↑

Properties

Label 36.0.41216642617...5504.2
Degree $36$
Signature $[0, 18]$
Discriminant $2^{72}\cdot 3^{90}$
Root discriminant $62.35$
Ramified primes $2, 3$
Class number $10298$ (GRH)
Class group $[10298]$ (GRH)
Galois group $C_2\times C_{18}$ (as 36T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 0, 0, 3321, 0, 0, 0, 77760, 0, 0, 0, 201654, 0, 0, 0, 175878, 0, 0, 0, 65367, 0, 0, 0, 12006, 0, 0, 0, 1143, 0, 0, 0, 54, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 + 54*x^32 + 1143*x^28 + 12006*x^24 + 65367*x^20 + 175878*x^16 + 201654*x^12 + 77760*x^8 + 3321*x^4 + 9)
 
gp: K = bnfinit(x^36 + 54*x^32 + 1143*x^28 + 12006*x^24 + 65367*x^20 + 175878*x^16 + 201654*x^12 + 77760*x^8 + 3321*x^4 + 9, 1)
 

Normalized defining polynomial

\( x^{36} + 54 x^{32} + 1143 x^{28} + 12006 x^{24} + 65367 x^{20} + 175878 x^{16} + 201654 x^{12} + 77760 x^{8} + 3321 x^{4} + 9 \)

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:  \(41216642617644769738384985747906299013992369570201489573102485504=2^{72}\cdot 3^{90}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.35$
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$$\chi_{216}(1,·)$, $\chi_{216}(131,·)$, $\chi_{216}(5,·)$, $\chi_{216}(7,·)$, $\chi_{216}(11,·)$, $\chi_{216}(145,·)$, $\chi_{216}(149,·)$, $\chi_{216}(151,·)$, $\chi_{216}(25,·)$, $\chi_{216}(155,·)$, $\chi_{216}(29,·)$, $\chi_{216}(31,·)$, $\chi_{216}(35,·)$, $\chi_{216}(169,·)$, $\chi_{216}(173,·)$, $\chi_{216}(175,·)$, $\chi_{216}(49,·)$, $\chi_{216}(179,·)$, $\chi_{216}(53,·)$, $\chi_{216}(55,·)$, $\chi_{216}(59,·)$, $\chi_{216}(193,·)$, $\chi_{216}(197,·)$, $\chi_{216}(199,·)$, $\chi_{216}(73,·)$, $\chi_{216}(203,·)$, $\chi_{216}(77,·)$, $\chi_{216}(79,·)$, $\chi_{216}(83,·)$, $\chi_{216}(97,·)$, $\chi_{216}(101,·)$, $\chi_{216}(103,·)$, $\chi_{216}(107,·)$, $\chi_{216}(121,·)$, $\chi_{216}(125,·)$, $\chi_{216}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{3} a^{18}$, $\frac{1}{3} a^{19}$, $\frac{1}{3} a^{20}$, $\frac{1}{3} a^{21}$, $\frac{1}{3} a^{22}$, $\frac{1}{3} a^{23}$, $\frac{1}{3} a^{24}$, $\frac{1}{3} a^{25}$, $\frac{1}{3} a^{26}$, $\frac{1}{3} a^{27}$, $\frac{1}{159} a^{28} + \frac{4}{53} a^{24} + \frac{2}{159} a^{20} - \frac{5}{53} a^{16} + \frac{21}{53} a^{12} - \frac{21}{53} a^{8} - \frac{26}{53} a^{4} + \frac{3}{53}$, $\frac{1}{159} a^{29} + \frac{4}{53} a^{25} + \frac{2}{159} a^{21} - \frac{5}{53} a^{17} + \frac{21}{53} a^{13} - \frac{21}{53} a^{9} - \frac{26}{53} a^{5} + \frac{3}{53} a$, $\frac{1}{159} a^{30} + \frac{4}{53} a^{26} + \frac{2}{159} a^{22} - \frac{5}{53} a^{18} + \frac{21}{53} a^{14} - \frac{21}{53} a^{10} - \frac{26}{53} a^{6} + \frac{3}{53} a^{2}$, $\frac{1}{159} a^{31} + \frac{4}{53} a^{27} + \frac{2}{159} a^{23} - \frac{5}{53} a^{19} + \frac{21}{53} a^{15} - \frac{21}{53} a^{11} - \frac{26}{53} a^{7} + \frac{3}{53} a^{3}$, $\frac{1}{7455571593376911} a^{32} + \frac{5630098654753}{7455571593376911} a^{28} + \frac{999833583149245}{7455571593376911} a^{24} - \frac{1008403650047437}{7455571593376911} a^{20} + \frac{1190232062959063}{2485190531125637} a^{16} - \frac{224175910915125}{2485190531125637} a^{12} + \frac{345618252810921}{2485190531125637} a^{8} - \frac{293657383928164}{2485190531125637} a^{4} + \frac{276490368671045}{2485190531125637}$, $\frac{1}{7455571593376911} a^{33} + \frac{5630098654753}{7455571593376911} a^{29} + \frac{999833583149245}{7455571593376911} a^{25} - \frac{1008403650047437}{7455571593376911} a^{21} + \frac{1190232062959063}{2485190531125637} a^{17} - \frac{224175910915125}{2485190531125637} a^{13} + \frac{345618252810921}{2485190531125637} a^{9} - \frac{293657383928164}{2485190531125637} a^{5} + \frac{276490368671045}{2485190531125637} a$, $\frac{1}{7455571593376911} a^{34} + \frac{5630098654753}{7455571593376911} a^{30} + \frac{999833583149245}{7455571593376911} a^{26} - \frac{1008403650047437}{7455571593376911} a^{22} + \frac{1085505657751552}{7455571593376911} a^{18} - \frac{224175910915125}{2485190531125637} a^{14} + \frac{345618252810921}{2485190531125637} a^{10} - \frac{293657383928164}{2485190531125637} a^{6} + \frac{276490368671045}{2485190531125637} a^{2}$, $\frac{1}{7455571593376911} a^{35} + \frac{5630098654753}{7455571593376911} a^{31} + \frac{999833583149245}{7455571593376911} a^{27} - \frac{1008403650047437}{7455571593376911} a^{23} + \frac{1085505657751552}{7455571593376911} a^{19} - \frac{224175910915125}{2485190531125637} a^{15} + \frac{345618252810921}{2485190531125637} a^{11} - \frac{293657383928164}{2485190531125637} a^{7} + \frac{276490368671045}{2485190531125637} a^{3}$

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

Class group and class number

$C_{10298}$, which has order $10298$ (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:  $17$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{11603575446050}{2485190531125637} a^{34} + \frac{1880042080186672}{7455571593376911} a^{30} + \frac{13267626299918010}{2485190531125637} a^{26} + \frac{139413118073095656}{2485190531125637} a^{22} + \frac{2278657102484531969}{7455571593376911} a^{18} + \frac{2046643213130018505}{2485190531125637} a^{14} + \frac{2355901502579224203}{2485190531125637} a^{10} + \frac{921736426969263426}{2485190531125637} a^{6} + \frac{48757664993510943}{2485190531125637} a^{2} \) (order $4$)
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:  \( 77725882445830.34 \) (assuming GRH)
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

\(\Q(\sqrt{-6}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{9})^+\), \(\Q(i, \sqrt{6})\), 6.0.10077696.1, 6.6.10077696.1, 6.0.419904.1, \(\Q(\zeta_{27})^+\), 12.0.6499837226778624.2, 18.0.396521139274783615537700143104.1, 18.18.396521139274783615537700143104.1, 18.0.258151783382020583032356864.7

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{4}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{36}$ $18^{2}$

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