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Properties

Label 36.0.15722901389...4816.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{54}\cdot 3^{90}$
Root discriminant $44.09$
Ramified primes $2, 3$
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, -512, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 512*x^18 + 262144)
 
gp: K = bnfinit(x^36 - 512*x^18 + 262144, 1)
 

Normalized defining polynomial

\( x^{36} - 512 x^{18} + 262144 \)

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:  \(157229013891772345498599951736092754417390325814062078754816=2^{54}\cdot 3^{90}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.09$
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:  \(216=2^{3}\cdot 3^{3}\)
Dirichlet character group:    $\lbrace$$\chi_{216}(1,·)$, $\chi_{216}(131,·)$, $\chi_{216}(65,·)$, $\chi_{216}(137,·)$, $\chi_{216}(11,·)$, $\chi_{216}(115,·)$, $\chi_{216}(17,·)$, $\chi_{216}(19,·)$, $\chi_{216}(25,·)$, $\chi_{216}(155,·)$, $\chi_{216}(161,·)$, $\chi_{216}(35,·)$, $\chi_{216}(41,·)$, $\chi_{216}(43,·)$, $\chi_{216}(49,·)$, $\chi_{216}(179,·)$, $\chi_{216}(185,·)$, $\chi_{216}(59,·)$, $\chi_{216}(193,·)$, $\chi_{216}(83,·)$, $\chi_{216}(67,·)$, $\chi_{216}(73,·)$, $\chi_{216}(203,·)$, $\chi_{216}(209,·)$, $\chi_{216}(163,·)$, $\chi_{216}(139,·)$, $\chi_{216}(89,·)$, $\chi_{216}(91,·)$, $\chi_{216}(97,·)$, $\chi_{216}(187,·)$, $\chi_{216}(145,·)$, $\chi_{216}(107,·)$, $\chi_{216}(113,·)$, $\chi_{216}(211,·)$, $\chi_{216}(169,·)$, $\chi_{216}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{128} a^{14}$, $\frac{1}{128} a^{15}$, $\frac{1}{256} a^{16}$, $\frac{1}{256} a^{17}$, $\frac{1}{512} a^{18}$, $\frac{1}{512} a^{19}$, $\frac{1}{1024} a^{20}$, $\frac{1}{1024} a^{21}$, $\frac{1}{2048} a^{22}$, $\frac{1}{2048} a^{23}$, $\frac{1}{4096} a^{24}$, $\frac{1}{4096} a^{25}$, $\frac{1}{8192} a^{26}$, $\frac{1}{8192} a^{27}$, $\frac{1}{16384} a^{28}$, $\frac{1}{16384} a^{29}$, $\frac{1}{32768} a^{30}$, $\frac{1}{32768} a^{31}$, $\frac{1}{65536} a^{32}$, $\frac{1}{65536} a^{33}$, $\frac{1}{131072} a^{34}$, $\frac{1}{131072} a^{35}$

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:  \( -\frac{1}{131072} a^{34} + \frac{1}{256} a^{16} \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\zeta_{9})\), 6.0.3359232.1, 6.6.10077696.1, \(\Q(\zeta_{27})^+\), 12.0.101559956668416.1, \(\Q(\zeta_{27})\), 18.0.132173713091594538512566714368.2, 18.18.396521139274783615537700143104.1

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 $18^{2}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
2Data not computed
3Data not computed