↑

Properties

Label 36.36.6502839601...3125.1
Degree $36$
Signature $[36, 0]$
Discriminant $3^{90}\cdot 5^{27}$
Root discriminant $52.12$
Ramified primes $3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_{36}$ (as 36T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 36, -324, -849, 8721, 7398, -93024, -30897, 523260, 72931, -1790712, -107406, 4056234, 104652, -6418656, -69768, 7354710, 32319, -6249100, -10395, 3996135, 2277, -1937520, -324, 712530, 27, -197316, -1, 40455, 0, -5952, 0, 594, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 36*x^34 + 594*x^32 - 5952*x^30 + 40455*x^28 - x^27 - 197316*x^26 + 27*x^25 + 712530*x^24 - 324*x^23 - 1937520*x^22 + 2277*x^21 + 3996135*x^20 - 10395*x^19 - 6249100*x^18 + 32319*x^17 + 7354710*x^16 - 69768*x^15 - 6418656*x^14 + 104652*x^13 + 4056234*x^12 - 107406*x^11 - 1790712*x^10 + 72931*x^9 + 523260*x^8 - 30897*x^7 - 93024*x^6 + 7398*x^5 + 8721*x^4 - 849*x^3 - 324*x^2 + 36*x + 1)
 
gp: K = bnfinit(x^36 - 36*x^34 + 594*x^32 - 5952*x^30 + 40455*x^28 - x^27 - 197316*x^26 + 27*x^25 + 712530*x^24 - 324*x^23 - 1937520*x^22 + 2277*x^21 + 3996135*x^20 - 10395*x^19 - 6249100*x^18 + 32319*x^17 + 7354710*x^16 - 69768*x^15 - 6418656*x^14 + 104652*x^13 + 4056234*x^12 - 107406*x^11 - 1790712*x^10 + 72931*x^9 + 523260*x^8 - 30897*x^7 - 93024*x^6 + 7398*x^5 + 8721*x^4 - 849*x^3 - 324*x^2 + 36*x + 1, 1)
 

Normalized defining polynomial

\( x^{36} - 36 x^{34} + 594 x^{32} - 5952 x^{30} + 40455 x^{28} - x^{27} - 197316 x^{26} + 27 x^{25} + 712530 x^{24} - 324 x^{23} - 1937520 x^{22} + 2277 x^{21} + 3996135 x^{20} - 10395 x^{19} - 6249100 x^{18} + 32319 x^{17} + 7354710 x^{16} - 69768 x^{15} - 6418656 x^{14} + 104652 x^{13} + 4056234 x^{12} - 107406 x^{11} - 1790712 x^{10} + 72931 x^{9} + 523260 x^{8} - 30897 x^{7} - 93024 x^{6} + 7398 x^{5} + 8721 x^{4} - 849 x^{3} - 324 x^{2} + 36 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $36$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[36, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(65028396011052373244549315269863064140390224754810333251953125=3^{90}\cdot 5^{27}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.12$
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:  $3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(135=3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{135}(128,·)$, $\chi_{135}(1,·)$, $\chi_{135}(2,·)$, $\chi_{135}(4,·)$, $\chi_{135}(8,·)$, $\chi_{135}(16,·)$, $\chi_{135}(17,·)$, $\chi_{135}(19,·)$, $\chi_{135}(23,·)$, $\chi_{135}(31,·)$, $\chi_{135}(32,·)$, $\chi_{135}(34,·)$, $\chi_{135}(38,·)$, $\chi_{135}(46,·)$, $\chi_{135}(47,·)$, $\chi_{135}(49,·)$, $\chi_{135}(53,·)$, $\chi_{135}(61,·)$, $\chi_{135}(62,·)$, $\chi_{135}(64,·)$, $\chi_{135}(68,·)$, $\chi_{135}(76,·)$, $\chi_{135}(77,·)$, $\chi_{135}(79,·)$, $\chi_{135}(83,·)$, $\chi_{135}(91,·)$, $\chi_{135}(92,·)$, $\chi_{135}(94,·)$, $\chi_{135}(98,·)$, $\chi_{135}(106,·)$, $\chi_{135}(107,·)$, $\chi_{135}(109,·)$, $\chi_{135}(113,·)$, $\chi_{135}(121,·)$, $\chi_{135}(122,·)$, $\chi_{135}(124,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $35$
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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 31157843880432600000 \) (assuming GRH)
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

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{15})^+\), 6.6.820125.1, \(\Q(\zeta_{27})^+\), \(\Q(\zeta_{45})^+\), 18.18.1923380668327365689220703125.1

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 $36$ R R $36$ $18^{2}$ $36$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $36$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{3}$ $18^{2}$ $36$ $36$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{4}$

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
5Data not computed