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Properties

Label 18.0.21873113464...8288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{32}\cdot 23^{9}$
Root discriminant $62.61$
Ramified primes $2, 3, 23$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9888146, -16205778, 26705808, -24788658, 23082102, -15746040, 10549596, -5608044, 2874348, -1203195, 481707, -157356, 49236, -12222, 2934, -516, 90, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 90*x^16 - 516*x^15 + 2934*x^14 - 12222*x^13 + 49236*x^12 - 157356*x^11 + 481707*x^10 - 1203195*x^9 + 2874348*x^8 - 5608044*x^7 + 10549596*x^6 - 15746040*x^5 + 23082102*x^4 - 24788658*x^3 + 26705808*x^2 - 16205778*x + 9888146)
 
gp: K = bnfinit(x^18 - 9*x^17 + 90*x^16 - 516*x^15 + 2934*x^14 - 12222*x^13 + 49236*x^12 - 157356*x^11 + 481707*x^10 - 1203195*x^9 + 2874348*x^8 - 5608044*x^7 + 10549596*x^6 - 15746040*x^5 + 23082102*x^4 - 24788658*x^3 + 26705808*x^2 - 16205778*x + 9888146, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 90 x^{16} - 516 x^{15} + 2934 x^{14} - 12222 x^{13} + 49236 x^{12} - 157356 x^{11} + 481707 x^{10} - 1203195 x^{9} + 2874348 x^{8} - 5608044 x^{7} + 10549596 x^{6} - 15746040 x^{5} + 23082102 x^{4} - 24788658 x^{3} + 26705808 x^{2} - 16205778 x + 9888146 \)

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:  \(-218731134641449487521369514508288=-\,2^{16}\cdot 3^{32}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $62.61$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} - \frac{1}{3}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{6} + \frac{1}{3}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{7} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{15} + \frac{1}{3} a^{9} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{31273064131183213794886194126114170500046818545855} a^{17} + \frac{378178711181396638028216706170053592855711366019}{10424354710394404598295398042038056833348939515285} a^{16} - \frac{1316626548495921406721441556055804836812924535951}{10424354710394404598295398042038056833348939515285} a^{15} + \frac{1075493440950056356975564324252604141484218931916}{31273064131183213794886194126114170500046818545855} a^{14} + \frac{108751580377882693679691947717311351406312241787}{6254612826236642758977238825222834100009363709171} a^{13} - \frac{477409729518379328681386834010665162651704423669}{10424354710394404598295398042038056833348939515285} a^{12} - \frac{690871186670913251987193420298400706953058413242}{10424354710394404598295398042038056833348939515285} a^{11} - \frac{2509299932353376611999687849033444688945062254622}{31273064131183213794886194126114170500046818545855} a^{10} - \frac{192867655381477537885215939996822765857402709560}{6254612826236642758977238825222834100009363709171} a^{9} - \frac{750904840398208827347480620635694266313492768306}{2084870942078880919659079608407611366669787903057} a^{8} + \frac{3601389062491702765162120777789002592905550140493}{31273064131183213794886194126114170500046818545855} a^{7} + \frac{5967317668029473074530864769023584945627949701654}{31273064131183213794886194126114170500046818545855} a^{6} + \frac{273908117239164996311562542843395133900380985662}{6254612826236642758977238825222834100009363709171} a^{5} - \frac{2360737447860493613104340811871260192812200938058}{6254612826236642758977238825222834100009363709171} a^{4} + \frac{4067572587859131156043967901405481620514511196202}{31273064131183213794886194126114170500046818545855} a^{3} - \frac{10351604949384690054484949313846307569287032992181}{31273064131183213794886194126114170500046818545855} a^{2} + \frac{516677713137427679206901252800535008338788231704}{10424354710394404598295398042038056833348939515285} a - \frac{14446318317871206740318348378889220366861413043736}{31273064131183213794886194126114170500046818545855}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 416075792.5032041 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

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 $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.108.1, 6.0.141915888.4, 9.1.11019960576.1

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

Sibling fields

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ $18$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
$2$2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
2.9.8.1$x^{9} - 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
$3$3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.4.2.1$x^{4} + 299 x^{2} + 25921$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
23.12.6.1$x^{12} + 365010 x^{6} - 6436343 x^{2} + 33308075025$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$