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Properties

Label 18.0.28634806194...9216.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{32}\cdot 11^{9}$
Root discriminant $43.30$
Ramified primes $2, 3, 11$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times D_9:C_3$ (as 18T45)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![78732, 0, 0, 69984, 0, 0, 26244, 0, 0, -272, 0, 0, 60, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 60*x^12 - 272*x^9 + 26244*x^6 + 69984*x^3 + 78732)
 
gp: K = bnfinit(x^18 + 60*x^12 - 272*x^9 + 26244*x^6 + 69984*x^3 + 78732, 1)
 

Normalized defining polynomial

\( x^{18} + 60 x^{12} - 272 x^{9} + 26244 x^{6} + 69984 x^{3} + 78732 \)

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:  \(-286348061945337113939091849216=-\,2^{16}\cdot 3^{32}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $43.30$
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:  $2, 3, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{12} a^{10} - \frac{1}{2} a^{4} - \frac{1}{6} a$, $\frac{1}{36} a^{11} + \frac{1}{6} a^{5} - \frac{1}{18} a^{2}$, $\frac{1}{108} a^{12} + \frac{1}{18} a^{6} - \frac{1}{54} a^{3}$, $\frac{1}{324} a^{13} + \frac{1}{54} a^{7} + \frac{53}{162} a^{4} + \frac{1}{3} a$, $\frac{1}{2916} a^{14} - \frac{1}{972} a^{13} + \frac{1}{324} a^{12} + \frac{1}{108} a^{11} - \frac{1}{36} a^{10} - \frac{71}{486} a^{8} + \frac{17}{162} a^{7} + \frac{1}{54} a^{6} + \frac{94}{729} a^{5} - \frac{13}{243} a^{4} + \frac{53}{162} a^{3} + \frac{17}{54} a^{2} - \frac{5}{18} a + \frac{1}{3}$, $\frac{1}{22483033596} a^{15} - \frac{2851675}{832704948} a^{12} + \frac{33758299}{3747172266} a^{9} + \frac{133162069}{5620758399} a^{6} + \frac{118826395}{416352474} a^{3} + \frac{3403633}{7710231}$, $\frac{1}{67449100788} a^{16} - \frac{2851675}{2498114844} a^{13} + \frac{33758299}{11241516798} a^{10} + \frac{133162069}{16862275197} a^{7} - \frac{297526079}{1249057422} a^{4} - \frac{4306598}{23130693} a$, $\frac{1}{202347302364} a^{17} - \frac{46933}{1249057422} a^{14} - \frac{1}{972} a^{13} + \frac{1}{324} a^{12} + \frac{692045309}{67449100788} a^{11} - \frac{1}{36} a^{10} + \frac{7968844907}{101173651182} a^{8} + \frac{17}{162} a^{7} + \frac{1}{54} a^{6} + \frac{200666957}{1249057422} a^{5} - \frac{13}{243} a^{4} + \frac{53}{162} a^{3} + \frac{65919037}{138784158} a^{2} - \frac{5}{18} a + \frac{1}{3}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 24719655.037054747 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times D_9:C_3$ (as 18T45):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 20 conjugacy class representatives for $C_2\times D_9:C_3$
Character table for $C_2\times D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-11}) \), 3.1.108.1, 6.0.15524784.3, 9.1.11019960576.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ $18$ R $18$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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
2Data not computed
$3$3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
3.9.16.6$x^{9} + 6 x^{8} + 3 x^{3} + 18 x + 6$$9$$1$$16$$(C_9:C_3):C_2$$[3/2, 2, 13/6]_{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.2.1$x^{4} + 143 x^{2} + 5929$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$