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Properties

Label 18.0.30613671805...5408.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}\cdot 17^{9}$
Root discriminant $120.93$
Ramified primes $2, 3, 17$
Class number $7799472$ (GRH)
Class group $[2, 2, 1949868]$ (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![314709501201, -2203524, 121352263884, 1016694, 21883306869, -79308, 2419311081, 1242, 180682038, -2, 9461124, 0, 348039, 0, 8703, 0, 135, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 135*x^16 + 8703*x^14 + 348039*x^12 + 9461124*x^10 - 2*x^9 + 180682038*x^8 + 1242*x^7 + 2419311081*x^6 - 79308*x^5 + 21883306869*x^4 + 1016694*x^3 + 121352263884*x^2 - 2203524*x + 314709501201)
 
gp: K = bnfinit(x^18 + 135*x^16 + 8703*x^14 + 348039*x^12 + 9461124*x^10 - 2*x^9 + 180682038*x^8 + 1242*x^7 + 2419311081*x^6 - 79308*x^5 + 21883306869*x^4 + 1016694*x^3 + 121352263884*x^2 - 2203524*x + 314709501201, 1)
 

Normalized defining polynomial

\( x^{18} + 135 x^{16} + 8703 x^{14} + 348039 x^{12} + 9461124 x^{10} - 2 x^{9} + 180682038 x^{8} + 1242 x^{7} + 2419311081 x^{6} - 79308 x^{5} + 21883306869 x^{4} + 1016694 x^{3} + 121352263884 x^{2} - 2203524 x + 314709501201 \)

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:  \(-30613671805187353870953069542862225408=-\,2^{18}\cdot 3^{44}\cdot 17^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $120.93$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1836=2^{2}\cdot 3^{3}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{1836}(1,·)$, $\chi_{1836}(67,·)$, $\chi_{1836}(1225,·)$, $\chi_{1836}(1291,·)$, $\chi_{1836}(205,·)$, $\chi_{1836}(271,·)$, $\chi_{1836}(1429,·)$, $\chi_{1836}(1495,·)$, $\chi_{1836}(409,·)$, $\chi_{1836}(475,·)$, $\chi_{1836}(1633,·)$, $\chi_{1836}(1699,·)$, $\chi_{1836}(613,·)$, $\chi_{1836}(679,·)$, $\chi_{1836}(817,·)$, $\chi_{1836}(883,·)$, $\chi_{1836}(1021,·)$, $\chi_{1836}(1087,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{163} a^{16} - \frac{48}{163} a^{15} - \frac{37}{163} a^{14} + \frac{4}{163} a^{13} + \frac{41}{163} a^{12} + \frac{27}{163} a^{11} + \frac{75}{163} a^{10} - \frac{36}{163} a^{9} + \frac{7}{163} a^{8} - \frac{37}{163} a^{7} + \frac{53}{163} a^{6} + \frac{8}{163} a^{5} - \frac{27}{163} a^{4} - \frac{20}{163} a^{3} + \frac{65}{163} a^{2} + \frac{8}{163} a - \frac{41}{163}$, $\frac{1}{53813324793526559704903811995580172193494334294813054508054068891} a^{17} + \frac{122315051647049117577649977104239002947953197088927122067285993}{53813324793526559704903811995580172193494334294813054508054068891} a^{16} - \frac{3524402545697280842208846104362033804520077705792229293020371074}{53813324793526559704903811995580172193494334294813054508054068891} a^{15} + \frac{2515087856515085641920383510010942254655569366680916887420991438}{53813324793526559704903811995580172193494334294813054508054068891} a^{14} - \frac{14664037179693672920881108917064761414722045327270130365062182570}{53813324793526559704903811995580172193494334294813054508054068891} a^{13} - \frac{4725660360811427255285698854629885405678651728169833216711524156}{53813324793526559704903811995580172193494334294813054508054068891} a^{12} + \frac{5357802614755405680665133926283374441698989179501932760094153357}{53813324793526559704903811995580172193494334294813054508054068891} a^{11} + \frac{20901184434208640126823714937555184123827027368443991814162640156}{53813324793526559704903811995580172193494334294813054508054068891} a^{10} + \frac{15776369635371244821524672662036454401202714589421589263330408967}{53813324793526559704903811995580172193494334294813054508054068891} a^{9} + \frac{19249433311381232909883960493203648466417538101205837765275565320}{53813324793526559704903811995580172193494334294813054508054068891} a^{8} - \frac{2943106500847653370585248077581448133633794506637062010721617883}{53813324793526559704903811995580172193494334294813054508054068891} a^{7} + \frac{25710128908201146239141035620088044274798739346776580991390692065}{53813324793526559704903811995580172193494334294813054508054068891} a^{6} - \frac{6177593783538000375860646738214288674614497138503616615313829115}{53813324793526559704903811995580172193494334294813054508054068891} a^{5} - \frac{539009742889182096414184394956730366973897380125179171884271888}{53813324793526559704903811995580172193494334294813054508054068891} a^{4} - \frac{26144679139286829152667402505640331199733646567718265317682653787}{53813324793526559704903811995580172193494334294813054508054068891} a^{3} + \frac{4262324472833281726955717310816336575067624141687661618530607616}{53813324793526559704903811995580172193494334294813054508054068891} a^{2} + \frac{7734968822587642584140950701586170930166479126030407039653498800}{53813324793526559704903811995580172193494334294813054508054068891} a - \frac{243736057260085400737373012835583944673951632869045688607747016}{1015345750821255843488751169727927777235742156505906688831208847}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1949868}$, which has order $7799472$ (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

\(\Q(\sqrt{-17}) \), \(\Q(\zeta_{9})^+\), 6.0.2062988352.6, \(\Q(\zeta_{27})^+\)

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 $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ $18$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
$17$17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
17.6.3.1$x^{6} - 34 x^{4} + 289 x^{2} - 44217$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$