↑

Properties

Label 18.0.13521742669...6603.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 110017^{6}$
Root discriminant $248.98$
Ramified primes $3, 110017$
Class number $1108$ (GRH)
Class group $[2, 554]$ (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![68719476736, 0, 0, -6291718144, 0, 0, 244702657, 0, 0, -5132480, 0, 0, 60865, 0, 0, -384, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 384*x^15 + 60865*x^12 - 5132480*x^9 + 244702657*x^6 - 6291718144*x^3 + 68719476736)
 
gp: K = bnfinit(x^18 - 384*x^15 + 60865*x^12 - 5132480*x^9 + 244702657*x^6 - 6291718144*x^3 + 68719476736, 1)
 

Normalized defining polynomial

\( x^{18} - 384 x^{15} + 60865 x^{12} - 5132480 x^{9} + 244702657 x^{6} - 6291718144 x^{3} + 68719476736 \)

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:  \(-13521742669994394164761026842904343927116603=-\,3^{27}\cdot 110017^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.98$
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:  $3, 110017$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{382} a^{9} - \frac{1}{382} a^{6} + \frac{63}{382} a^{3} - \frac{46}{191}$, $\frac{1}{382} a^{10} - \frac{1}{382} a^{7} + \frac{63}{382} a^{4} - \frac{46}{191} a$, $\frac{1}{382} a^{11} - \frac{1}{382} a^{8} + \frac{63}{382} a^{5} - \frac{46}{191} a^{2}$, $\frac{1}{382} a^{12} + \frac{31}{191} a^{6} - \frac{29}{382} a^{3} - \frac{46}{191}$, $\frac{1}{12224} a^{13} - \frac{511}{12224} a^{7} - \frac{87}{191} a^{4} - \frac{5631}{12224} a$, $\frac{1}{782336} a^{14} - \frac{3}{6112} a^{11} + \frac{85441}{782336} a^{8} - \frac{3139}{12224} a^{5} + \frac{335297}{782336} a^{2}$, $\frac{1}{1369951698944} a^{15} - \frac{12032003}{10702747648} a^{12} - \frac{1260720703}{1369951698944} a^{9} + \frac{2363449021}{21405495296} a^{6} + \frac{175118998977}{1369951698944} a^{3} - \frac{183335}{5225951}$, $\frac{1}{87676908732416} a^{16} - \frac{12032003}{684975849472} a^{13} - \frac{47882113599}{87676908732416} a^{10} - \frac{275179530563}{1369951698944} a^{7} - \frac{31531014431295}{87676908732416} a^{4} - \frac{33843509}{83615216} a$, $\frac{1}{5611322158874624} a^{17} - \frac{12032003}{43838454366208} a^{14} - \frac{3720213369407}{5611322158874624} a^{11} + \frac{14166693484221}{87676908732416} a^{8} + \frac{920750384340417}{5611322158874624} a^{5} - \frac{2011058813}{5351373824} a^{2}$

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

Class group and class number

$C_{2}\times C_{554}$, which has order $1108$ (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:  \( \frac{268543}{43838454366208} a^{17} - \frac{660717}{342487924736} a^{14} + \frac{10421726015}{43838454366208} a^{11} - \frac{9899748765}{684975849472} a^{8} + \frac{19211859524415}{43838454366208} a^{5} - \frac{112909692511}{21405495296} a^{2} \) (order $18$)
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:  \( 829326601048.2684 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), Deg 6, 6.0.326800987803.1, \(\Q(\zeta_{9})\), 6.0.238237920108387.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
110017Data not computed