↑

Properties

Label 18.0.16190942733...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{9}\cdot 211^{9}$
Root discriminant $32.48$
Ramified primes $5, 211$
Class number $4$ (GRH)
Class group $[4]$ (GRH)
Galois group $D_9$ (as 18T5)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![625, 125, -905, 4210, 3841, -2402, 5218, 9636, 7706, 2672, 440, 158, 110, -38, -9, 1, 13, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 13*x^16 + x^15 - 9*x^14 - 38*x^13 + 110*x^12 + 158*x^11 + 440*x^10 + 2672*x^9 + 7706*x^8 + 9636*x^7 + 5218*x^6 - 2402*x^5 + 3841*x^4 + 4210*x^3 - 905*x^2 + 125*x + 625)
 
gp: K = bnfinit(x^18 - 2*x^17 + 13*x^16 + x^15 - 9*x^14 - 38*x^13 + 110*x^12 + 158*x^11 + 440*x^10 + 2672*x^9 + 7706*x^8 + 9636*x^7 + 5218*x^6 - 2402*x^5 + 3841*x^4 + 4210*x^3 - 905*x^2 + 125*x + 625, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 13 x^{16} + x^{15} - 9 x^{14} - 38 x^{13} + 110 x^{12} + 158 x^{11} + 440 x^{10} + 2672 x^{9} + 7706 x^{8} + 9636 x^{7} + 5218 x^{6} - 2402 x^{5} + 3841 x^{4} + 4210 x^{3} - 905 x^{2} + 125 x + 625 \)

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:  \(-1619094273320941745099609375=-\,5^{9}\cdot 211^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.48$
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:  $5, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{10} a^{10} - \frac{1}{5} a^{6} - \frac{1}{2} a^{3} - \frac{2}{5} a^{2}$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{7} - \frac{1}{2} a^{4} - \frac{2}{5} a^{3}$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{8} - \frac{1}{2} a^{5} - \frac{2}{5} a^{4}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{9} - \frac{1}{2} a^{6} - \frac{2}{5} a^{5}$, $\frac{1}{20} a^{14} - \frac{2}{5} a^{6} - \frac{2}{5} a^{2} - \frac{1}{4}$, $\frac{1}{20} a^{15} + \frac{1}{10} a^{7} - \frac{2}{5} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{100} a^{16} - \frac{1}{50} a^{14} + \frac{1}{50} a^{13} + \frac{1}{50} a^{11} + \frac{1}{25} a^{10} - \frac{7}{50} a^{9} + \frac{1}{50} a^{8} - \frac{1}{25} a^{7} + \frac{19}{50} a^{6} + \frac{8}{25} a^{5} - \frac{19}{50} a^{4} + \frac{3}{25} a^{3} + \frac{1}{4} a^{2} - \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{63534487194341334425932144900} a^{17} - \frac{141674997020158569114188479}{31767243597170667212966072450} a^{16} - \frac{1463912246526784347737613487}{63534487194341334425932144900} a^{15} + \frac{304224633922860782422472822}{15883621798585333606483036225} a^{14} - \frac{543953627456314015238365653}{31767243597170667212966072450} a^{13} - \frac{212452699510120615616189272}{15883621798585333606483036225} a^{12} - \frac{639829203573309288534716678}{15883621798585333606483036225} a^{11} - \frac{634154780076525430109131139}{15883621798585333606483036225} a^{10} + \frac{2174515902700678512797489047}{31767243597170667212966072450} a^{9} + \frac{1222359850705449681098396831}{6353448719434133442593214490} a^{8} - \frac{54759380443360712826188975}{635344871943413344259321449} a^{7} + \frac{13671112017603630392199592719}{31767243597170667212966072450} a^{6} + \frac{5837451491213796062760909704}{15883621798585333606483036225} a^{5} - \frac{467269011257318093331006356}{15883621798585333606483036225} a^{4} - \frac{22586803496162107436066264041}{63534487194341334425932144900} a^{3} + \frac{1412924187493103568463854546}{3176724359717066721296607245} a^{2} + \frac{1217891397786378074276469201}{12706897438868266885186428980} a + \frac{63684947229612715427360050}{635344871943413344259321449}$

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

Class group and class number

$C_{4}$, which has order $4$ (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:  \( 3094239.39528 \) (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

\(\Q(\sqrt{-1055}) \), 3.1.1055.1 x3, 6.0.1174241375.1, 9.1.1238824650625.1 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
211Data not computed