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Properties

Label 18.0.87946075369...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{37}\cdot 5^{9}$
Root discriminant $21.39$
Ramified primes $3, 5$
Class number $2$
Class group $[2]$
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![25, 0, 0, -210, 576, -594, 399, 792, -486, -602, 243, 81, -6, -9, 27, -3, -9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^16 - 3*x^15 + 27*x^14 - 9*x^13 - 6*x^12 + 81*x^11 + 243*x^10 - 602*x^9 - 486*x^8 + 792*x^7 + 399*x^6 - 594*x^5 + 576*x^4 - 210*x^3 + 25)
 
gp: K = bnfinit(x^18 - 9*x^16 - 3*x^15 + 27*x^14 - 9*x^13 - 6*x^12 + 81*x^11 + 243*x^10 - 602*x^9 - 486*x^8 + 792*x^7 + 399*x^6 - 594*x^5 + 576*x^4 - 210*x^3 + 25, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{16} - 3 x^{15} + 27 x^{14} - 9 x^{13} - 6 x^{12} + 81 x^{11} + 243 x^{10} - 602 x^{9} - 486 x^{8} + 792 x^{7} + 399 x^{6} - 594 x^{5} + 576 x^{4} - 210 x^{3} + 25 \)

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:  \(-879460753693354224609375=-\,3^{37}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.39$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{15} a^{10} + \frac{1}{5} a^{6} + \frac{2}{5} a^{2} + \frac{1}{3} a$, $\frac{1}{45} a^{11} + \frac{1}{45} a^{10} - \frac{1}{9} a^{9} - \frac{1}{3} a^{8} + \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{2}{15} a^{3} - \frac{4}{45} a^{2} + \frac{1}{9} a + \frac{4}{9}$, $\frac{1}{45} a^{12} + \frac{1}{9} a^{9} + \frac{2}{5} a^{8} - \frac{1}{3} a^{6} - \frac{1}{5} a^{4} - \frac{2}{9} a^{3} + \frac{2}{9}$, $\frac{1}{315} a^{13} - \frac{2}{315} a^{12} - \frac{2}{315} a^{11} - \frac{2}{105} a^{10} + \frac{1}{105} a^{9} - \frac{32}{105} a^{8} - \frac{52}{105} a^{7} + \frac{44}{105} a^{6} + \frac{32}{105} a^{5} - \frac{16}{45} a^{4} + \frac{8}{315} a^{3} - \frac{136}{315} a^{2} + \frac{2}{7} a + \frac{2}{7}$, $\frac{1}{1575} a^{14} + \frac{2}{1575} a^{13} - \frac{1}{525} a^{12} - \frac{1}{225} a^{10} - \frac{32}{225} a^{9} + \frac{212}{525} a^{8} - \frac{17}{35} a^{7} + \frac{82}{525} a^{6} - \frac{778}{1575} a^{5} + \frac{652}{1575} a^{4} - \frac{2}{35} a^{3} - \frac{13}{105} a^{2} - \frac{17}{63} a + \frac{20}{63}$, $\frac{1}{1575} a^{15} - \frac{2}{1575} a^{13} - \frac{4}{1575} a^{12} - \frac{17}{1575} a^{11} - \frac{2}{105} a^{10} + \frac{7}{225} a^{9} + \frac{211}{525} a^{8} - \frac{193}{525} a^{7} + \frac{4}{315} a^{6} - \frac{22}{75} a^{5} - \frac{379}{1575} a^{4} + \frac{1}{63} a^{3} + \frac{109}{315} a^{2} - \frac{4}{21} a + \frac{20}{63}$, $\frac{1}{143325} a^{16} + \frac{4}{143325} a^{15} + \frac{22}{143325} a^{14} - \frac{73}{47775} a^{13} - \frac{304}{28665} a^{12} + \frac{272}{143325} a^{11} + \frac{2306}{143325} a^{10} + \frac{7288}{143325} a^{9} - \frac{23201}{47775} a^{8} + \frac{5372}{20475} a^{7} - \frac{57643}{143325} a^{6} + \frac{65921}{143325} a^{5} - \frac{562}{15925} a^{4} - \frac{9314}{28665} a^{3} + \frac{12914}{28665} a^{2} - \frac{2222}{5733} a - \frac{2542}{5733}$, $\frac{1}{1458331875} a^{17} - \frac{3748}{1458331875} a^{16} + \frac{664}{26515125} a^{15} - \frac{328463}{1458331875} a^{14} - \frac{103108}{486110625} a^{13} + \frac{10161493}{1458331875} a^{12} + \frac{103997}{22435875} a^{11} + \frac{37495816}{1458331875} a^{10} + \frac{235603}{4487175} a^{9} + \frac{30632114}{208333125} a^{8} - \frac{679846}{22435875} a^{7} + \frac{5465951}{39414375} a^{6} - \frac{506908777}{1458331875} a^{5} + \frac{86356709}{486110625} a^{4} - \frac{2174582}{7882875} a^{3} - \frac{17061922}{58333275} a^{2} - \frac{12027394}{58333275} a - \frac{128809}{8333325}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 93546.1279517 \)
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

\(\Q(\sqrt{-15}) \), 3.1.135.1 x3, 6.0.273375.1, 9.1.242137805625.3 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$