↑

Properties

Label 18.0.45028390589...0000.7
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 5^{12}$
Root discriminant $44.40$
Ramified primes $2, 3, 5$
Class number $81$ (GRH)
Class group $[3, 3, 3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![204304, -178992, -119808, 291912, -58824, -126324, 74517, -10035, 5733, -11714, 6543, -639, -834, 315, 99, -114, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 114*x^15 + 99*x^14 + 315*x^13 - 834*x^12 - 639*x^11 + 6543*x^10 - 11714*x^9 + 5733*x^8 - 10035*x^7 + 74517*x^6 - 126324*x^5 - 58824*x^4 + 291912*x^3 - 119808*x^2 - 178992*x + 204304)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 114*x^15 + 99*x^14 + 315*x^13 - 834*x^12 - 639*x^11 + 6543*x^10 - 11714*x^9 + 5733*x^8 - 10035*x^7 + 74517*x^6 - 126324*x^5 - 58824*x^4 + 291912*x^3 - 119808*x^2 - 178992*x + 204304, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 114 x^{15} + 99 x^{14} + 315 x^{13} - 834 x^{12} - 639 x^{11} + 6543 x^{10} - 11714 x^{9} + 5733 x^{8} - 10035 x^{7} + 74517 x^{6} - 126324 x^{5} - 58824 x^{4} + 291912 x^{3} - 119808 x^{2} - 178992 x + 204304 \)

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:  \(-450283905890997363000000000000=-\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.40$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{9} + \frac{1}{3} a^{6} - \frac{1}{6} a^{3} + \frac{1}{9}$, $\frac{1}{18} a^{10} + \frac{1}{3} a^{7} - \frac{1}{6} a^{4} + \frac{1}{9} a$, $\frac{1}{18} a^{11} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{7}{18} a^{2}$, $\frac{1}{684} a^{12} - \frac{1}{114} a^{11} + \frac{1}{342} a^{10} - \frac{1}{684} a^{9} + \frac{7}{38} a^{8} - \frac{26}{57} a^{7} - \frac{33}{76} a^{6} - \frac{17}{38} a^{5} + \frac{23}{114} a^{4} + \frac{149}{684} a^{3} - \frac{29}{114} a^{2} + \frac{10}{171} a + \frac{28}{171}$, $\frac{1}{4104} a^{13} - \frac{1}{4104} a^{12} - \frac{11}{684} a^{11} - \frac{67}{4104} a^{10} - \frac{107}{4104} a^{9} - \frac{13}{57} a^{8} - \frac{35}{152} a^{7} - \frac{199}{456} a^{6} + \frac{5}{228} a^{5} + \frac{1067}{4104} a^{4} + \frac{571}{4104} a^{3} + \frac{62}{171} a^{2} - \frac{473}{1026} a - \frac{145}{1026}$, $\frac{1}{660744} a^{14} - \frac{11}{94392} a^{13} - \frac{11}{23598} a^{12} - \frac{1333}{94392} a^{11} - \frac{2245}{94392} a^{10} + \frac{895}{47196} a^{9} + \frac{8515}{73416} a^{8} - \frac{3691}{73416} a^{7} - \frac{203}{2622} a^{6} - \frac{19267}{94392} a^{5} - \frac{16795}{94392} a^{4} + \frac{12481}{47196} a^{3} - \frac{32023}{82593} a^{2} - \frac{107}{8694} a - \frac{40780}{82593}$, $\frac{1}{11232648} a^{15} - \frac{1}{1872108} a^{14} - \frac{5}{59432} a^{13} - \frac{179}{1604664} a^{12} - \frac{1277}{66861} a^{11} + \frac{2345}{178296} a^{10} + \frac{10891}{11232648} a^{9} - \frac{129433}{624036} a^{8} + \frac{124129}{1248072} a^{7} - \frac{8629}{1604664} a^{6} + \frac{5437}{66861} a^{5} - \frac{43409}{178296} a^{4} + \frac{102223}{1404081} a^{3} - \frac{8842}{468027} a^{2} + \frac{26849}{312018} a + \frac{278896}{1404081}$, $\frac{1}{157257072} a^{16} - \frac{1}{157257072} a^{15} + \frac{113}{157257072} a^{14} - \frac{166}{1404081} a^{13} - \frac{5125}{22465296} a^{12} + \frac{181753}{22465296} a^{11} - \frac{11177}{569772} a^{10} + \frac{3151775}{157257072} a^{9} + \frac{3977671}{17473008} a^{8} - \frac{5803907}{19657134} a^{7} - \frac{3014579}{22465296} a^{6} + \frac{1708255}{22465296} a^{5} + \frac{56061847}{157257072} a^{4} + \frac{2992301}{11232648} a^{3} + \frac{5934065}{39314268} a^{2} - \frac{19865}{267444} a - \frac{8966935}{19657134}$, $\frac{1}{33735958493369328} a^{17} - \frac{267203}{1405664937223722} a^{16} + \frac{515118263}{16867979246684664} a^{15} - \frac{1179853039}{11245319497789776} a^{14} - \frac{294082050149}{4819422641909904} a^{13} + \frac{1674269457389}{2409711320954952} a^{12} - \frac{1562655459079}{33735958493369328} a^{11} + \frac{41895682575833}{4819422641909904} a^{10} - \frac{84669851901527}{8433989623342332} a^{9} + \frac{344358610940543}{1775576762808912} a^{8} - \frac{2734870554649439}{11245319497789776} a^{7} + \frac{978864490960723}{2409711320954952} a^{6} + \frac{1094390280344891}{5622659748894888} a^{5} + \frac{242090317896487}{1466780804059536} a^{4} + \frac{5771742995117359}{16867979246684664} a^{3} + \frac{976297413703613}{4216994811671166} a^{2} - \frac{2086513087677917}{8433989623342332} a + \frac{12800721443945}{37318538156382}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $81$ (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:  \( -\frac{272746849}{36236260465488} a^{17} + \frac{250176793}{4529532558186} a^{16} - \frac{4388541383}{18118130232744} a^{15} + \frac{14637653759}{36236260465488} a^{14} + \frac{7928152547}{36236260465488} a^{13} - \frac{26473623211}{9059065116372} a^{12} + \frac{100387563077}{36236260465488} a^{11} + \frac{22076899069}{2131544733264} a^{10} - \frac{693570669725}{18118130232744} a^{9} + \frac{911501994481}{36236260465488} a^{8} + \frac{1496642207677}{36236260465488} a^{7} + \frac{20547117563}{532886183316} a^{6} - \frac{920102951975}{2264766279093} a^{5} + \frac{6660715405075}{36236260465488} a^{4} + \frac{6051126037043}{4529532558186} a^{3} - \frac{3282983412755}{2264766279093} a^{2} - \frac{7240253437679}{9059065116372} a + \frac{41326196150}{20042179461} \) (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:  \( 2421999.57121 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

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 $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24300.1 x3, 3.1.243.1 x3, 3.1.2700.1 x3, 3.1.24300.3 x3, 6.0.1771470000.3, 6.0.177147.2, 6.0.21870000.2, 6.0.1771470000.1, 9.1.387420489000000.9 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 R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$