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Properties

Label 18.0.21186825847...8527.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,1087^{9}$
Root discriminant $32.97$
Ramified prime $1087$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1087, 0, 4506, 0, 8021, 0, 1486, 0, -1609, 0, -364, 0, 292, 0, -31, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 - 31*x^14 + 292*x^12 - 364*x^10 - 1609*x^8 + 1486*x^6 + 8021*x^4 + 4506*x^2 + 1087)
 
gp: K = bnfinit(x^18 - 6*x^16 - 31*x^14 + 292*x^12 - 364*x^10 - 1609*x^8 + 1486*x^6 + 8021*x^4 + 4506*x^2 + 1087, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} - 31 x^{14} + 292 x^{12} - 364 x^{10} - 1609 x^{8} + 1486 x^{6} + 8021 x^{4} + 4506 x^{2} + 1087 \)

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:  \(-2118682584788822909365118527=-\,1087^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.97$
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:  $1087$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{30} a^{10} + \frac{1}{30} a^{8} + \frac{1}{5} a^{6} - \frac{1}{2} a^{5} + \frac{2}{5} a^{4} - \frac{1}{2} a^{3} + \frac{1}{15} a^{2} - \frac{1}{2} a + \frac{1}{15}$, $\frac{1}{30} a^{11} + \frac{1}{30} a^{9} + \frac{1}{5} a^{7} - \frac{1}{2} a^{6} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{15} a^{3} - \frac{1}{2} a^{2} + \frac{1}{15} a$, $\frac{1}{90} a^{12} + \frac{1}{18} a^{8} - \frac{1}{2} a^{7} + \frac{1}{15} a^{6} - \frac{1}{9} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a - \frac{16}{45}$, $\frac{1}{90} a^{13} + \frac{1}{18} a^{9} - \frac{1}{6} a^{8} + \frac{1}{15} a^{7} - \frac{1}{9} a^{5} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{16}{45} a - \frac{1}{3}$, $\frac{1}{3510} a^{14} - \frac{1}{1755} a^{12} - \frac{5}{702} a^{10} - \frac{152}{1755} a^{8} - \frac{1}{2} a^{7} - \frac{731}{1755} a^{6} + \frac{7}{54} a^{4} - \frac{1}{2} a^{3} - \frac{1367}{3510} a^{2} + \frac{1669}{3510}$, $\frac{1}{3510} a^{15} - \frac{1}{1755} a^{13} - \frac{5}{702} a^{11} + \frac{281}{3510} a^{9} - \frac{1}{6} a^{8} + \frac{293}{3510} a^{7} + \frac{7}{54} a^{5} - \frac{1367}{3510} a^{3} - \frac{1}{2} a^{2} - \frac{671}{3510} a + \frac{1}{6}$, $\frac{1}{309739950} a^{16} - \frac{18332}{154869975} a^{14} - \frac{5119}{51623325} a^{12} + \frac{4231909}{309739950} a^{10} - \frac{10424113}{154869975} a^{8} + \frac{110985289}{309739950} a^{6} - \frac{6599056}{51623325} a^{4} - \frac{1}{2} a^{3} + \frac{36580477}{154869975} a^{2} - \frac{67833191}{309739950}$, $\frac{1}{309739950} a^{17} - \frac{18332}{154869975} a^{15} - \frac{5119}{51623325} a^{13} + \frac{4231909}{309739950} a^{11} - \frac{10424113}{154869975} a^{9} + \frac{110985289}{309739950} a^{7} - \frac{6599056}{51623325} a^{5} - \frac{1}{2} a^{4} + \frac{36580477}{154869975} a^{3} - \frac{67833191}{309739950} a$

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:  $8$
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:  \( 3275556.82428 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-1087}) \), 3.1.1087.1 x3, 6.0.1284365503.1, 9.1.1396105301761.1 x9

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

Sibling fields

Degree 9 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
1087Data not computed