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Properties

Label 18.0.13724760780...9888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{26}\cdot 3^{25}\cdot 17^{6}$
Root discriminant $32.18$
Ramified primes $2, 3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times S_3\wr C_2$ (as 18T63)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 96, 504, 984, 2124, 738, 3945, 300, 4815, -1718, 3126, -684, 843, -126, 162, -12, 15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 15*x^16 - 12*x^15 + 162*x^14 - 126*x^13 + 843*x^12 - 684*x^11 + 3126*x^10 - 1718*x^9 + 4815*x^8 + 300*x^7 + 3945*x^6 + 738*x^5 + 2124*x^4 + 984*x^3 + 504*x^2 + 96*x + 16)
 
gp: K = bnfinit(x^18 + 15*x^16 - 12*x^15 + 162*x^14 - 126*x^13 + 843*x^12 - 684*x^11 + 3126*x^10 - 1718*x^9 + 4815*x^8 + 300*x^7 + 3945*x^6 + 738*x^5 + 2124*x^4 + 984*x^3 + 504*x^2 + 96*x + 16, 1)
 

Normalized defining polynomial

\( x^{18} + 15 x^{16} - 12 x^{15} + 162 x^{14} - 126 x^{13} + 843 x^{12} - 684 x^{11} + 3126 x^{10} - 1718 x^{9} + 4815 x^{8} + 300 x^{7} + 3945 x^{6} + 738 x^{5} + 2124 x^{4} + 984 x^{3} + 504 x^{2} + 96 x + 16 \)

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:  \(-1372476078024604464225189888=-\,2^{26}\cdot 3^{25}\cdot 17^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.18$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} + \frac{1}{3} a$, $\frac{1}{18} a^{11} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{7}{18} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{9} + \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1}{3} a^{4} - \frac{7}{18} a^{3} + \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{18} a^{13} - \frac{1}{18} a^{10} - \frac{1}{6} a^{8} - \frac{1}{6} a^{5} - \frac{7}{18} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{9} a$, $\frac{1}{18} a^{14} + \frac{1}{18} a^{10} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{3} a^{7} - \frac{2}{9} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{13} - \frac{1}{18} a^{10} - \frac{1}{36} a^{9} - \frac{1}{6} a^{8} - \frac{1}{6} a^{7} + \frac{1}{18} a^{6} - \frac{5}{12} a^{5} - \frac{2}{9} a^{4} + \frac{1}{4} a^{3} + \frac{1}{3} a^{2} + \frac{2}{9} a + \frac{4}{9}$, $\frac{1}{72} a^{16} - \frac{1}{72} a^{14} - \frac{1}{36} a^{12} - \frac{1}{36} a^{11} - \frac{1}{72} a^{10} - \frac{1}{18} a^{9} - \frac{1}{12} a^{8} - \frac{5}{36} a^{7} + \frac{1}{24} a^{6} + \frac{7}{18} a^{5} - \frac{1}{24} a^{4} - \frac{5}{36} a^{3} + \frac{1}{9} a^{2} + \frac{1}{18} a - \frac{4}{9}$, $\frac{1}{210775059321787368} a^{17} + \frac{124396099640525}{105387529660893684} a^{16} + \frac{290249614063251}{23419451035754152} a^{15} - \frac{543218868083455}{35129176553631228} a^{14} - \frac{455786732644043}{105387529660893684} a^{13} - \frac{491626147584011}{105387529660893684} a^{12} - \frac{27978319956827}{70258353107262456} a^{11} + \frac{6788630821791671}{105387529660893684} a^{10} - \frac{8656109597120531}{105387529660893684} a^{9} - \frac{428284155664073}{105387529660893684} a^{8} - \frac{88861488173751925}{210775059321787368} a^{7} + \frac{11867377287515023}{35129176553631228} a^{6} + \frac{4648122352140419}{70258353107262456} a^{5} - \frac{4934482321028507}{52693764830446842} a^{4} - \frac{10344878070656500}{26346882415223421} a^{3} - \frac{1126175775282121}{8782294138407807} a^{2} + \frac{8395804731101401}{26346882415223421} a + \frac{530878911835610}{26346882415223421}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{1097937536208845}{105387529660893684} a^{17} - \frac{11625616927711}{70258353107262456} a^{16} + \frac{5520264539438959}{35129176553631228} a^{15} - \frac{26962061984973107}{210775059321787368} a^{14} + \frac{14951091202312900}{8782294138407807} a^{13} - \frac{15880721778335647}{11709725517877076} a^{12} + \frac{235653036595353107}{26346882415223421} a^{11} - \frac{522623394247095685}{70258353107262456} a^{10} + \frac{586874475411311309}{17564588276815614} a^{9} - \frac{2036227389924529745}{105387529660893684} a^{8} + \frac{467183243039216555}{8782294138407807} a^{7} - \frac{19115336396740577}{70258353107262456} a^{6} + \frac{4747406253632993449}{105387529660893684} a^{5} + \frac{399459899913120229}{70258353107262456} a^{4} + \frac{285435385243534383}{11709725517877076} a^{3} + \frac{421837396604080349}{52693764830446842} a^{2} + \frac{106622983802244241}{17564588276815614} a + \frac{10216334086907942}{8782294138407807} \) (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:  \( 5741990.78627 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\wr C_2$ (as 18T63):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 18 conjugacy class representatives for $C_2\times S_3\wr C_2$
Character table for $C_2\times S_3\wr C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 9.9.21389063233536.1

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.22.60$x^{12} - 84 x^{10} + 444 x^{8} + 32 x^{6} - 272 x^{4} - 320 x^{2} + 64$$6$$2$$22$$D_6$$[3]_{3}^{2}$
3Data not computed
$17$17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$