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Properties

Label 18.2.12878054313...6448.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{16}\cdot 3^{32}\cdot 13^{9}$
Root discriminant $47.07$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (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, -87480, 0, 0, 26244, 0, 0, -3620, 0, 0, 924, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 924*x^12 - 3620*x^9 + 26244*x^6 - 87480*x^3 - 78732)
 
gp: K = bnfinit(x^18 + 924*x^12 - 3620*x^9 + 26244*x^6 - 87480*x^3 - 78732, 1)
 

Normalized defining polynomial

\( x^{18} + 924 x^{12} - 3620 x^{9} + 26244 x^{6} - 87480 x^{3} - 78732 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1287805431371247745393996136448=2^{16}\cdot 3^{32}\cdot 13^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.07$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{24} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{5}{12} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{4}$, $\frac{1}{24} a^{10} + \frac{1}{4} a^{4} + \frac{5}{12} a$, $\frac{1}{72} a^{11} + \frac{1}{12} a^{5} + \frac{17}{36} a^{2}$, $\frac{1}{216} a^{12} + \frac{1}{36} a^{6} + \frac{53}{108} a^{3}$, $\frac{1}{648} a^{13} + \frac{1}{108} a^{7} - \frac{55}{324} a^{4} + \frac{1}{3} a$, $\frac{1}{5832} a^{14} + \frac{1}{1944} a^{13} + \frac{1}{648} a^{12} - \frac{1}{216} a^{11} - \frac{1}{72} a^{10} + \frac{73}{972} a^{8} - \frac{35}{324} a^{7} + \frac{1}{108} a^{6} + \frac{317}{729} a^{5} - \frac{88}{243} a^{4} - \frac{55}{324} a^{3} + \frac{19}{108} a^{2} + \frac{7}{36} a + \frac{1}{3}$, $\frac{1}{104713262568} a^{15} + \frac{7839589}{3878268984} a^{12} - \frac{80160775}{34904420856} a^{9} + \frac{2429005387}{26178315642} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{57485513}{969567246} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{24646807}{71819796}$, $\frac{1}{314139787704} a^{16} + \frac{7839589}{11634806952} a^{13} - \frac{383627911}{26178315642} a^{10} + \frac{11155110601}{78534946926} a^{7} + \frac{1339379843}{5817403476} a^{4} + \frac{32563969}{107729694} a$, $\frac{1}{942419363112} a^{17} + \frac{309101}{5817403476} a^{14} - \frac{1}{1944} a^{13} - \frac{1}{648} a^{12} - \frac{80160775}{314139787704} a^{11} + \frac{1}{72} a^{10} + \frac{39277648007}{471209681556} a^{8} + \frac{35}{324} a^{7} - \frac{1}{108} a^{6} - \frac{2729571031}{5817403476} a^{5} + \frac{88}{243} a^{4} + \frac{55}{324} a^{3} + \frac{238692445}{646378164} a^{2} - \frac{7}{36} a - \frac{1}{3}$

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:  $9$
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:  \( 159629939.02206567 \) (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

\(\Q(\sqrt{13}) \), 3.1.108.1, 6.2.25625808.3, 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$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}$
13Data not computed