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Properties

Label 18.0.50912632855...4976.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{31}\cdot 17^{3}$
Root discriminant $26.80$
Ramified primes $2, 3, 17$
Class number $4$
Class group $[2, 2]$
Galois group $C_2\times S_3\times A_4$ (as 18T60)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5041, 9978, 14298, 16230, 15561, 11526, 7647, -294, -1107, -2302, 60, -30, 348, -18, 15, -24, 3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 24*x^15 + 15*x^14 - 18*x^13 + 348*x^12 - 30*x^11 + 60*x^10 - 2302*x^9 - 1107*x^8 - 294*x^7 + 7647*x^6 + 11526*x^5 + 15561*x^4 + 16230*x^3 + 14298*x^2 + 9978*x + 5041)
 
gp: K = bnfinit(x^18 + 3*x^16 - 24*x^15 + 15*x^14 - 18*x^13 + 348*x^12 - 30*x^11 + 60*x^10 - 2302*x^9 - 1107*x^8 - 294*x^7 + 7647*x^6 + 11526*x^5 + 15561*x^4 + 16230*x^3 + 14298*x^2 + 9978*x + 5041, 1)
 

Normalized defining polynomial

\( x^{18} + 3 x^{16} - 24 x^{15} + 15 x^{14} - 18 x^{13} + 348 x^{12} - 30 x^{11} + 60 x^{10} - 2302 x^{9} - 1107 x^{8} - 294 x^{7} + 7647 x^{6} + 11526 x^{5} + 15561 x^{4} + 16230 x^{3} + 14298 x^{2} + 9978 x + 5041 \)

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:  \(-50912632855685765032574976=-\,2^{24}\cdot 3^{31}\cdot 17^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $26.80$
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 not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{31947187383917396134985141932158593} a^{17} - \frac{80689210258899214012753506893294}{31947187383917396134985141932158593} a^{16} + \frac{2599087074212315279382001227937609}{31947187383917396134985141932158593} a^{15} + \frac{8788521361070764018194679335380159}{31947187383917396134985141932158593} a^{14} - \frac{9679440006159899826522906438192583}{31947187383917396134985141932158593} a^{13} + \frac{10863451524096262185641891565031664}{31947187383917396134985141932158593} a^{12} + \frac{2775864670671788340491829626811640}{31947187383917396134985141932158593} a^{11} - \frac{9837476714469656368201496981062208}{31947187383917396134985141932158593} a^{10} - \frac{3054570936743698196201117002558756}{31947187383917396134985141932158593} a^{9} - \frac{1658029275184937936162775241149664}{31947187383917396134985141932158593} a^{8} + \frac{8764032207403788352818601464992733}{31947187383917396134985141932158593} a^{7} + \frac{13904142091585769525591409673871425}{31947187383917396134985141932158593} a^{6} - \frac{677023212734995950036137581695434}{31947187383917396134985141932158593} a^{5} - \frac{2833481733193423216171614753526656}{31947187383917396134985141932158593} a^{4} + \frac{14445755643024181934903150526156098}{31947187383917396134985141932158593} a^{3} - \frac{11653570078836918641208832299989756}{31947187383917396134985141932158593} a^{2} + \frac{5507601459427138745960448185499968}{31947187383917396134985141932158593} a - \frac{479152885665014147505542297273643}{31947187383917396134985141932158593}$

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

Class group and class number

$C_{2}\times C_{2}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 73200.0215742 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times S_3\times A_4$ (as 18T60):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 144
The 24 conjugacy class representatives for $C_2\times S_3\times A_4$
Character table for $C_2\times S_3\times A_4$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.108.1, 6.0.21415104.1, 9.3.918330048.1

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

Sibling fields

Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.1.1$x^{2} - 17$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.4.2.1$x^{4} + 85 x^{2} + 2601$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$