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Properties

Label 36.0.14212734556...0601.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 7^{18}$
Root discriminant $41.24$
Ramified primes $3, 7$
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![262144, 0, 0, 0, 0, 0, 0, 0, 0, -2560, 0, 0, 0, 0, 0, 0, 0, 0, -487, 0, 0, 0, 0, 0, 0, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144)
 
gp: K = bnfinit(x^36 - 5*x^27 - 487*x^18 - 2560*x^9 + 262144, 1)
 

Normalized defining polynomial

\( x^{36} - 5 x^{27} - 487 x^{18} - 2560 x^{9} + 262144 \)

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:  \(14212734556341031905549296191351828189377245025195450200601=3^{90}\cdot 7^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.24$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(1,·)$, $\chi_{189}(134,·)$, $\chi_{189}(8,·)$, $\chi_{189}(139,·)$, $\chi_{189}(13,·)$, $\chi_{189}(146,·)$, $\chi_{189}(20,·)$, $\chi_{189}(22,·)$, $\chi_{189}(155,·)$, $\chi_{189}(29,·)$, $\chi_{189}(160,·)$, $\chi_{189}(34,·)$, $\chi_{189}(167,·)$, $\chi_{189}(41,·)$, $\chi_{189}(43,·)$, $\chi_{189}(176,·)$, $\chi_{189}(50,·)$, $\chi_{189}(181,·)$, $\chi_{189}(55,·)$, $\chi_{189}(188,·)$, $\chi_{189}(62,·)$, $\chi_{189}(64,·)$, $\chi_{189}(71,·)$, $\chi_{189}(76,·)$, $\chi_{189}(83,·)$, $\chi_{189}(85,·)$, $\chi_{189}(92,·)$, $\chi_{189}(97,·)$, $\chi_{189}(104,·)$, $\chi_{189}(106,·)$, $\chi_{189}(113,·)$, $\chi_{189}(118,·)$, $\chi_{189}(169,·)$, $\chi_{189}(148,·)$, $\chi_{189}(125,·)$, $\chi_{189}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{17} a^{18} + \frac{6}{17} a^{9} + \frac{2}{17}$, $\frac{1}{34} a^{19} - \frac{11}{34} a^{10} - \frac{15}{34} a$, $\frac{1}{68} a^{20} + \frac{23}{68} a^{11} - \frac{15}{68} a^{2}$, $\frac{1}{136} a^{21} - \frac{45}{136} a^{12} - \frac{15}{136} a^{3}$, $\frac{1}{272} a^{22} + \frac{91}{272} a^{13} + \frac{121}{272} a^{4}$, $\frac{1}{544} a^{23} + \frac{91}{544} a^{14} + \frac{121}{544} a^{5}$, $\frac{1}{1088} a^{24} - \frac{453}{1088} a^{15} - \frac{423}{1088} a^{6}$, $\frac{1}{2176} a^{25} + \frac{635}{2176} a^{16} + \frac{665}{2176} a^{7}$, $\frac{1}{4352} a^{26} - \frac{1541}{4352} a^{17} - \frac{1511}{4352} a^{8}$, $\frac{1}{4238848} a^{27} + \frac{205}{8704} a^{18} - \frac{2561}{8704} a^{9} + \frac{1943}{8279}$, $\frac{1}{8477696} a^{28} + \frac{205}{17408} a^{19} - \frac{2561}{17408} a^{10} - \frac{3168}{8279} a$, $\frac{1}{16955392} a^{29} + \frac{205}{34816} a^{20} - \frac{2561}{34816} a^{11} + \frac{5111}{16558} a^{2}$, $\frac{1}{33910784} a^{30} + \frac{205}{69632} a^{21} + \frac{32255}{69632} a^{12} + \frac{5111}{33116} a^{3}$, $\frac{1}{67821568} a^{31} + \frac{205}{139264} a^{22} - \frac{37377}{139264} a^{13} - \frac{28005}{66232} a^{4}$, $\frac{1}{135643136} a^{32} + \frac{205}{278528} a^{23} + \frac{101887}{278528} a^{14} + \frac{38227}{132464} a^{5}$, $\frac{1}{271286272} a^{33} + \frac{205}{557056} a^{24} - \frac{176641}{557056} a^{15} - \frac{94237}{264928} a^{6}$, $\frac{1}{542572544} a^{34} + \frac{205}{1114112} a^{25} - \frac{176641}{1114112} a^{16} + \frac{170691}{529856} a^{7}$, $\frac{1}{1085145088} a^{35} + \frac{205}{2228224} a^{26} + \frac{937471}{2228224} a^{17} - \frac{359165}{1059712} a^{8}$

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:  \( -\frac{4049}{542572544} a^{34} - \frac{93}{1114112} a^{25} + \frac{4049}{1114112} a^{16} + \frac{20245}{1059712} a^{7} \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\zeta_{9})\), 6.0.2250423.1, 6.6.6751269.1, \(\Q(\zeta_{27})^+\), 12.0.45579633110361.1, \(\Q(\zeta_{27})\), 18.0.39739057971752889532465351767.1, 18.18.119217173915258668597396055301.1

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 $18^{2}$ R $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{12}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
7Data not computed