computed
13Data not computed