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Properties

Label 18.0.68606847607...5039.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 137^{9}$
Root discriminant $30.97$
Ramified primes $7, 137$
Class number $4$
Class group $[4]$
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![46991, 0, 37877, 0, -350, 0, -7570, 0, 263, 0, 1223, 0, -520, 0, 118, 0, -16, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 16*x^16 + 118*x^14 - 520*x^12 + 1223*x^10 + 263*x^8 - 7570*x^6 - 350*x^4 + 37877*x^2 + 46991)
 
gp: K = bnfinit(x^18 - 16*x^16 + 118*x^14 - 520*x^12 + 1223*x^10 + 263*x^8 - 7570*x^6 - 350*x^4 + 37877*x^2 + 46991, 1)
 

Normalized defining polynomial

\( x^{18} - 16 x^{16} + 118 x^{14} - 520 x^{12} + 1223 x^{10} + 263 x^{8} - 7570 x^{6} - 350 x^{4} + 37877 x^{2} + 46991 \)

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:  \(-686068476073813329411695039=-\,7^{9}\cdot 137^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.97$
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:  $7, 137$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{14} a^{10} + \frac{1}{7} a^{8} + \frac{3}{14} a^{6} - \frac{1}{2} a^{5} + \frac{1}{7} a^{4} + \frac{1}{14} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{11} + \frac{1}{7} a^{9} + \frac{3}{14} a^{7} - \frac{1}{2} a^{6} + \frac{1}{7} a^{5} + \frac{1}{14} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{12} - \frac{1}{14} a^{8} - \frac{1}{2} a^{7} - \frac{2}{7} a^{6} - \frac{3}{14} a^{4} - \frac{1}{2} a^{3} - \frac{1}{7} a^{2}$, $\frac{1}{14} a^{13} - \frac{1}{14} a^{9} - \frac{1}{2} a^{8} - \frac{2}{7} a^{7} - \frac{3}{14} a^{5} - \frac{1}{2} a^{4} - \frac{1}{7} a^{3}$, $\frac{1}{14} a^{14} - \frac{1}{7} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} + \frac{1}{14} a^{2} - \frac{1}{2}$, $\frac{1}{98} a^{15} - \frac{1}{49} a^{13} - \frac{1}{98} a^{11} - \frac{8}{49} a^{9} + \frac{19}{98} a^{7} - \frac{3}{98} a^{5} + \frac{9}{49} a^{3} - \frac{1}{2} a^{2} - \frac{5}{14} a$, $\frac{1}{3182654571412442} a^{16} + \frac{44787720192116}{1591327285706221} a^{14} + \frac{46455725404917}{1591327285706221} a^{12} + \frac{1809736622783}{3182654571412442} a^{10} - \frac{543365663296212}{1591327285706221} a^{8} - \frac{1}{2} a^{7} - \frac{431475485026465}{1591327285706221} a^{6} - \frac{1}{2} a^{5} - \frac{1192484497501629}{3182654571412442} a^{4} - \frac{40439594156050}{227332469386603} a^{2} - \frac{1}{2} a + \frac{4221778179868}{32476067055229}$, $\frac{1}{3182654571412442} a^{17} - \frac{7852760781455}{3182654571412442} a^{15} + \frac{60435383754605}{3182654571412442} a^{13} + \frac{49618968894235}{1591327285706221} a^{11} + \frac{699452361445171}{3182654571412442} a^{9} + \frac{1377897656757871}{3182654571412442} a^{7} - \frac{1}{2} a^{6} - \frac{218202485844759}{3182654571412442} a^{5} - \frac{1}{2} a^{4} + \frac{94104112215613}{227332469386603} a^{3} + \frac{13082994510483}{64952134110458} a$

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

Class group and class number

$C_{4}$, which has order $4$

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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 659446.094846 \)
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

\(\Q(\sqrt{-959}) \), 3.1.959.1 x3, 6.0.881974079.1, 9.1.845813141761.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$137$137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$
137.2.1.1$x^{2} - 137$$2$$1$$1$$C_2$$[\ ]_{2}$