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Properties

Label 36.0.12274208875...2041.1
Degree $36$
Signature $[0, 18]$
Discriminant $3^{90}\cdot 17^{18}$
Root discriminant $64.27$
Ramified primes $3, 17$
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![68719476736, 0, 0, 0, 0, 0, 0, 0, 0, 1230241792, 0, 0, 0, 0, 0, 0, 0, 0, 22286393, 0, 0, 0, 0, 0, 0, 0, 0, -4693, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^36 - 4693*x^27 + 22286393*x^18 + 1230241792*x^9 + 68719476736)
 
gp: K = bnfinit(x^36 - 4693*x^27 + 22286393*x^18 + 1230241792*x^9 + 68719476736, 1)
 

Normalized defining polynomial

\( x^{36} - 4693 x^{27} + 22286393 x^{18} + 1230241792 x^{9} + 68719476736 \)

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:  \(122742088752587853242976134017548025824688225088579999619212332041=3^{90}\cdot 17^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $64.27$
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:  $3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(459=3^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{459}(256,·)$, $\chi_{459}(1,·)$, $\chi_{459}(392,·)$, $\chi_{459}(137,·)$, $\chi_{459}(271,·)$, $\chi_{459}(16,·)$, $\chi_{459}(407,·)$, $\chi_{459}(152,·)$, $\chi_{459}(409,·)$, $\chi_{459}(154,·)$, $\chi_{459}(290,·)$, $\chi_{459}(35,·)$, $\chi_{459}(424,·)$, $\chi_{459}(169,·)$, $\chi_{459}(305,·)$, $\chi_{459}(50,·)$, $\chi_{459}(307,·)$, $\chi_{459}(52,·)$, $\chi_{459}(443,·)$, $\chi_{459}(188,·)$, $\chi_{459}(322,·)$, $\chi_{459}(67,·)$, $\chi_{459}(458,·)$, $\chi_{459}(203,·)$, $\chi_{459}(205,·)$, $\chi_{459}(341,·)$, $\chi_{459}(86,·)$, $\chi_{459}(220,·)$, $\chi_{459}(356,·)$, $\chi_{459}(101,·)$, $\chi_{459}(358,·)$, $\chi_{459}(103,·)$, $\chi_{459}(239,·)$, $\chi_{459}(373,·)$, $\chi_{459}(118,·)$, $\chi_{459}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{1165} a^{18} + \frac{566}{1165} a^{9} - \frac{19}{1165}$, $\frac{1}{4660} a^{19} + \frac{1731}{4660} a^{10} - \frac{19}{4660} a$, $\frac{1}{18640} a^{20} - \frac{7589}{18640} a^{11} - \frac{4679}{18640} a^{2}$, $\frac{1}{74560} a^{21} + \frac{11051}{74560} a^{12} - \frac{4679}{74560} a^{3}$, $\frac{1}{298240} a^{22} - \frac{138069}{298240} a^{13} + \frac{144441}{298240} a^{4}$, $\frac{1}{1192960} a^{23} + \frac{160171}{1192960} a^{14} + \frac{144441}{1192960} a^{5}$, $\frac{1}{4771840} a^{24} - \frac{1032789}{4771840} a^{15} - \frac{1048519}{4771840} a^{6}$, $\frac{1}{19087360} a^{25} - \frac{5804629}{19087360} a^{16} + \frac{3723321}{19087360} a^{7}$, $\frac{1}{76349440} a^{26} + \frac{32370091}{76349440} a^{17} + \frac{22810681}{76349440} a^{8}$, $\frac{1}{6806214500679680} a^{27} - \frac{23293}{305397760} a^{18} - \frac{122159103}{305397760} a^{9} - \frac{2077090889}{5192729569}$, $\frac{1}{27224858002718720} a^{28} - \frac{23293}{1221591040} a^{19} - \frac{122159103}{1221591040} a^{10} - \frac{2077090889}{20770918276} a$, $\frac{1}{108899432010874880} a^{29} - \frac{23293}{4886364160} a^{20} - \frac{1343750143}{4886364160} a^{11} - \frac{2077090889}{83083673104} a^{2}$, $\frac{1}{435597728043499520} a^{30} - \frac{23293}{19545456640} a^{21} - \frac{6230114303}{19545456640} a^{12} + \frac{164090255319}{332334692416} a^{3}$, $\frac{1}{1742390912173998080} a^{31} - \frac{23293}{78181826560} a^{22} + \frac{32860798977}{78181826560} a^{13} - \frac{168244437097}{1329338769664} a^{4}$, $\frac{1}{6969563648695992320} a^{32} - \frac{23293}{312727306240} a^{23} - \frac{123502854143}{312727306240} a^{14} - \frac{168244437097}{5317355078656} a^{5}$, $\frac{1}{27878254594783969280} a^{33} - \frac{23293}{1250909224960} a^{24} - \frac{123502854143}{1250909224960} a^{15} - \frac{5485599515753}{21269420314624} a^{6}$, $\frac{1}{111513018379135877120} a^{34} - \frac{23293}{5003636899840} a^{25} + \frac{1127406370817}{5003636899840} a^{16} + \frac{15783820798871}{85077681258496} a^{7}$, $\frac{1}{446052073516543508480} a^{35} - \frac{23293}{20014547599360} a^{26} + \frac{6131043270657}{20014547599360} a^{17} - \frac{69293860459625}{340310725033984} a^{8}$

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:  \( -\frac{233059073}{1742390912173998080} a^{31} + \frac{49661}{78181826560} a^{22} - \frac{235418369}{78181826560} a^{13} + \frac{50852864}{25963647845} a^{4} \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\zeta_{9})\), 6.6.32234193.1, 6.0.96702579.1, \(\Q(\zeta_{27})^+\), 12.0.9351388785251241.1, \(\Q(\zeta_{27})\), 18.18.116781890125989356502353933497857.1, 18.0.350345670377968069507061800493571.1

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 $18^{2}$ R $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{4}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{12}$ $18^{2}$ $18^{2}$ $18^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{6}$ $18^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{4}$ $18^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{18}$ $18^{2}$

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
3Data not computed
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$