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Properties

Label 18.14.1124739256...9689.2
Degree $18$
Signature $[14, 2]$
Discriminant $3^{24}\cdot 73^{5}\cdot 577^{3}$
Root discriminant $41.11$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T702

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 243, 4509, -3132, -39096, 93699, -67416, -53073, 97215, -17779, -33030, 15654, 2491, -2796, 261, 172, -36, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 36*x^16 + 172*x^15 + 261*x^14 - 2796*x^13 + 2491*x^12 + 15654*x^11 - 33030*x^10 - 17779*x^9 + 97215*x^8 - 53073*x^7 - 67416*x^6 + 93699*x^5 - 39096*x^4 - 3132*x^3 + 4509*x^2 + 243*x - 27)
 
gp: K = bnfinit(x^18 - 3*x^17 - 36*x^16 + 172*x^15 + 261*x^14 - 2796*x^13 + 2491*x^12 + 15654*x^11 - 33030*x^10 - 17779*x^9 + 97215*x^8 - 53073*x^7 - 67416*x^6 + 93699*x^5 - 39096*x^4 - 3132*x^3 + 4509*x^2 + 243*x - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 36 x^{16} + 172 x^{15} + 261 x^{14} - 2796 x^{13} + 2491 x^{12} + 15654 x^{11} - 33030 x^{10} - 17779 x^{9} + 97215 x^{8} - 53073 x^{7} - 67416 x^{6} + 93699 x^{5} - 39096 x^{4} - 3132 x^{3} + 4509 x^{2} + 243 x - 27 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(112473925614060022797462679689=3^{24}\cdot 73^{5}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $41.11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} + \frac{1}{3} a^{8} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{12} + \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{2}{9} a^{9} - \frac{1}{3} a^{8} - \frac{4}{9} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{13} + \frac{1}{3} a^{11} - \frac{2}{9} a^{10} + \frac{1}{3} a^{9} - \frac{4}{9} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3}$, $\frac{1}{172196506800802255749357326420497041} a^{17} - \frac{2159498669381041774317303384769961}{172196506800802255749357326420497041} a^{16} + \frac{189821259529621463263713037316074}{57398835600267418583119108806832347} a^{15} - \frac{16471894420942268349053639745160766}{172196506800802255749357326420497041} a^{14} - \frac{7075418636709491617722607575248054}{172196506800802255749357326420497041} a^{13} - \frac{382915827072599235511103123028940}{57398835600267418583119108806832347} a^{12} + \frac{53512128211958780754786084690313582}{172196506800802255749357326420497041} a^{11} + \frac{50041527113161395053415470737752586}{172196506800802255749357326420497041} a^{10} + \frac{2642163190804861464479073947881688}{57398835600267418583119108806832347} a^{9} + \frac{24047639325913379761787062059003821}{172196506800802255749357326420497041} a^{8} + \frac{46169850237764996392367191342360136}{172196506800802255749357326420497041} a^{7} - \frac{2429631032963465647444147452189370}{57398835600267418583119108806832347} a^{6} - \frac{9051872458272399623880047795212868}{57398835600267418583119108806832347} a^{5} - \frac{15928705482275070779704081861342534}{57398835600267418583119108806832347} a^{4} + \frac{7915710732484233216427460431745761}{57398835600267418583119108806832347} a^{3} + \frac{6186956240232136608115963121282766}{19132945200089139527706369602277449} a^{2} - \frac{1728930604183572168977346032417459}{19132945200089139527706369602277449} a - \frac{5399317049420021878269851049128360}{19132945200089139527706369602277449}$

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:  $15$
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:  \( 159344616.259 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T702:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 55 conjugacy class representatives for t18n702 are not computed
Character table for t18n702 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 12 sibling: data not computed
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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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