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Properties

Label 18.0.39803953177...5379.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,419^{9}$
Root discriminant $20.47$
Ramified prime $419$
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![256, -192, -652, 402, 881, -522, -781, 450, 384, -350, 4, 396, 60, -114, 12, 28, -5, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 - 5*x^16 + 28*x^15 + 12*x^14 - 114*x^13 + 60*x^12 + 396*x^11 + 4*x^10 - 350*x^9 + 384*x^8 + 450*x^7 - 781*x^6 - 522*x^5 + 881*x^4 + 402*x^3 - 652*x^2 - 192*x + 256)
 
gp: K = bnfinit(x^18 - 2*x^17 - 5*x^16 + 28*x^15 + 12*x^14 - 114*x^13 + 60*x^12 + 396*x^11 + 4*x^10 - 350*x^9 + 384*x^8 + 450*x^7 - 781*x^6 - 522*x^5 + 881*x^4 + 402*x^3 - 652*x^2 - 192*x + 256, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} - 5 x^{16} + 28 x^{15} + 12 x^{14} - 114 x^{13} + 60 x^{12} + 396 x^{11} + 4 x^{10} - 350 x^{9} + 384 x^{8} + 450 x^{7} - 781 x^{6} - 522 x^{5} + 881 x^{4} + 402 x^{3} - 652 x^{2} - 192 x + 256 \)

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:  \(-398039531776795387285379=-\,419^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.47$
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:  $419$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{9} - \frac{1}{8} a^{4} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{5} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{12} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} + \frac{3}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{5}{16} a^{3} + \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{13} - \frac{1}{32} a^{12} - \frac{1}{16} a^{11} - \frac{1}{32} a^{10} + \frac{1}{32} a^{9} + \frac{1}{16} a^{8} + \frac{3}{32} a^{7} - \frac{7}{32} a^{6} + \frac{1}{16} a^{5} - \frac{7}{32} a^{4} + \frac{7}{32} a^{3} - \frac{1}{16} a^{2} + \frac{1}{8} a$, $\frac{1}{64} a^{14} + \frac{1}{64} a^{12} - \frac{3}{64} a^{11} + \frac{7}{64} a^{9} - \frac{3}{64} a^{8} + \frac{3}{16} a^{7} - \frac{9}{64} a^{6} + \frac{11}{64} a^{5} - \frac{31}{64} a^{3} + \frac{13}{32} a^{2} + \frac{5}{16} a - \frac{1}{2}$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{13} - \frac{1}{64} a^{12} - \frac{1}{16} a^{11} + \frac{1}{64} a^{10} - \frac{5}{64} a^{9} - \frac{1}{8} a^{8} - \frac{15}{64} a^{7} - \frac{7}{64} a^{6} + \frac{1}{16} a^{5} - \frac{9}{64} a^{4} - \frac{5}{16} a^{3} - \frac{3}{8} a^{2} + \frac{3}{8} a$, $\frac{1}{59840} a^{16} + \frac{49}{7480} a^{15} - \frac{351}{59840} a^{14} - \frac{439}{59840} a^{13} + \frac{5}{374} a^{12} - \frac{283}{5440} a^{11} - \frac{237}{5440} a^{10} + \frac{29}{272} a^{9} + \frac{5207}{59840} a^{8} + \frac{8783}{59840} a^{7} - \frac{457}{3740} a^{6} + \frac{749}{11968} a^{5} - \frac{357}{1760} a^{4} + \frac{2999}{14960} a^{3} + \frac{497}{3740} a^{2} + \frac{543}{1870} a - \frac{9}{935}$, $\frac{1}{51302627200} a^{17} - \frac{11247}{5130262720} a^{16} - \frac{36990829}{10260525440} a^{15} - \frac{85915503}{12825656800} a^{14} + \frac{5196976}{400801775} a^{13} + \frac{326103341}{25651313600} a^{12} + \frac{7062061}{1165968800} a^{11} - \frac{7725133}{145746100} a^{10} + \frac{111769181}{1603207100} a^{9} - \frac{253969503}{25651313600} a^{8} - \frac{2792072777}{12825656800} a^{7} + \frac{1366551147}{25651313600} a^{6} - \frac{5091833773}{51302627200} a^{5} + \frac{1370897371}{25651313600} a^{4} + \frac{951653849}{2052105088} a^{3} - \frac{5557089099}{25651313600} a^{2} + \frac{3216829353}{12825656800} a + \frac{626741481}{1603207100}$

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:  \( 144526.096265 \) (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

\(\Q(\sqrt{-419}) \), 3.1.419.1 x3, 6.0.73560059.1, 9.1.30821664721.1 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
419Data not computed