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Properties

Label 18.0.13187127617...2752.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{32}\cdot 13^{9}$
Root discriminant $69.18$
Ramified primes $2, 3, 13$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![80707214, 0, 0, 0, 0, 0, 74088, 0, 0, 11666, 0, 0, 6174, 0, 0, -114, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 114*x^15 + 6174*x^12 + 11666*x^9 + 74088*x^6 + 80707214)
 
gp: K = bnfinit(x^18 - 114*x^15 + 6174*x^12 + 11666*x^9 + 74088*x^6 + 80707214, 1)
 

Normalized defining polynomial

\( x^{18} - 114 x^{15} + 6174 x^{12} + 11666 x^{9} + 74088 x^{6} + 80707214 \)

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:  \(-1318712761724157691283452043722752=-\,2^{26}\cdot 3^{32}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $69.18$
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 not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5} a^{9} - \frac{2}{5} a^{6} - \frac{2}{5}$, $\frac{1}{105} a^{10} + \frac{1}{15} a^{9} + \frac{1}{3} a^{8} - \frac{2}{105} a^{7} + \frac{1}{5} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{17}{105} a - \frac{7}{15}$, $\frac{1}{735} a^{11} + \frac{1}{15} a^{9} + \frac{278}{735} a^{8} + \frac{1}{3} a^{7} - \frac{2}{15} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{64}{245} a^{2} - \frac{1}{3} a - \frac{2}{15}$, $\frac{1}{5145} a^{12} + \frac{229}{5145} a^{9} - \frac{4}{15} a^{6} - \frac{682}{5145} a^{3} + \frac{1}{15}$, $\frac{1}{36015} a^{13} - \frac{38}{12005} a^{10} - \frac{1}{15} a^{9} - \frac{1}{3} a^{8} - \frac{17}{105} a^{7} - \frac{1}{5} a^{6} + \frac{1}{3} a^{5} + \frac{4463}{36015} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{16}{35} a + \frac{7}{15}$, $\frac{1}{252105} a^{14} - \frac{38}{84035} a^{11} - \frac{1}{15} a^{9} + \frac{6}{245} a^{8} - \frac{1}{3} a^{7} + \frac{2}{15} a^{6} + \frac{9491}{84035} a^{5} + \frac{1}{3} a^{4} + \frac{118}{735} a^{2} + \frac{1}{3} a + \frac{2}{15}$, $\frac{1}{8190733646639985} a^{15} + \frac{281431701913}{8190733646639985} a^{12} + \frac{1247056688387}{23879689931895} a^{9} - \frac{3851081784730519}{8190733646639985} a^{6} + \frac{93906905063}{4775937986379} a^{3} + \frac{9309820634}{23206695755}$, $\frac{1}{57335135526479895} a^{16} + \frac{281431701913}{57335135526479895} a^{13} - \frac{344922640406}{167157829523265} a^{10} - \frac{1}{15} a^{9} - \frac{1}{3} a^{8} + \frac{21813216974741434}{57335135526479895} a^{7} - \frac{1}{5} a^{6} + \frac{1}{3} a^{5} - \frac{14233907054074}{33431565904653} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{106832227469}{487340610855} a + \frac{7}{15}$, $\frac{1}{401345948685359265} a^{17} + \frac{281431701913}{401345948685359265} a^{14} - \frac{344922640406}{1170104806662855} a^{11} - \frac{1}{15} a^{9} + \frac{32753354781127098}{133781982895119755} a^{8} - \frac{1}{3} a^{7} + \frac{2}{15} a^{6} - \frac{25377762355625}{234020961332571} a^{5} + \frac{1}{3} a^{4} - \frac{180985084557}{1137128091995} a^{2} + \frac{1}{3} a + \frac{2}{15}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 1592843982.5010967 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-13}) \), 3.1.108.1, 6.0.410012928.1, 9.1.11019960576.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
13Data not computed