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Properties

Label 18.0.27375704250...6272.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}\cdot 13^{9}$
Root discriminant $105.75$
Ramified primes $2, 3, 13$
Class number $1163474$ (GRH)
Class group $[1163474]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![36926027009, -838908, 17095168548, 482646, 3750130377, -47556, 510655881, 954, 47554038, -2, 3144492, 0, 148083, 0, 4815, 0, 99, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 99*x^16 + 4815*x^14 + 148083*x^12 + 3144492*x^10 - 2*x^9 + 47554038*x^8 + 954*x^7 + 510655881*x^6 - 47556*x^5 + 3750130377*x^4 + 482646*x^3 + 17095168548*x^2 - 838908*x + 36926027009)
 
gp: K = bnfinit(x^18 + 99*x^16 + 4815*x^14 + 148083*x^12 + 3144492*x^10 - 2*x^9 + 47554038*x^8 + 954*x^7 + 510655881*x^6 - 47556*x^5 + 3750130377*x^4 + 482646*x^3 + 17095168548*x^2 - 838908*x + 36926027009, 1)
 

Normalized defining polynomial

\( x^{18} + 99 x^{16} + 4815 x^{14} + 148083 x^{12} + 3144492 x^{10} - 2 x^{9} + 47554038 x^{8} + 954 x^{7} + 510655881 x^{6} - 47556 x^{5} + 3750130377 x^{4} + 482646 x^{3} + 17095168548 x^{2} - 838908 x + 36926027009 \)

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:  \(-2737570425013469092239722803000246272=-\,2^{18}\cdot 3^{44}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.75$
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 Galois and abelian over $\Q$.
Conductor:  \(1404=2^{2}\cdot 3^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1404}(1,·)$, $\chi_{1404}(259,·)$, $\chi_{1404}(1093,·)$, $\chi_{1404}(1351,·)$, $\chi_{1404}(781,·)$, $\chi_{1404}(1039,·)$, $\chi_{1404}(469,·)$, $\chi_{1404}(727,·)$, $\chi_{1404}(157,·)$, $\chi_{1404}(415,·)$, $\chi_{1404}(1249,·)$, $\chi_{1404}(103,·)$, $\chi_{1404}(937,·)$, $\chi_{1404}(1195,·)$, $\chi_{1404}(625,·)$, $\chi_{1404}(883,·)$, $\chi_{1404}(313,·)$, $\chi_{1404}(571,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1062294642302073588506179226288782531663820541785786909138901649} a^{17} - \frac{416414629074629946612192311192678509202343321232147383467306677}{1062294642302073588506179226288782531663820541785786909138901649} a^{16} + \frac{452833945919686128177834994367638780354741181289205684536964547}{1062294642302073588506179226288782531663820541785786909138901649} a^{15} - \frac{242092265188370350951257716713234828396031500785842786992672536}{1062294642302073588506179226288782531663820541785786909138901649} a^{14} - \frac{416995958725678371215422868414649552353401915046315099842102440}{1062294642302073588506179226288782531663820541785786909138901649} a^{13} - \frac{14513113679613117722270868321713882395352347222271993455274040}{1062294642302073588506179226288782531663820541785786909138901649} a^{12} + \frac{174626230342277785106365276456908107943483276101655464181622858}{1062294642302073588506179226288782531663820541785786909138901649} a^{11} + \frac{199561901505051650592787656950483523765453653917541861846380962}{1062294642302073588506179226288782531663820541785786909138901649} a^{10} + \frac{206601020310028225912701895044852370501659087256314696104797874}{1062294642302073588506179226288782531663820541785786909138901649} a^{9} + \frac{200250060368891245824454764934414176141945225739889066102064453}{1062294642302073588506179226288782531663820541785786909138901649} a^{8} - \frac{225019326747450235484192310898057190176408444503724641317231205}{1062294642302073588506179226288782531663820541785786909138901649} a^{7} - \frac{436882142861656306602527332965699106668839400965391773354976917}{1062294642302073588506179226288782531663820541785786909138901649} a^{6} + \frac{3694612735189795797865081927149530547250034943944809583930089}{20043295137774973368041117477146840220072085694071451115828333} a^{5} + \frac{86078619589567561876791563337865544770925072692502681507036184}{1062294642302073588506179226288782531663820541785786909138901649} a^{4} + \frac{70066963904657930427003731059733794537555930803082972543054950}{1062294642302073588506179226288782531663820541785786909138901649} a^{3} + \frac{15790235813039757803486676501885394023739000180496829905644428}{1062294642302073588506179226288782531663820541785786909138901649} a^{2} - \frac{200967772844849665186804218111266462280673762180933156322631273}{1062294642302073588506179226288782531663820541785786909138901649} a - \frac{2605848399219595060563058299946663190664140389607461712963449}{6517145044797997475498032063121365224931414366783968767723323}$

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

Class group and class number

$C_{1163474}$, which has order $1163474$ (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:  \( 40934.03294431194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

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

Intermediate fields

\(\Q(\sqrt{-13}) \), \(\Q(\zeta_{9})^+\), 6.0.922529088.2, \(\Q(\zeta_{27})^+\)

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

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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