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Properties

Label 18.0.80969502434...2816.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{20}\cdot 7^{12}$
Root discriminant $31.25$
Ramified primes $2, 3, 7$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2\times C_3:S_3$ (as 18T12)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![729, 0, 0, 0, 0, 0, 999, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 29*x^12 + 999*x^6 + 729)
 
gp: K = bnfinit(x^18 - 29*x^12 + 999*x^6 + 729, 1)
 

Normalized defining polynomial

\( x^{18} - 29 x^{12} + 999 x^{6} + 729 \)

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:  \(-809695024340348725484322816=-\,2^{24}\cdot 3^{20}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.25$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{6} + \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{18} a^{10} - \frac{1}{2} a^{7} + \frac{7}{18} a^{4} - \frac{1}{2} a$, $\frac{1}{54} a^{11} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{25}{54} a^{5} + \frac{7}{18} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{2430} a^{12} + \frac{16}{243} a^{6} - \frac{13}{90}$, $\frac{1}{7290} a^{13} + \frac{259}{729} a^{7} - \frac{103}{270} a$, $\frac{1}{7290} a^{14} + \frac{16}{729} a^{8} + \frac{77}{270} a^{2}$, $\frac{1}{21870} a^{15} - \frac{211}{4374} a^{9} - \frac{1}{2} a^{6} + \frac{16}{405} a^{3} - \frac{1}{2}$, $\frac{1}{65610} a^{16} + \frac{1}{21870} a^{14} + \frac{1}{7290} a^{12} - \frac{211}{13122} a^{10} + \frac{259}{2187} a^{8} - \frac{1}{2} a^{7} + \frac{259}{729} a^{6} + \frac{16}{1215} a^{4} + \frac{167}{810} a^{2} - \frac{1}{2} a - \frac{103}{270}$, $\frac{1}{65610} a^{17} + \frac{16}{6561} a^{11} + \frac{1}{18} a^{9} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} + \frac{1157}{2430} a^{5} + \frac{7}{18} a^{3} - \frac{1}{3} a - \frac{1}{2}$

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:  \( \frac{5}{4374} a^{15} - \frac{86}{2187} a^{9} + \frac{187}{162} a^{3} \) (order $4$)
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:  \( 3131117.232313821 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_3:S_3$ (as 18T12):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_2\times C_3:S_3$
Character table for $C_2\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-1}) \), 3.1.108.1, 3.1.588.1, 3.1.1323.1, 3.1.5292.1, 6.0.186624.1, 6.0.112021056.1, 6.0.5531904.1, 6.0.448084224.1, 9.1.444611571264.1

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

Sibling fields

Galois closure: data not computed
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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
2.12.16.13$x^{12} + 12 x^{10} + 12 x^{8} + 8 x^{6} + 32 x^{4} - 16 x^{2} + 16$$6$$2$$16$$D_6$$[2]_{3}^{2}$
$3$3.6.6.3$x^{6} + 3 x^{4} + 9$$3$$2$$6$$D_{6}$$[3/2]_{2}^{2}$
3.12.14.6$x^{12} + 3 x^{11} + 3 x^{10} - 6 x^{9} + 3 x^{8} + 9 x^{7} + 9 x^{4} + 9 x^{3} + 9$$6$$2$$14$$D_6$$[3/2]_{2}^{2}$
7Data not computed