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Properties

Label 18.0.86555874828...3583.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167^{3}$
Root discriminant $67.58$
Ramified primes $3, 7, 29, 167$
Class number $2016$ (GRH)
Class group $[12, 168]$ (GRH)
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12643463, 13292223, 2671494, 2018409, 8110695, 6557943, 769082, -1369188, -479694, 183507, 126270, -1179, -14358, -1764, 888, 159, -33, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 33*x^16 + 159*x^15 + 888*x^14 - 1764*x^13 - 14358*x^12 - 1179*x^11 + 126270*x^10 + 183507*x^9 - 479694*x^8 - 1369188*x^7 + 769082*x^6 + 6557943*x^5 + 8110695*x^4 + 2018409*x^3 + 2671494*x^2 + 13292223*x + 12643463)
 
gp: K = bnfinit(x^18 - 6*x^17 - 33*x^16 + 159*x^15 + 888*x^14 - 1764*x^13 - 14358*x^12 - 1179*x^11 + 126270*x^10 + 183507*x^9 - 479694*x^8 - 1369188*x^7 + 769082*x^6 + 6557943*x^5 + 8110695*x^4 + 2018409*x^3 + 2671494*x^2 + 13292223*x + 12643463, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 33 x^{16} + 159 x^{15} + 888 x^{14} - 1764 x^{13} - 14358 x^{12} - 1179 x^{11} + 126270 x^{10} + 183507 x^{9} - 479694 x^{8} - 1369188 x^{7} + 769082 x^{6} + 6557943 x^{5} + 8110695 x^{4} + 2018409 x^{3} + 2671494 x^{2} + 13292223 x + 12643463 \)

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:  \(-865558748289981242920101560093583=-\,3^{18}\cdot 7^{14}\cdot 29^{4}\cdot 167^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.58$
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:  $3, 7, 29, 167$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{7} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{256949140988726751472820790847626784690960451786812} a^{17} + \frac{14283073303616013541406979000485120966220727746945}{256949140988726751472820790847626784690960451786812} a^{16} + \frac{30136706063157814402128098829181407707972053403669}{128474570494363375736410395423813392345480225893406} a^{15} - \frac{34863157059647650098053813658838348453691712992465}{256949140988726751472820790847626784690960451786812} a^{14} + \frac{49703498304792469361129955134415944899061209692803}{256949140988726751472820790847626784690960451786812} a^{13} - \frac{7143435797989185331560215419235479547061875011921}{256949140988726751472820790847626784690960451786812} a^{12} + \frac{23743639099032985973829522328510222061578074941547}{256949140988726751472820790847626784690960451786812} a^{11} - \frac{32014957267427055278117521266462852402830421571027}{128474570494363375736410395423813392345480225893406} a^{10} + \frac{17941479206946262914563106770308896567733278573351}{128474570494363375736410395423813392345480225893406} a^{9} + \frac{102465272189526069314739392499262545246686875911553}{256949140988726751472820790847626784690960451786812} a^{8} - \frac{74350147526944523423995984844313786607728360953441}{256949140988726751472820790847626784690960451786812} a^{7} + \frac{64917638335451454849234007007057261248805296108933}{256949140988726751472820790847626784690960451786812} a^{6} + \frac{82365809767467084452212495770044777814227827467405}{256949140988726751472820790847626784690960451786812} a^{5} - \frac{205442241369579668653164300576888950637648190848}{773943195749176962267532502553092724972772445141} a^{4} + \frac{1385972717730253505013869276081958398532905405651}{256949140988726751472820790847626784690960451786812} a^{3} + \frac{28313226374015889547563068681181631520703385506485}{64237285247181687868205197711906696172740112946703} a^{2} - \frac{338326208763623187611841296207792942740346594762}{773943195749176962267532502553092724972772445141} a + \frac{97974968551400109100278044916831164416378827980803}{256949140988726751472820790847626784690960451786812}$

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

Class group and class number

$C_{12}\times C_{168}$, which has order $2016$ (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:  \( -1 \) (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:  \( 1152993.7459 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.0.400967.1, 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R $18$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{7}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
$3$3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
3.9.9.2$x^{9} + 18 x^{3} + 27 x + 27$$3$$3$$9$$C_3^2 : S_3 $$[3/2, 3/2]_{2}^{3}$
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
29.3.2.1$x^{3} - 29$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
167Data not computed