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Properties

Label 18.2.21106009685...2841.1
Degree $18$
Signature $[2, 8]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $25.52$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-269, -189, 237, -300, -588, 561, 461, 156, -120, -98, -237, -24, -27, -63, -6, -11, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 11*x^15 - 6*x^14 - 63*x^13 - 27*x^12 - 24*x^11 - 237*x^10 - 98*x^9 - 120*x^8 + 156*x^7 + 461*x^6 + 561*x^5 - 588*x^4 - 300*x^3 + 237*x^2 - 189*x - 269)
 
gp: K = bnfinit(x^18 + 3*x^16 - 11*x^15 - 6*x^14 - 63*x^13 - 27*x^12 - 24*x^11 - 237*x^10 - 98*x^9 - 120*x^8 + 156*x^7 + 461*x^6 + 561*x^5 - 588*x^4 - 300*x^3 + 237*x^2 - 189*x - 269, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 11 x^{15} - 6 x^{14} - 63 x^{13} - 27 x^{12} - 24 x^{11} - 237 x^{10} - 98 x^{9} - 120 x^{8} + 156 x^{7} + 461 x^{6} + 561 x^{5} - 588 x^{4} - 300 x^{3} + 237 x^{2} - 189 x - 269 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21106009685505727678262841=3^{24}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.52$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{13} + \frac{1}{3} a^{11} - \frac{1}{9} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{3731548639167477175650666111} a^{17} + \frac{61656809520376338506079874}{1243849546389159058550222037} a^{16} - \frac{186539432964214455033622501}{3731548639167477175650666111} a^{15} - \frac{427691825766075962459303590}{3731548639167477175650666111} a^{14} + \frac{181091012811527394265572641}{1243849546389159058550222037} a^{13} - \frac{451845461094426677627527994}{3731548639167477175650666111} a^{12} - \frac{108018595436299896433233991}{3731548639167477175650666111} a^{11} - \frac{92158343557491303225905220}{414616515463053019516740679} a^{10} - \frac{284657398751534154118025177}{3731548639167477175650666111} a^{9} + \frac{441988962380322139831436522}{1243849546389159058550222037} a^{8} + \frac{54612986677311331439731342}{1243849546389159058550222037} a^{7} - \frac{494667483275954858034343418}{1243849546389159058550222037} a^{6} + \frac{1562895527059413398450114876}{3731548639167477175650666111} a^{5} - \frac{120702870934826217729944784}{414616515463053019516740679} a^{4} - \frac{1486272841732130846297079164}{3731548639167477175650666111} a^{3} + \frac{357093144588220262000512145}{3731548639167477175650666111} a^{2} - \frac{582037574437502550371099743}{1243849546389159058550222037} a + \frac{1045996808540533522862182054}{3731548639167477175650666111}$

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:  $9$
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:  \( 163681.359064 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$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