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Properties

Label 18.0.20296288612...9728.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 13^{15}$
Root discriminant $373.80$
Ramified primes $2, 3, 13$
Class number $155704548$ (GRH)
Class group $[2, 77852274]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7725011394048, 0, 16914992230656, 0, 4337177495040, 0, 428157265536, 0, 21172612032, 0, 588128112, 0, 9596496, 0, 91260, 0, 468, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 468*x^16 + 91260*x^14 + 9596496*x^12 + 588128112*x^10 + 21172612032*x^8 + 428157265536*x^6 + 4337177495040*x^4 + 16914992230656*x^2 + 7725011394048)
 
gp: K = bnfinit(x^18 + 468*x^16 + 91260*x^14 + 9596496*x^12 + 588128112*x^10 + 21172612032*x^8 + 428157265536*x^6 + 4337177495040*x^4 + 16914992230656*x^2 + 7725011394048, 1)
 

Normalized defining polynomial

\( x^{18} + 468 x^{16} + 91260 x^{14} + 9596496 x^{12} + 588128112 x^{10} + 21172612032 x^{8} + 428157265536 x^{6} + 4337177495040 x^{4} + 16914992230656 x^{2} + 7725011394048 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-20296288612744454761949989145129188927343689728=-\,2^{27}\cdot 3^{45}\cdot 13^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $373.80$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2808=2^{3}\cdot 3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2808}(1,·)$, $\chi_{2808}(1733,·)$, $\chi_{2808}(1537,·)$, $\chi_{2808}(841,·)$, $\chi_{2808}(1037,·)$, $\chi_{2808}(2573,·)$, $\chi_{2808}(1873,·)$, $\chi_{2808}(2713,·)$, $\chi_{2808}(601,·)$, $\chi_{2808}(797,·)$, $\chi_{2808}(101,·)$, $\chi_{2808}(1637,·)$, $\chi_{2808}(2473,·)$, $\chi_{2808}(2669,·)$, $\chi_{2808}(1777,·)$, $\chi_{2808}(1973,·)$, $\chi_{2808}(937,·)$, $\chi_{2808}(701,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{104} a^{6}$, $\frac{1}{104} a^{7}$, $\frac{1}{208} a^{8}$, $\frac{1}{314704} a^{9} + \frac{9}{12104} a^{7} + \frac{351}{6052} a^{5} + \frac{531}{3026} a^{3} + \frac{104}{1513} a$, $\frac{1}{2219922016} a^{10} + \frac{1081097}{554980504} a^{8} + \frac{733829}{554980504} a^{6} + \frac{1828235}{21345404} a^{4} - \frac{2364715}{10672702} a^{2} + \frac{966}{3527}$, $\frac{1}{2219922016} a^{11} + \frac{11}{85381616} a^{9} - \frac{1417641}{554980504} a^{7} + \frac{177544}{5336351} a^{5} + \frac{1119010}{5336351} a^{3} + \frac{734996}{5336351} a$, $\frac{1}{57717972416} a^{12} + \frac{736129}{1109961008} a^{8} - \frac{253199}{69372563} a^{6} + \frac{659610}{5336351} a^{4} - \frac{1127653}{5336351} a^{2} - \frac{45}{3527}$, $\frac{1}{57717972416} a^{13} - \frac{39}{42690808} a^{9} + \frac{611661}{138745126} a^{7} + \frac{46095}{21345404} a^{5} + \frac{13615}{119918} a^{3} - \frac{2022043}{5336351} a$, $\frac{1}{115435944832} a^{14} + \frac{1671099}{1109961008} a^{8} - \frac{2387299}{554980504} a^{6} - \frac{2008123}{21345404} a^{4} + \frac{1514897}{10672702} a^{2} - \frac{982}{3527}$, $\frac{1}{115435944832} a^{15} - \frac{699}{1109961008} a^{9} - \frac{271339}{277490252} a^{7} - \frac{1810611}{21345404} a^{5} - \frac{375575}{10672702} a^{3} + \frac{746825}{5336351} a$, $\frac{1}{230871889664} a^{16} + \frac{643595}{1109961008} a^{8} - \frac{383941}{138745126} a^{6} + \frac{2172801}{21345404} a^{4} + \frac{1039925}{5336351} a^{2} + \frac{1577}{3527}$, $\frac{1}{230871889664} a^{17} + \frac{1681}{1109961008} a^{9} - \frac{482697}{138745126} a^{7} + \frac{987729}{21345404} a^{5} - \frac{2586371}{10672702} a^{3} - \frac{336843}{5336351} a$

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

Class group and class number

$C_{2}\times C_{77852274}$, which has order $155704548$ (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:  $8$
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:  \( 59652214.53290313 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-78}) \), \(\Q(\zeta_{9})^+\), 6.0.22140698112.9, 9.9.151470380950257681.1

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 $18$ $18$ $18$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
13Data not computed