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Properties

Label 18.10.1561944598...0864.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{2}$
Root discriminant $54.07$
Ramified primes $2, 3, 7, 12547$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T268

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-98331, -35109, 153729, 13578, -42750, 19161, -34608, -2175, 11151, -4649, 3996, 882, -1134, 81, -90, -9, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 - 9*x^15 - 90*x^14 + 81*x^13 - 1134*x^12 + 882*x^11 + 3996*x^10 - 4649*x^9 + 11151*x^8 - 2175*x^7 - 34608*x^6 + 19161*x^5 - 42750*x^4 + 13578*x^3 + 153729*x^2 - 35109*x - 98331)
 
gp: K = bnfinit(x^18 + 12*x^16 - 9*x^15 - 90*x^14 + 81*x^13 - 1134*x^12 + 882*x^11 + 3996*x^10 - 4649*x^9 + 11151*x^8 - 2175*x^7 - 34608*x^6 + 19161*x^5 - 42750*x^4 + 13578*x^3 + 153729*x^2 - 35109*x - 98331, 1)
 

Normalized defining polynomial

\( x^{18} + 12 x^{16} - 9 x^{15} - 90 x^{14} + 81 x^{13} - 1134 x^{12} + 882 x^{11} + 3996 x^{10} - 4649 x^{9} + 11151 x^{8} - 2175 x^{7} - 34608 x^{6} + 19161 x^{5} - 42750 x^{4} + 13578 x^{3} + 153729 x^{2} - 35109 x - 98331 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(15619445986872425452605886500864=2^{12}\cdot 3^{30}\cdot 7^{6}\cdot 12547^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.07$
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:  $2, 3, 7, 12547$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{18} a^{12} - \frac{1}{2} a^{10} + \frac{7}{18} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a + \frac{1}{6}$, $\frac{1}{18} a^{13} - \frac{1}{2} a^{11} + \frac{7}{18} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{6} a$, $\frac{1}{18} a^{14} + \frac{7}{18} a^{11} - \frac{1}{2} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{324} a^{15} - \frac{1}{36} a^{14} + \frac{1}{54} a^{13} - \frac{1}{324} a^{12} + \frac{1}{6} a^{11} - \frac{1}{3} a^{10} + \frac{73}{324} a^{9} - \frac{17}{36} a^{8} + \frac{17}{54} a^{7} + \frac{49}{108} a^{6} + \frac{5}{12} a^{5} + \frac{1}{12} a^{4} + \frac{7}{108} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{17}{108}$, $\frac{1}{324} a^{16} - \frac{1}{108} a^{14} - \frac{1}{324} a^{13} - \frac{1}{36} a^{12} + \frac{2}{9} a^{11} - \frac{143}{324} a^{10} + \frac{1}{18} a^{9} - \frac{29}{108} a^{8} + \frac{31}{108} a^{7} - \frac{1}{6} a^{5} - \frac{5}{27} a^{4} - \frac{1}{36} a^{3} - \frac{1}{2} a^{2} - \frac{17}{108} a + \frac{1}{12}$, $\frac{1}{6844092343407898931610490644} a^{17} - \frac{6435887750757234215658155}{6844092343407898931610490644} a^{16} + \frac{1242023890272814490367715}{6844092343407898931610490644} a^{15} + \frac{67621169160983060214782893}{3422046171703949465805245322} a^{14} + \frac{69673864403840116056813985}{3422046171703949465805245322} a^{13} + \frac{92846318079510113647392125}{6844092343407898931610490644} a^{12} - \frac{2923483268102292463840184087}{6844092343407898931610490644} a^{11} - \frac{203533974482020770374839505}{6844092343407898931610490644} a^{10} - \frac{2420535807539224730860464275}{6844092343407898931610490644} a^{9} - \frac{183062584315117895581958377}{570341028617324910967540887} a^{8} - \frac{561729810377412242728005169}{2281364114469299643870163548} a^{7} - \frac{44037092888185307540917202}{570341028617324910967540887} a^{6} - \frac{133077016115189322721913084}{570341028617324910967540887} a^{5} - \frac{1055014031946621039356245445}{2281364114469299643870163548} a^{4} + \frac{1090257240681356872763242057}{2281364114469299643870163548} a^{3} + \frac{1036592097116792750316686437}{2281364114469299643870163548} a^{2} + \frac{21355274598444457830177466}{570341028617324910967540887} a + \frac{40007949214257606656349235}{2281364114469299643870163548}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $13$
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:  \( 2068119064.87 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T268:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1152
The 24 conjugacy class representatives for t18n268
Character table for t18n268 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.756.1, 9.9.314987206464.1

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

Sibling fields

Degree 12 siblings: data not computed
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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.9.15.13$x^{9} + 3 x^{7} + 6 x^{6} + 6 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
3.9.15.13$x^{9} + 3 x^{7} + 6 x^{6} + 6 x^{3} + 3$$9$$1$$15$$S_3\times C_3$$[3/2, 2]_{2}$
$7$7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
12547Data not computed