↑

Properties

Label 18.0.14989323840...0448.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 37^{12}$
Root discriminant $132.09$
Ramified primes $2, 3, 37$
Class number $243$ (GRH)
Class group $[3, 3, 3, 9]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![66996, -520884, 1822716, -3847170, 5131062, -3954150, 1476721, -156159, 31989, 44796, -41133, 14733, -2756, 225, 69, -90, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 2756*x^12 + 14733*x^11 - 41133*x^10 + 44796*x^9 + 31989*x^8 - 156159*x^7 + 1476721*x^6 - 3954150*x^5 + 5131062*x^4 - 3847170*x^3 + 1822716*x^2 - 520884*x + 66996)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 90*x^15 + 69*x^14 + 225*x^13 - 2756*x^12 + 14733*x^11 - 41133*x^10 + 44796*x^9 + 31989*x^8 - 156159*x^7 + 1476721*x^6 - 3954150*x^5 + 5131062*x^4 - 3847170*x^3 + 1822716*x^2 - 520884*x + 66996, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 90 x^{15} + 69 x^{14} + 225 x^{13} - 2756 x^{12} + 14733 x^{11} - 41133 x^{10} + 44796 x^{9} + 31989 x^{8} - 156159 x^{7} + 1476721 x^{6} - 3954150 x^{5} + 5131062 x^{4} - 3847170 x^{3} + 1822716 x^{2} - 520884 x + 66996 \)

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:  \(-149893238403862212943036837746306920448=-\,2^{12}\cdot 3^{33}\cdot 37^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $132.09$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{5}{18} a^{3} + \frac{1}{3}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{10} + \frac{1}{3} a^{8} - \frac{1}{18} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{5}{18} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{7}{18} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{1}{18} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{2889774} a^{15} - \frac{1408}{481629} a^{14} + \frac{16538}{1444887} a^{13} + \frac{22373}{2889774} a^{12} - \frac{19793}{481629} a^{11} - \frac{36535}{1444887} a^{10} - \frac{9503}{321086} a^{9} - \frac{16582}{160543} a^{8} - \frac{471851}{1444887} a^{7} - \frac{302165}{2889774} a^{6} - \frac{161630}{481629} a^{5} - \frac{224375}{1444887} a^{4} + \frac{238723}{1444887} a^{3} + \frac{45943}{160543} a^{2} + \frac{141241}{481629} a - \frac{32531}{481629}$, $\frac{1}{14379229336374} a^{16} + \frac{200702}{7189614668187} a^{15} + \frac{13560040811}{7189614668187} a^{14} + \frac{142600973237}{7189614668187} a^{13} + \frac{332749856629}{14379229336374} a^{12} - \frac{140000966594}{7189614668187} a^{11} - \frac{149210008246}{7189614668187} a^{10} + \frac{213139686497}{4793076445458} a^{9} + \frac{2168381851222}{7189614668187} a^{8} + \frac{41094516802}{194313909951} a^{7} - \frac{4747618921453}{14379229336374} a^{6} + \frac{2173219307054}{7189614668187} a^{5} - \frac{997509987659}{14379229336374} a^{4} - \frac{4076492873461}{14379229336374} a^{3} + \frac{1121235211085}{2396538222729} a^{2} - \frac{883035544582}{2396538222729} a + \frac{140695430191}{2396538222729}$, $\frac{1}{991254021593777182113041358} a^{17} + \frac{12317873463347}{991254021593777182113041358} a^{16} + \frac{21645182329268426647}{141607717370539597444720194} a^{15} - \frac{25641159281949095939178067}{991254021593777182113041358} a^{14} + \frac{9420292691364434356917775}{495627010796888591056520679} a^{13} + \frac{8579313834891329329293590}{495627010796888591056520679} a^{12} + \frac{398289982977434774927819}{991254021593777182113041358} a^{11} - \frac{13504764018047575052030192}{495627010796888591056520679} a^{10} + \frac{4287812671237951447891577}{70803858685269798722360097} a^{9} + \frac{421225559664785929744983463}{991254021593777182113041358} a^{8} - \frac{98849244759214074637459831}{495627010796888591056520679} a^{7} - \frac{127681026413429062055274362}{495627010796888591056520679} a^{6} + \frac{9271800059980690400167505}{23601286228423266240786699} a^{5} + \frac{401534392543382747275073765}{991254021593777182113041358} a^{4} + \frac{258951282576120003684928123}{991254021593777182113041358} a^{3} + \frac{22577711887030000009406795}{55069667866320954561835631} a^{2} + \frac{47264847584917394136450829}{165209003598962863685506893} a - \frac{44060561666371839811248319}{165209003598962863685506893}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $243$ (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:  \( \frac{51784871992364}{323635252456957263} a^{17} - \frac{1328905490563906}{970905757370871789} a^{16} + \frac{147655908976202}{26240696145158697} a^{15} - \frac{3840759206132648}{323635252456957263} a^{14} + \frac{5491248338790685}{970905757370871789} a^{13} + \frac{2030949844197793}{52481392290317394} a^{12} - \frac{137138408867869331}{323635252456957263} a^{11} + \frac{233845061592584852}{107878417485652421} a^{10} - \frac{10888164825063543955}{1941811514741743578} a^{9} + \frac{499513027422845450}{107878417485652421} a^{8} + \frac{7058410337807289437}{970905757370871789} a^{7} - \frac{42312706842974400517}{1941811514741743578} a^{6} + \frac{73276185866066971909}{323635252456957263} a^{5} - \frac{515893669607217327986}{970905757370871789} a^{4} + \frac{375336094325324897273}{647270504913914526} a^{3} - \frac{37667826952933503084}{107878417485652421} a^{2} + \frac{41516951539339702252}{323635252456957263} a - \frac{2350161032465792150}{107878417485652421} \) (order $6$)
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:  \( 25473645362.465027 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.4107.1 x3, 6.0.5312031978672.11, 6.0.5312031978672.10, 6.0.2834352.4, 6.0.50602347.2

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$