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Properties

Label 18.10.6489745507...0553.1
Degree $18$
Signature $[10, 4]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{4}$
Root discriminant $58.53$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-224019, -73305, 1477521, -122463, -1230822, -539649, -429672, -148635, 2514, -15539, 8847, 3957, 86, 759, -105, -54, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 54*x^15 - 105*x^14 + 759*x^13 + 86*x^12 + 3957*x^11 + 8847*x^10 - 15539*x^9 + 2514*x^8 - 148635*x^7 - 429672*x^6 - 539649*x^5 - 1230822*x^4 - 122463*x^3 + 1477521*x^2 - 73305*x - 224019)
 
gp: K = bnfinit(x^18 - 3*x^17 - 54*x^15 - 105*x^14 + 759*x^13 + 86*x^12 + 3957*x^11 + 8847*x^10 - 15539*x^9 + 2514*x^8 - 148635*x^7 - 429672*x^6 - 539649*x^5 - 1230822*x^4 - 122463*x^3 + 1477521*x^2 - 73305*x - 224019, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 54 x^{15} - 105 x^{14} + 759 x^{13} + 86 x^{12} + 3957 x^{11} + 8847 x^{10} - 15539 x^{9} + 2514 x^{8} - 148635 x^{7} - 429672 x^{6} - 539649 x^{5} - 1230822 x^{4} - 122463 x^{3} + 1477521 x^{2} - 73305 x - 224019 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(64897455079312633154135966180553=3^{24}\cdot 73^{5}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $58.53$
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:  $3, 73, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} + \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{4}{9} a^{9} + \frac{1}{3} a^{8} + \frac{4}{9} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{3} a^{11} - \frac{4}{9} a^{10} + \frac{1}{3} a^{9} + \frac{4}{9} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{20049626208849591047109845578674617427070368090229389} a^{17} + \frac{862225951513079230964019203990955654151616757532670}{20049626208849591047109845578674617427070368090229389} a^{16} + \frac{223813137443221150390039199074376055777319878233392}{20049626208849591047109845578674617427070368090229389} a^{15} - \frac{679818053336043976036086582927301412363792517695560}{6683208736283197015703281859558205809023456030076463} a^{14} + \frac{180857739944601951948666756943417609128908348289991}{2227736245427732338567760619852735269674485343358821} a^{13} + \frac{1081343214583585824785527933554899027556573493517347}{6683208736283197015703281859558205809023456030076463} a^{12} + \frac{6723676027548206441123240422097529381871046532362120}{20049626208849591047109845578674617427070368090229389} a^{11} + \frac{1547034558344378395047895514724054957756586048193401}{20049626208849591047109845578674617427070368090229389} a^{10} + \frac{6543733858405434012311639889474902988292041121397261}{20049626208849591047109845578674617427070368090229389} a^{9} - \frac{5783856347016865137047219820589147593533048470631030}{20049626208849591047109845578674617427070368090229389} a^{8} - \frac{1711968634897274103002066988324864295195283955879706}{20049626208849591047109845578674617427070368090229389} a^{7} - \frac{4479162955461930543069654375866092784507530355682527}{20049626208849591047109845578674617427070368090229389} a^{6} + \frac{1064768223467586080047517807933097064634504704449522}{2227736245427732338567760619852735269674485343358821} a^{5} + \frac{506031199102514121097055035600880170769149794468219}{2227736245427732338567760619852735269674485343358821} a^{4} - \frac{1112895285439957817353738293900908562636336068976127}{2227736245427732338567760619852735269674485343358821} a^{3} + \frac{347491980979516807968265104400223523903977746387006}{2227736245427732338567760619852735269674485343358821} a^{2} - \frac{399680460878848559710680169082451862810638930943681}{2227736245427732338567760619852735269674485343358821} a + \frac{589759447792129438190887509432093747146169625474901}{2227736245427732338567760619852735269674485343358821}$

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:  $13$
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:  \( 1831456300.14 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed