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Properties

Label 18.0.48036106180...0128.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 7^{12}$
Root discriminant $26.71$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![225, 585, 846, 675, 261, -117, -165, 30, 375, 363, 108, -102, -80, -25, 10, -2, 1, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + x^16 - 2*x^15 + 10*x^14 - 25*x^13 - 80*x^12 - 102*x^11 + 108*x^10 + 363*x^9 + 375*x^8 + 30*x^7 - 165*x^6 - 117*x^5 + 261*x^4 + 675*x^3 + 846*x^2 + 585*x + 225)
 
gp: K = bnfinit(x^18 + x^16 - 2*x^15 + 10*x^14 - 25*x^13 - 80*x^12 - 102*x^11 + 108*x^10 + 363*x^9 + 375*x^8 + 30*x^7 - 165*x^6 - 117*x^5 + 261*x^4 + 675*x^3 + 846*x^2 + 585*x + 225, 1)
 

Normalized defining polynomial

\( x^{18} + x^{16} - 2 x^{15} + 10 x^{14} - 25 x^{13} - 80 x^{12} - 102 x^{11} + 108 x^{10} + 363 x^{9} + 375 x^{8} + 30 x^{7} - 165 x^{6} - 117 x^{5} + 261 x^{4} + 675 x^{3} + 846 x^{2} + 585 x + 225 \)

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:  \(-48036106180347836985520128=-\,2^{12}\cdot 3^{25}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.71$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{63} a^{12} - \frac{3}{7} a^{11} - \frac{8}{63} a^{10} + \frac{22}{63} a^{9} - \frac{26}{63} a^{8} - \frac{1}{63} a^{7} + \frac{31}{63} a^{6} - \frac{2}{21} a^{5} + \frac{1}{7} a^{4} + \frac{2}{21} a^{3} - \frac{5}{21} a^{2} + \frac{2}{21} a - \frac{2}{21}$, $\frac{1}{63} a^{13} + \frac{19}{63} a^{11} - \frac{5}{63} a^{10} + \frac{1}{63} a^{9} - \frac{10}{63} a^{8} + \frac{4}{63} a^{7} + \frac{4}{21} a^{6} - \frac{3}{7} a^{5} - \frac{1}{21} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{10}{21} a + \frac{3}{7}$, $\frac{1}{189} a^{14} + \frac{1}{189} a^{12} - \frac{86}{189} a^{11} - \frac{44}{189} a^{10} - \frac{13}{27} a^{9} + \frac{94}{189} a^{8} - \frac{11}{63} a^{7} + \frac{5}{21} a^{6} - \frac{1}{9} a^{5} + \frac{16}{63} a^{4} - \frac{1}{63} a^{3} + \frac{16}{63} a^{2} - \frac{3}{7} a - \frac{3}{7}$, $\frac{1}{189} a^{15} + \frac{1}{189} a^{13} + \frac{1}{189} a^{12} + \frac{64}{189} a^{11} - \frac{31}{189} a^{10} - \frac{71}{189} a^{9} - \frac{1}{7} a^{8} - \frac{2}{9} a^{7} + \frac{10}{63} a^{6} + \frac{31}{63} a^{5} + \frac{8}{63} a^{4} + \frac{1}{63} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{5}{21}$, $\frac{1}{6615} a^{16} - \frac{16}{6615} a^{15} - \frac{13}{6615} a^{14} - \frac{8}{2205} a^{13} + \frac{19}{6615} a^{12} + \frac{1706}{6615} a^{11} - \frac{349}{735} a^{10} + \frac{61}{135} a^{9} + \frac{2389}{6615} a^{8} + \frac{211}{735} a^{7} - \frac{908}{2205} a^{6} - \frac{73}{245} a^{5} + \frac{48}{245} a^{4} - \frac{116}{2205} a^{3} - \frac{737}{2205} a^{2} - \frac{1}{35} a - \frac{11}{147}$, $\frac{1}{2399488446285} a^{17} - \frac{5338834}{88869942455} a^{16} - \frac{5343444236}{2399488446285} a^{15} - \frac{2026982306}{799829482095} a^{14} + \frac{15787513372}{2399488446285} a^{13} + \frac{30788542}{48969151965} a^{12} - \frac{193091236661}{799829482095} a^{11} - \frac{745678055509}{2399488446285} a^{10} - \frac{746575834604}{2399488446285} a^{9} + \frac{132552556103}{342784063755} a^{8} - \frac{357798558349}{799829482095} a^{7} - \frac{209623558831}{799829482095} a^{6} + \frac{228267665021}{799829482095} a^{5} + \frac{44753551207}{159965896419} a^{4} + \frac{10559079341}{159965896419} a^{3} - \frac{284164443589}{799829482095} a^{2} - \frac{17533335238}{266609827365} a - \frac{7045681574}{17773988491}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{8222758}{48969151965} a^{17} - \frac{2083697}{1813672295} a^{16} + \frac{253334497}{48969151965} a^{15} - \frac{505530464}{48969151965} a^{14} + \frac{1040501191}{48969151965} a^{13} - \frac{2314488691}{48969151965} a^{12} + \frac{5159676151}{48969151965} a^{11} - \frac{2882825389}{16323050655} a^{10} + \frac{554039267}{5441016885} a^{9} - \frac{4768154179}{16323050655} a^{8} + \frac{7908952523}{16323050655} a^{7} + \frac{182394982}{16323050655} a^{6} + \frac{4722233098}{16323050655} a^{5} - \frac{295233656}{1088203377} a^{4} + \frac{227745688}{1088203377} a^{3} - \frac{366021613}{1813672295} a^{2} + \frac{5628007861}{5441016885} a + \frac{800959222}{1088203377} \) (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:  \( 1035241.6408124544 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.588.1 x3, 6.0.1037232.1, 9.3.1333834713792.2

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.11.11$x^{6} + 9 x^{3} + 21$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
3.6.11.11$x^{6} + 9 x^{3} + 21$$6$$1$$11$$S_3\times C_3$$[5/2]_{2}^{3}$
$7$7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.3$x^{3} - 28$$3$$1$$2$$C_3$$[\ ]_{3}$