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Properties

Label 18.0.54291600394...2083.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 19^{6}\cdot 73^{6}$
Root discriminant $57.95$
Ramified primes $3, 19, 73$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![13997521, 0, 0, 11238213, 0, 0, 2773366, 0, 0, -57335, 0, 0, 3718, 0, 0, -43, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 43*x^15 + 3718*x^12 - 57335*x^9 + 2773366*x^6 + 11238213*x^3 + 13997521)
 
gp: K = bnfinit(x^18 - 43*x^15 + 3718*x^12 - 57335*x^9 + 2773366*x^6 + 11238213*x^3 + 13997521, 1)
 

Normalized defining polynomial

\( x^{18} - 43 x^{15} + 3718 x^{12} - 57335 x^{9} + 2773366 x^{6} + 11238213 x^{3} + 13997521 \)

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:  \(-54291600394481055845270041612083=-\,3^{27}\cdot 19^{6}\cdot 73^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.95$
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:  $3, 19, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{6} - \frac{1}{2} a^{3} - \frac{1}{4}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{7} - \frac{1}{2} a^{4} - \frac{1}{4} a$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{8} - \frac{1}{2} a^{5} - \frac{1}{4} a^{2}$, $\frac{1}{1789468374251925484} a^{15} + \frac{1961292607169593}{68825706701997134} a^{12} + \frac{3615005508345853}{137651413403994268} a^{9} + \frac{37487708616590143}{447367093562981371} a^{6} + \frac{261517022601436879}{1789468374251925484} a^{3} + \frac{139335441776316975}{447367093562981371}$, $\frac{1}{431261878194714041644} a^{16} + \frac{1171998306541120871}{16586995315181309294} a^{13} - \frac{5089487290439442063}{33173990630362618588} a^{10} + \frac{932221895742552885}{107815469548678510411} a^{7} + \frac{186366227944801687215}{431261878194714041644} a^{4} + \frac{46665513172326379559}{107815469548678510411} a$, $\frac{1}{103934112644926084036204} a^{17} + \frac{76985475531398133565}{7994931741917391079708} a^{14} + \frac{1355044128554427920045}{7994931741917391079708} a^{11} - \frac{10885633536833559339971}{103934112644926084036204} a^{8} - \frac{17279739838941116999367}{103934112644926084036204} a^{5} - \frac{25796866108542215490815}{103934112644926084036204} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( \frac{17372760042092093}{103934112644926084036204} a^{17} - \frac{60334685465171153}{7994931741917391079708} a^{14} + \frac{5096490511728453187}{7994931741917391079708} a^{11} - \frac{1143490527043905070027}{103934112644926084036204} a^{8} + \frac{50666845682732881557587}{103934112644926084036204} a^{5} + \frac{70358138985580163383475}{103934112644926084036204} a^{2} \) (order $18$)
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:  \( 362211450.25290346 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 6.0.37865545227.2, Deg 6, \(\Q(\zeta_{9})\), 6.0.51941763.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
3Data not computed
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$
73.3.2.2$x^{3} + 365$$3$$1$$2$$C_3$$[\ ]_{3}$