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Properties

Label 36.0.79962223364...8784.3
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{18}\cdot 19^{34}$
Root discriminant $55.89$
Ramified primes $2, 3, 19$
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, 129140163, 0, 43046721, 0, 14348907, 0, 4782969, 0, 1594323, 0, 531441, 0, 177147, 0, 59049, 0, 19683, 0, 6561, 0, 2187, 0, 729, 0, 243, 0, 81, 0, 27, 0, 9, 0, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 3*x^34 + 9*x^32 + 27*x^30 + 81*x^28 + 243*x^26 + 729*x^24 + 2187*x^22 + 6561*x^20 + 19683*x^18 + 59049*x^16 + 177147*x^14 + 531441*x^12 + 1594323*x^10 + 4782969*x^8 + 14348907*x^6 + 43046721*x^4 + 129140163*x^2 + 387420489)
 
gp: K = bnfinit(x^36 + 3*x^34 + 9*x^32 + 27*x^30 + 81*x^28 + 243*x^26 + 729*x^24 + 2187*x^22 + 6561*x^20 + 19683*x^18 + 59049*x^16 + 177147*x^14 + 531441*x^12 + 1594323*x^10 + 4782969*x^8 + 14348907*x^6 + 43046721*x^4 + 129140163*x^2 + 387420489, 1)
 

Normalized defining polynomial

\( x^{36} + 3 x^{34} + 9 x^{32} + 27 x^{30} + 81 x^{28} + 243 x^{26} + 729 x^{24} + 2187 x^{22} + 6561 x^{20} + 19683 x^{18} + 59049 x^{16} + 177147 x^{14} + 531441 x^{12} + 1594323 x^{10} + 4782969 x^{8} + 14348907 x^{6} + 43046721 x^{4} + 129140163 x^{2} + 387420489 \)

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:  \(799622233646074762983150698451178476894456963777140963130998784=2^{36}\cdot 3^{18}\cdot 19^{34}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $55.89$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(228=2^{2}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{228}(1,·)$, $\chi_{228}(131,·)$, $\chi_{228}(11,·)$, $\chi_{228}(13,·)$, $\chi_{228}(143,·)$, $\chi_{228}(145,·)$, $\chi_{228}(23,·)$, $\chi_{228}(25,·)$, $\chi_{228}(155,·)$, $\chi_{228}(157,·)$, $\chi_{228}(35,·)$, $\chi_{228}(37,·)$, $\chi_{228}(167,·)$, $\chi_{228}(169,·)$, $\chi_{228}(47,·)$, $\chi_{228}(49,·)$, $\chi_{228}(179,·)$, $\chi_{228}(181,·)$, $\chi_{228}(59,·)$, $\chi_{228}(61,·)$, $\chi_{228}(191,·)$, $\chi_{228}(193,·)$, $\chi_{228}(71,·)$, $\chi_{228}(73,·)$, $\chi_{228}(203,·)$, $\chi_{228}(205,·)$, $\chi_{228}(83,·)$, $\chi_{228}(85,·)$, $\chi_{228}(215,·)$, $\chi_{228}(217,·)$, $\chi_{228}(97,·)$, $\chi_{228}(227,·)$, $\chi_{228}(107,·)$, $\chi_{228}(109,·)$, $\chi_{228}(119,·)$, $\chi_{228}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{3} a^{2}$, $\frac{1}{3} a^{3}$, $\frac{1}{9} a^{4}$, $\frac{1}{9} a^{5}$, $\frac{1}{27} a^{6}$, $\frac{1}{27} a^{7}$, $\frac{1}{81} a^{8}$, $\frac{1}{81} a^{9}$, $\frac{1}{243} a^{10}$, $\frac{1}{243} a^{11}$, $\frac{1}{729} a^{12}$, $\frac{1}{729} a^{13}$, $\frac{1}{2187} a^{14}$, $\frac{1}{2187} a^{15}$, $\frac{1}{6561} a^{16}$, $\frac{1}{6561} a^{17}$, $\frac{1}{19683} a^{18}$, $\frac{1}{19683} a^{19}$, $\frac{1}{59049} a^{20}$, $\frac{1}{59049} a^{21}$, $\frac{1}{177147} a^{22}$, $\frac{1}{177147} a^{23}$, $\frac{1}{531441} a^{24}$, $\frac{1}{531441} a^{25}$, $\frac{1}{1594323} a^{26}$, $\frac{1}{1594323} a^{27}$, $\frac{1}{4782969} a^{28}$, $\frac{1}{4782969} a^{29}$, $\frac{1}{14348907} a^{30}$, $\frac{1}{14348907} a^{31}$, $\frac{1}{43046721} a^{32}$, $\frac{1}{43046721} a^{33}$, $\frac{1}{129140163} a^{34}$, $\frac{1}{129140163} a^{35}$

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:  \( -\frac{1}{1594323} a^{26} \) (order $38$)
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

\(\Q(\sqrt{3}) \), \(\Q(\sqrt{-57}) \), \(\Q(\sqrt{-19}) \), 3.3.361.1, \(\Q(\sqrt{3}, \sqrt{-19})\), 6.6.225194688.1, 6.0.4278699072.1, 6.0.2476099.1, \(\Q(\zeta_{19})^+\), 12.0.18307265748733661184.3, 18.18.1488294338429317924379721203712.1, 18.0.28277592430157040563214702870528.1, \(\Q(\zeta_{19})\)

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 $18^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{18}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$

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
19Data not computed