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Properties

Label 18.0.13637978599...8839.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{37}\cdot 13^{13}$
Root discriminant $60.99$
Ramified primes $3, 13$
Class number $108$ (GRH)
Class group $[3, 36]$ (GRH)
Galois group $C_2\times C_3:S_3.C_2$ (as 18T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8272, -67293, 255672, -618753, 1073547, -1418553, 1486950, -1260243, 885375, -516022, 255627, -105237, 37410, -10773, 2727, -516, 90, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 90*x^16 - 516*x^15 + 2727*x^14 - 10773*x^13 + 37410*x^12 - 105237*x^11 + 255627*x^10 - 516022*x^9 + 885375*x^8 - 1260243*x^7 + 1486950*x^6 - 1418553*x^5 + 1073547*x^4 - 618753*x^3 + 255672*x^2 - 67293*x + 8272)
 
gp: K = bnfinit(x^18 - 9*x^17 + 90*x^16 - 516*x^15 + 2727*x^14 - 10773*x^13 + 37410*x^12 - 105237*x^11 + 255627*x^10 - 516022*x^9 + 885375*x^8 - 1260243*x^7 + 1486950*x^6 - 1418553*x^5 + 1073547*x^4 - 618753*x^3 + 255672*x^2 - 67293*x + 8272, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 90 x^{16} - 516 x^{15} + 2727 x^{14} - 10773 x^{13} + 37410 x^{12} - 105237 x^{11} + 255627 x^{10} - 516022 x^{9} + 885375 x^{8} - 1260243 x^{7} + 1486950 x^{6} - 1418553 x^{5} + 1073547 x^{4} - 618753 x^{3} + 255672 x^{2} - 67293 x + 8272 \)

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:  \(-136379785993511844880006339228839=-\,3^{37}\cdot 13^{13}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $60.99$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{2} - \frac{1}{9}$, $\frac{1}{117} a^{12} - \frac{2}{39} a^{11} + \frac{16}{117} a^{9} + \frac{4}{39} a^{8} - \frac{5}{39} a^{7} + \frac{2}{39} a^{6} - \frac{1}{3} a^{5} - \frac{19}{39} a^{4} - \frac{14}{117} a^{3} - \frac{10}{39} a^{2} + \frac{16}{39} a + \frac{22}{117}$, $\frac{1}{351} a^{13} + \frac{1}{351} a^{12} - \frac{1}{117} a^{11} + \frac{16}{351} a^{10} - \frac{32}{351} a^{9} - \frac{1}{39} a^{8} + \frac{19}{117} a^{7} + \frac{1}{117} a^{6} + \frac{11}{39} a^{5} + \frac{16}{351} a^{4} - \frac{128}{351} a^{3} - \frac{28}{117} a^{2} - \frac{71}{351} a + \frac{115}{351}$, $\frac{1}{702} a^{14} - \frac{1}{702} a^{13} - \frac{1}{351} a^{12} - \frac{35}{702} a^{11} - \frac{25}{702} a^{10} + \frac{32}{351} a^{9} + \frac{37}{234} a^{8} - \frac{1}{18} a^{7} - \frac{1}{117} a^{6} - \frac{23}{54} a^{5} - \frac{97}{702} a^{4} - \frac{4}{27} a^{3} + \frac{319}{702} a^{2} - \frac{145}{702} a - \frac{160}{351}$, $\frac{1}{9126} a^{15} - \frac{1}{9126} a^{14} - \frac{1}{234} a^{12} - \frac{463}{9126} a^{11} + \frac{1}{507} a^{10} + \frac{107}{9126} a^{9} + \frac{503}{3042} a^{8} + \frac{37}{507} a^{7} + \frac{841}{9126} a^{6} + \frac{1505}{9126} a^{5} + \frac{223}{507} a^{4} + \frac{1063}{3042} a^{3} - \frac{2473}{9126} a^{2} + \frac{136}{507} a + \frac{2116}{4563}$, $\frac{1}{23809734} a^{16} - \frac{4}{11904867} a^{15} + \frac{844}{3968289} a^{14} + \frac{139}{101751} a^{13} - \frac{11983}{3968289} a^{12} + \frac{182255}{3968289} a^{11} - \frac{603866}{11904867} a^{10} - \frac{1717661}{11904867} a^{9} - \frac{52808}{440921} a^{8} - \frac{1777912}{11904867} a^{7} + \frac{1081424}{11904867} a^{6} + \frac{1615396}{3968289} a^{5} - \frac{1910621}{3968289} a^{4} - \frac{220793}{1322763} a^{3} - \frac{1561363}{3968289} a^{2} - \frac{2020663}{23809734} a - \frac{5051714}{11904867}$, $\frac{1}{166668138} a^{17} + \frac{1}{83334069} a^{16} + \frac{356}{11904867} a^{15} + \frac{7666}{83334069} a^{14} - \frac{42904}{83334069} a^{13} + \frac{51607}{83334069} a^{12} + \frac{1370333}{83334069} a^{11} + \frac{59668}{11904867} a^{10} - \frac{10292761}{83334069} a^{9} - \frac{466498}{6410313} a^{8} - \frac{3164813}{83334069} a^{7} - \frac{12524599}{83334069} a^{6} - \frac{427157}{11904867} a^{5} + \frac{252485}{83334069} a^{4} - \frac{2916413}{83334069} a^{3} + \frac{1503679}{166668138} a^{2} + \frac{2888815}{83334069} a + \frac{32172506}{83334069}$

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

Class group and class number

$C_{3}\times C_{36}$, which has order $108$ (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:  \( 46928148.8521 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3.C_2$ (as 18T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 12 conjugacy class representatives for $C_2\times C_3:S_3.C_2$
Character table for $C_2\times C_3:S_3.C_2$

Intermediate fields

\(\Q(\sqrt{-39}) \), 9.9.1870004703089601.1

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

Sibling fields

Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
3Data not computed
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$
13.4.3.2$x^{4} - 52$$4$$1$$3$$C_4$$[\ ]_{4}$