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Properties

Label 18.0.25815178338...6864.8
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{44}$
Root discriminant $29.33$
Ramified primes $2, 3$
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![12544, 0, 1296, 0, 2808, 0, 3465, 0, 2754, 0, 1503, 0, 564, 0, 135, 0, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 + 135*x^14 + 564*x^12 + 1503*x^10 + 2754*x^8 + 3465*x^6 + 2808*x^4 + 1296*x^2 + 12544)
 
gp: K = bnfinit(x^18 + 18*x^16 + 135*x^14 + 564*x^12 + 1503*x^10 + 2754*x^8 + 3465*x^6 + 2808*x^4 + 1296*x^2 + 12544, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} + 135 x^{14} + 564 x^{12} + 1503 x^{10} + 2754 x^{8} + 3465 x^{6} + 2808 x^{4} + 1296 x^{2} + 12544 \)

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:  \(-258151783382020583032356864=-\,2^{18}\cdot 3^{44}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.33$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{7} - \frac{1}{8} a^{3}$, $\frac{1}{16} a^{8} - \frac{1}{16} a^{7} + \frac{3}{16} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{112} a^{9} - \frac{5}{112} a^{7} - \frac{1}{112} a^{5} - \frac{1}{4} a^{4} + \frac{25}{112} a^{3} - \frac{1}{4} a^{2} + \frac{9}{28} a$, $\frac{1}{224} a^{10} - \frac{5}{224} a^{8} + \frac{27}{224} a^{6} + \frac{53}{224} a^{4} + \frac{9}{56} a^{2} - \frac{1}{2} a$, $\frac{1}{224} a^{11} - \frac{1}{224} a^{9} + \frac{1}{32} a^{7} - \frac{1}{32} a^{5} - \frac{1}{4} a^{4} - \frac{1}{7} a^{3} + \frac{1}{4} a^{2} + \frac{1}{7} a$, $\frac{1}{1792} a^{12} - \frac{1}{448} a^{10} + \frac{11}{896} a^{8} - \frac{1}{16} a^{7} + \frac{5}{112} a^{6} + \frac{425}{1792} a^{4} + \frac{1}{16} a^{3} - \frac{19}{448} a^{2} - \frac{1}{2} a - \frac{1}{4}$, $\frac{1}{1792} a^{13} - \frac{1}{448} a^{11} + \frac{3}{896} a^{9} + \frac{3}{112} a^{7} - \frac{1}{256} a^{5} - \frac{1}{4} a^{4} - \frac{13}{64} a^{3} + \frac{1}{4} a^{2} - \frac{9}{28} a$, $\frac{1}{3584} a^{14} - \frac{1}{3584} a^{13} - \frac{1}{896} a^{11} - \frac{1}{1792} a^{10} + \frac{1}{1792} a^{9} + \frac{3}{112} a^{8} - \frac{13}{448} a^{7} - \frac{367}{3584} a^{6} + \frac{9}{512} a^{5} - \frac{15}{112} a^{4} - \frac{69}{896} a^{3} + \frac{103}{224} a^{2} + \frac{19}{56} a - \frac{1}{2}$, $\frac{1}{3584} a^{15} - \frac{1}{3584} a^{13} - \frac{3}{1792} a^{11} + \frac{1}{1792} a^{9} + \frac{9}{3584} a^{7} - \frac{1}{8} a^{6} - \frac{321}{3584} a^{5} - \frac{1}{4} a^{4} + \frac{191}{896} a^{3} + \frac{3}{8} a^{2} - \frac{1}{8} a$, $\frac{1}{14336} a^{16} + \frac{1}{14336} a^{14} - \frac{1}{7168} a^{12} + \frac{15}{7168} a^{10} - \frac{57}{2048} a^{8} + \frac{561}{14336} a^{6} + \frac{113}{896} a^{4} - \frac{1}{4} a^{3} + \frac{99}{896} a^{2} - \frac{1}{4} a - \frac{3}{8}$, $\frac{1}{28672} a^{17} - \frac{1}{28672} a^{16} + \frac{1}{28672} a^{15} - \frac{1}{28672} a^{14} - \frac{1}{14336} a^{13} - \frac{3}{14336} a^{12} + \frac{15}{14336} a^{11} - \frac{31}{14336} a^{10} + \frac{113}{28672} a^{9} - \frac{353}{28672} a^{8} - \frac{207}{28672} a^{7} + \frac{655}{28672} a^{6} - \frac{143}{1792} a^{5} - \frac{515}{3584} a^{4} - \frac{333}{1792} a^{3} + \frac{691}{1792} a^{2} - \frac{5}{112} a + \frac{5}{16}$

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:  \( -\frac{1}{112} a^{9} - \frac{9}{112} a^{7} - \frac{27}{112} a^{5} - \frac{39}{112} a^{3} - \frac{9}{28} a \) (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:  \( 326412091.549 \) (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

\(\Q(\sqrt{-1}) \), 3.1.324.1 x3, 6.0.419904.2, 9.1.8033551259904.2 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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
3Data not computed