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Properties

Label 18.10.3826859639...2896.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{6}\cdot 3^{32}\cdot 19^{9}$
Root discriminant $38.72$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T585

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-27, 0, 243, -135, -432, 432, -108, 81, -72, -46, 21, -42, 135, -51, -60, 39, 3, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 3*x^16 + 39*x^15 - 60*x^14 - 51*x^13 + 135*x^12 - 42*x^11 + 21*x^10 - 46*x^9 - 72*x^8 + 81*x^7 - 108*x^6 + 432*x^5 - 432*x^4 - 135*x^3 + 243*x^2 - 27)
 
gp: K = bnfinit(x^18 - 6*x^17 + 3*x^16 + 39*x^15 - 60*x^14 - 51*x^13 + 135*x^12 - 42*x^11 + 21*x^10 - 46*x^9 - 72*x^8 + 81*x^7 - 108*x^6 + 432*x^5 - 432*x^4 - 135*x^3 + 243*x^2 - 27, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 3 x^{16} + 39 x^{15} - 60 x^{14} - 51 x^{13} + 135 x^{12} - 42 x^{11} + 21 x^{10} - 46 x^{9} - 72 x^{8} + 81 x^{7} - 108 x^{6} + 432 x^{5} - 432 x^{4} - 135 x^{3} + 243 x^{2} - 27 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(38268596395430935911408712896=2^{6}\cdot 3^{32}\cdot 19^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.72$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{6} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{14} + \frac{1}{3} a^{11} + \frac{1}{6} a^{10} + \frac{1}{6} a^{9} - \frac{1}{3} a^{7} + \frac{1}{6} a^{6} - \frac{1}{18} a^{5} + \frac{1}{6} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{342} a^{15} - \frac{1}{38} a^{13} - \frac{5}{38} a^{12} + \frac{20}{57} a^{11} - \frac{10}{57} a^{10} - \frac{8}{19} a^{9} + \frac{49}{114} a^{8} + \frac{26}{57} a^{7} + \frac{13}{171} a^{6} + \frac{43}{114} a^{5} + \frac{23}{57} a^{4} - \frac{23}{114} a^{3} - \frac{2}{19} a^{2} - \frac{3}{38} a + \frac{17}{38}$, $\frac{1}{23598} a^{16} - \frac{13}{874} a^{14} - \frac{88}{1311} a^{13} + \frac{211}{7866} a^{12} + \frac{1861}{7866} a^{11} - \frac{1289}{2622} a^{10} + \frac{1987}{7866} a^{9} + \frac{2503}{7866} a^{8} + \frac{881}{23598} a^{7} + \frac{335}{3933} a^{6} - \frac{34}{3933} a^{5} + \frac{1271}{3933} a^{4} + \frac{28}{57} a^{3} + \frac{464}{1311} a^{2} + \frac{815}{2622} a - \frac{7}{46}$, $\frac{1}{5804211276} a^{17} + \frac{116377}{5804211276} a^{16} + \frac{188941}{161228091} a^{15} + \frac{990759}{214970788} a^{14} + \frac{86577199}{1934737092} a^{13} + \frac{34786772}{483684273} a^{12} + \frac{770227}{1934737092} a^{11} + \frac{598165687}{1934737092} a^{10} + \frac{236633336}{483684273} a^{9} + \frac{200551541}{1451052819} a^{8} - \frac{206304923}{2902105638} a^{7} - \frac{964625551}{1934737092} a^{6} + \frac{6366869}{1934737092} a^{5} + \frac{743406493}{1934737092} a^{4} - \frac{86504339}{214970788} a^{3} - \frac{9421334}{53742697} a^{2} - \frac{18129535}{644912364} a - \frac{48176735}{214970788}$

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:  $13$
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:  \( 171649877.818 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T585:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 13824
The 96 conjugacy class representatives for t18n585 are not computed
Character table for t18n585 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.5609891727441.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
$3$3.9.16.1$x^{9} + 3 x^{8} + 3 x^{6} + 3$$9$$1$$16$$C_3^2:C_3$$[2, 2]^{3}$
3.9.16.1$x^{9} + 3 x^{8} + 3 x^{6} + 3$$9$$1$$16$$C_3^2:C_3$$[2, 2]^{3}$
$19$19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.0.1$x^{3} - x + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
19.3.2.3$x^{3} - 304$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.3$x^{6} - 4864$$6$$1$$5$$C_6$$[\ ]_{6}$