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Properties

Label 18.18.3719662119...8576.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{30}\cdot 3^{24}\cdot 13^{6}\cdot 71^{4}$
Root discriminant $83.29$
Ramified primes $2, 3, 13, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T463

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-25411681, 0, 51539184, 0, -43145919, 0, 19353109, 0, -5062413, 0, 790428, 0, -72752, 0, 3771, 0, -99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 99*x^16 + 3771*x^14 - 72752*x^12 + 790428*x^10 - 5062413*x^8 + 19353109*x^6 - 43145919*x^4 + 51539184*x^2 - 25411681)
 
gp: K = bnfinit(x^18 - 99*x^16 + 3771*x^14 - 72752*x^12 + 790428*x^10 - 5062413*x^8 + 19353109*x^6 - 43145919*x^4 + 51539184*x^2 - 25411681, 1)
 

Normalized defining polynomial

\( x^{18} - 99 x^{16} + 3771 x^{14} - 72752 x^{12} + 790428 x^{10} - 5062413 x^{8} + 19353109 x^{6} - 43145919 x^{4} + 51539184 x^{2} - 25411681 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(37196621191331827761794933720088576=2^{30}\cdot 3^{24}\cdot 13^{6}\cdot 71^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $83.29$
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:  $2, 3, 13, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{71} a^{12} - \frac{28}{71} a^{10} + \frac{8}{71} a^{8} + \frac{23}{71} a^{6} - \frac{15}{71} a^{4} + \frac{29}{71} a^{2}$, $\frac{1}{71} a^{13} - \frac{28}{71} a^{11} + \frac{8}{71} a^{9} + \frac{23}{71} a^{7} - \frac{15}{71} a^{5} + \frac{29}{71} a^{3}$, $\frac{1}{5041} a^{14} - \frac{28}{5041} a^{12} + \frac{1783}{5041} a^{10} - \frac{1610}{5041} a^{8} + \frac{624}{5041} a^{6} - \frac{2314}{5041} a^{4} - \frac{32}{71} a^{2}$, $\frac{1}{5041} a^{15} - \frac{28}{5041} a^{13} + \frac{1783}{5041} a^{11} - \frac{1610}{5041} a^{9} + \frac{624}{5041} a^{7} - \frac{2314}{5041} a^{5} - \frac{32}{71} a^{3}$, $\frac{1}{9732599250849802477} a^{16} - \frac{555512575021407}{9732599250849802477} a^{14} + \frac{20640331203311088}{9732599250849802477} a^{12} - \frac{4616176315157110236}{9732599250849802477} a^{10} - \frac{4746374992690551340}{9732599250849802477} a^{8} - \frac{163275418461736591}{9732599250849802477} a^{6} + \frac{24494182744192179}{137078862688025387} a^{4} + \frac{549146219893628}{1930688206873597} a^{2} - \frac{7932604282482}{27192791646107}$, $\frac{1}{9732599250849802477} a^{17} - \frac{555512575021407}{9732599250849802477} a^{15} + \frac{20640331203311088}{9732599250849802477} a^{13} - \frac{4616176315157110236}{9732599250849802477} a^{11} - \frac{4746374992690551340}{9732599250849802477} a^{9} - \frac{163275418461736591}{9732599250849802477} a^{7} + \frac{24494182744192179}{137078862688025387} a^{5} + \frac{549146219893628}{1930688206873597} a^{3} - \frac{7932604282482}{27192791646107} a$

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:  $17$
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:  \( 342981847768 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T463:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 48 conjugacy class representatives for t18n463
Character table for t18n463 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.4212.1, 9.9.74724856128.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
2Data not computed
$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}$
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.12.6.1$x^{12} + 338 x^{8} + 8788 x^{6} + 28561 x^{4} + 19307236$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
71Data not computed