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Properties

Label 18.0.10591107618...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{15}$
Root discriminant $27.91$
Ramified primes $2, 3, 5$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_2\times C_3^2:S_3$ (as 18T52)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![216, 216, -1800, 216, 4680, -4464, 2106, -3150, 1866, -45, 1065, 255, 180, 0, -5, -9, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 9*x^15 - 5*x^14 + 180*x^12 + 255*x^11 + 1065*x^10 - 45*x^9 + 1866*x^8 - 3150*x^7 + 2106*x^6 - 4464*x^5 + 4680*x^4 + 216*x^3 - 1800*x^2 + 216*x + 216)
 
gp: K = bnfinit(x^18 + x^16 - 9*x^15 - 5*x^14 + 180*x^12 + 255*x^11 + 1065*x^10 - 45*x^9 + 1866*x^8 - 3150*x^7 + 2106*x^6 - 4464*x^5 + 4680*x^4 + 216*x^3 - 1800*x^2 + 216*x + 216, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 9 x^{15} - 5 x^{14} + 180 x^{12} + 255 x^{11} + 1065 x^{10} - 45 x^{9} + 1866 x^{8} - 3150 x^{7} + 2106 x^{6} - 4464 x^{5} + 4680 x^{4} + 216 x^{3} - 1800 x^{2} + 216 x + 216 \)

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:  \(-105911076180375000000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.91$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{3} a^{8} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{4}{9} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} + \frac{1}{3} a^{2}$, $\frac{1}{54} a^{12} + \frac{1}{54} a^{10} + \frac{1}{18} a^{9} + \frac{13}{54} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{7}{18} a^{5} + \frac{5}{18} a^{4} + \frac{1}{6} a^{3} + \frac{4}{9} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{54} a^{13} + \frac{1}{54} a^{11} + \frac{1}{18} a^{10} + \frac{2}{27} a^{9} + \frac{2}{9} a^{8} + \frac{2}{9} a^{7} - \frac{1}{18} a^{6} - \frac{2}{9} a^{5} + \frac{1}{6} a^{4} - \frac{1}{18} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{54} a^{14} - \frac{1}{18} a^{10} + \frac{1}{18} a^{9} + \frac{23}{54} a^{8} - \frac{1}{9} a^{7} + \frac{2}{9} a^{6} + \frac{7}{18} a^{5} - \frac{1}{6} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{108} a^{15} - \frac{1}{108} a^{13} - \frac{1}{108} a^{12} - \frac{1}{108} a^{11} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{35}{108} a^{8} + \frac{5}{36} a^{7} - \frac{11}{36} a^{6} - \frac{1}{9} a^{5} + \frac{5}{18} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3}$, $\frac{1}{108} a^{16} - \frac{1}{108} a^{14} - \frac{1}{108} a^{13} - \frac{1}{108} a^{12} + \frac{1}{54} a^{11} - \frac{1}{54} a^{10} - \frac{7}{108} a^{9} + \frac{1}{36} a^{8} + \frac{7}{36} a^{7} - \frac{5}{18} a^{6} - \frac{1}{18} a^{5} + \frac{7}{18} a^{4} - \frac{1}{18} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{16368726039394542804} a^{17} - \frac{57158991465262019}{16368726039394542804} a^{16} + \frac{3982358991803095}{1364060503282878567} a^{15} - \frac{49224788552191217}{8184363019697271402} a^{14} - \frac{129011833071480337}{16368726039394542804} a^{13} - \frac{20766405503935226}{4092181509848635701} a^{12} + \frac{31431536851225879}{16368726039394542804} a^{11} + \frac{146809046228651275}{5456242013131514268} a^{10} + \frac{292548760047639776}{4092181509848635701} a^{9} - \frac{1976322922558631353}{16368726039394542804} a^{8} - \frac{209803539771038471}{454686834427626189} a^{7} + \frac{294230149378399123}{5456242013131514268} a^{6} + \frac{182624969020025357}{909373668855252378} a^{5} + \frac{178904072218913755}{454686834427626189} a^{4} - \frac{625390110676663991}{1364060503282878567} a^{3} + \frac{186740797893014659}{1364060503282878567} a^{2} + \frac{62531746039122973}{454686834427626189} a - \frac{173080360243078933}{454686834427626189}$

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:  \( 396879.84357445274 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3^2:S_3$ (as 18T52):

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 $C_2\times C_3^2:S_3$
Character table for $C_2\times C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.300.1, 6.0.1350000.1, 9.3.177147000000.2

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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.12.22.67$x^{12} + 99 x^{11} + 117 x^{10} - 114 x^{9} - 81 x^{8} + 9 x^{7} - 15 x^{6} + 54 x^{5} - 108 x^{4} + 45 x^{3} - 81 x^{2} - 108 x - 72$$6$$2$$22$$D_6$$[5/2]_{2}^{2}$
5Data not computed