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Properties

Label 36.0.20526871859...6272.1
Degree $36$
Signature $[0, 18]$
Discriminant $2^{36}\cdot 3^{18}\cdot 37^{35}$
Root discriminant $115.94$
Ramified primes $2, 3, 37$
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![14334558093, 0, 272356603767, 0, 1543354088013, 0, 4115610901368, 0, 6287738877090, 0, 6135308843706, 0, 4090205895804, 0, 1947717093240, 0, 682655745915, 0, 179646248925, 0, 35929249785, 0, 5491163340, 0, 640635723, 0, 56580363, 0, 3716280, 0, 175824, 0, 5661, 0, 111, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 111*x^34 + 5661*x^32 + 175824*x^30 + 3716280*x^28 + 56580363*x^26 + 640635723*x^24 + 5491163340*x^22 + 35929249785*x^20 + 179646248925*x^18 + 682655745915*x^16 + 1947717093240*x^14 + 4090205895804*x^12 + 6135308843706*x^10 + 6287738877090*x^8 + 4115610901368*x^6 + 1543354088013*x^4 + 272356603767*x^2 + 14334558093)
 
gp: K = bnfinit(x^36 + 111*x^34 + 5661*x^32 + 175824*x^30 + 3716280*x^28 + 56580363*x^26 + 640635723*x^24 + 5491163340*x^22 + 35929249785*x^20 + 179646248925*x^18 + 682655745915*x^16 + 1947717093240*x^14 + 4090205895804*x^12 + 6135308843706*x^10 + 6287738877090*x^8 + 4115610901368*x^6 + 1543354088013*x^4 + 272356603767*x^2 + 14334558093, 1)
 

Normalized defining polynomial

\( x^{36} + 111 x^{34} + 5661 x^{32} + 175824 x^{30} + 3716280 x^{28} + 56580363 x^{26} + 640635723 x^{24} + 5491163340 x^{22} + 35929249785 x^{20} + 179646248925 x^{18} + 682655745915 x^{16} + 1947717093240 x^{14} + 4090205895804 x^{12} + 6135308843706 x^{10} + 6287738877090 x^{8} + 4115610901368 x^{6} + 1543354088013 x^{4} + 272356603767 x^{2} + 14334558093 \)

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:  \(205268718591960451664107537071777407379898873740754655345797836302054326272=2^{36}\cdot 3^{18}\cdot 37^{35}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $115.94$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(444=2^{2}\cdot 3\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{444}(1,·)$, $\chi_{444}(131,·)$, $\chi_{444}(397,·)$, $\chi_{444}(143,·)$, $\chi_{444}(145,·)$, $\chi_{444}(23,·)$, $\chi_{444}(25,·)$, $\chi_{444}(157,·)$, $\chi_{444}(289,·)$, $\chi_{444}(35,·)$, $\chi_{444}(167,·)$, $\chi_{444}(169,·)$, $\chi_{444}(431,·)$, $\chi_{444}(433,·)$, $\chi_{444}(179,·)$, $\chi_{444}(181,·)$, $\chi_{444}(311,·)$, $\chi_{444}(59,·)$, $\chi_{444}(191,·)$, $\chi_{444}(49,·)$, $\chi_{444}(73,·)$, $\chi_{444}(203,·)$, $\chi_{444}(335,·)$, $\chi_{444}(337,·)$, $\chi_{444}(85,·)$, $\chi_{444}(347,·)$, $\chi_{444}(349,·)$, $\chi_{444}(227,·)$, $\chi_{444}(229,·)$, $\chi_{444}(361,·)$, $\chi_{444}(239,·)$, $\chi_{444}(373,·)$, $\chi_{444}(119,·)$, $\chi_{444}(121,·)$, $\chi_{444}(251,·)$, $\chi_{444}(383,·)$$\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:  \( -1 \) (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

\(\Q(\sqrt{37}) \), 3.3.1369.1, 4.0.7294032.1, 6.6.69343957.1, 9.9.3512479453921.1, 12.0.531259171951520321851392.1, \(\Q(\zeta_{37})^+\)

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 $36$ $18^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{6}$ $36$ $36$ $36$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{9}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{12}$ $18^{2}$ $36$

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
37Data not computed