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Properties

Label 18.0.66086513059...5824.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 379^{9}$
Root discriminant $30.90$
Ramified primes $2, 379$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![14373, -12606, -5470, 18166, -5428, -19882, 18315, -6544, 5588, -4548, 2184, -1236, 595, -238, 114, -34, 16, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 16*x^16 - 34*x^15 + 114*x^14 - 238*x^13 + 595*x^12 - 1236*x^11 + 2184*x^10 - 4548*x^9 + 5588*x^8 - 6544*x^7 + 18315*x^6 - 19882*x^5 - 5428*x^4 + 18166*x^3 - 5470*x^2 - 12606*x + 14373)
 
gp: K = bnfinit(x^18 - 2*x^17 + 16*x^16 - 34*x^15 + 114*x^14 - 238*x^13 + 595*x^12 - 1236*x^11 + 2184*x^10 - 4548*x^9 + 5588*x^8 - 6544*x^7 + 18315*x^6 - 19882*x^5 - 5428*x^4 + 18166*x^3 - 5470*x^2 - 12606*x + 14373, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 16 x^{16} - 34 x^{15} + 114 x^{14} - 238 x^{13} + 595 x^{12} - 1236 x^{11} + 2184 x^{10} - 4548 x^{9} + 5588 x^{8} - 6544 x^{7} + 18315 x^{6} - 19882 x^{5} - 5428 x^{4} + 18166 x^{3} - 5470 x^{2} - 12606 x + 14373 \)

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:  \(-660865130598477469667405824=-\,2^{12}\cdot 379^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.90$
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:  $2, 379$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} + \frac{1}{3} a^{5} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} + \frac{1}{3} a^{6} - \frac{1}{2} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{18} a^{7} + \frac{1}{6} a^{6} - \frac{1}{18} a^{5} - \frac{1}{18} a^{4} - \frac{5}{18} a^{3} - \frac{7}{18} a^{2} - \frac{1}{6} a$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} + \frac{1}{9} a^{8} + \frac{1}{18} a^{7} + \frac{5}{18} a^{6} + \frac{1}{3} a^{5} + \frac{5}{18} a^{4} - \frac{5}{18} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{14} + \frac{1}{54} a^{13} - \frac{1}{54} a^{12} + \frac{1}{54} a^{10} + \frac{2}{27} a^{9} - \frac{1}{9} a^{8} + \frac{1}{54} a^{7} - \frac{2}{27} a^{6} - \frac{2}{9} a^{5} + \frac{25}{54} a^{4} + \frac{2}{27} a^{3} - \frac{25}{54} a^{2} + \frac{2}{9} a + \frac{1}{6}$, $\frac{1}{756} a^{15} + \frac{1}{126} a^{14} - \frac{5}{189} a^{13} + \frac{1}{756} a^{12} - \frac{19}{378} a^{11} - \frac{1}{42} a^{10} + \frac{1}{378} a^{9} + \frac{11}{378} a^{8} - \frac{1}{378} a^{7} - \frac{11}{189} a^{6} - \frac{25}{378} a^{5} + \frac{47}{126} a^{4} + \frac{7}{108} a^{3} + \frac{61}{189} a^{2} + \frac{13}{126} a - \frac{13}{84}$, $\frac{1}{11174436} a^{16} + \frac{7045}{11174436} a^{15} - \frac{12331}{1862406} a^{14} - \frac{268235}{11174436} a^{13} - \frac{89945}{3724812} a^{12} - \frac{245281}{5587218} a^{11} + \frac{19051}{399087} a^{10} + \frac{232250}{2793609} a^{9} - \frac{192485}{5587218} a^{8} - \frac{14005}{5587218} a^{7} + \frac{453385}{1862406} a^{6} + \frac{716939}{5587218} a^{5} + \frac{1279885}{3724812} a^{4} + \frac{150919}{11174436} a^{3} - \frac{87499}{2793609} a^{2} - \frac{1502597}{3724812} a + \frac{557281}{1241604}$, $\frac{1}{362304655614062078649732} a^{17} + \frac{256991739457975}{13418690948668965875916} a^{16} + \frac{750034590153650437}{1927152423479053609839} a^{15} - \frac{292897423582852993189}{32936786874005643513612} a^{14} + \frac{5097115480702816480859}{362304655614062078649732} a^{13} - \frac{3215341902433650335191}{181152327807031039324866} a^{12} - \frac{209421975214734293867}{2744732239500470292801} a^{11} + \frac{585682499318167507765}{12939451986216502808919} a^{10} - \frac{291832914417682774511}{10064018211501724406937} a^{9} + \frac{852835366318379655355}{181152327807031039324866} a^{8} - \frac{2424594130508219845313}{181152327807031039324866} a^{7} - \frac{645932115679326952664}{90576163903515519662433} a^{6} + \frac{9004053018126652034363}{27869588893389390665364} a^{5} - \frac{20680470753100665494783}{51757807944866011235676} a^{4} + \frac{18998376705756116016958}{90576163903515519662433} a^{3} + \frac{94063442240529705476605}{362304655614062078649732} a^{2} + \frac{5649796050688522741985}{120768218538020692883244} a + \frac{2771378977120758627625}{20128036423003448813874}$

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:  $8$
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:  \( 3306484.46564 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-379}) \), 3.1.379.1 x3, 6.0.54439939.1, 9.1.1320495160384.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
379Data not computed