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Properties

Label 18.0.17322019250...4663.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,823^{9}$
Root discriminant $28.69$
Ramified prime $823$
Class number $1$
Class group Trivial
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![2025, -4995, 10881, -21435, 32042, -35630, 28762, -16577, 8548, -4960, 2163, -421, 86, -35, -30, 13, 9, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 9*x^16 + 13*x^15 - 30*x^14 - 35*x^13 + 86*x^12 - 421*x^11 + 2163*x^10 - 4960*x^9 + 8548*x^8 - 16577*x^7 + 28762*x^6 - 35630*x^5 + 32042*x^4 - 21435*x^3 + 10881*x^2 - 4995*x + 2025)
 
gp: K = bnfinit(x^18 - 6*x^17 + 9*x^16 + 13*x^15 - 30*x^14 - 35*x^13 + 86*x^12 - 421*x^11 + 2163*x^10 - 4960*x^9 + 8548*x^8 - 16577*x^7 + 28762*x^6 - 35630*x^5 + 32042*x^4 - 21435*x^3 + 10881*x^2 - 4995*x + 2025, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 9 x^{16} + 13 x^{15} - 30 x^{14} - 35 x^{13} + 86 x^{12} - 421 x^{11} + 2163 x^{10} - 4960 x^{9} + 8548 x^{8} - 16577 x^{7} + 28762 x^{6} - 35630 x^{5} + 32042 x^{4} - 21435 x^{3} + 10881 x^{2} - 4995 x + 2025 \)

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:  \(-173220192505318905457564663=-\,823^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $28.69$
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:  $823$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} - \frac{4}{9} a^{7} + \frac{4}{9} a^{6} + \frac{1}{9} a^{5} + \frac{2}{9} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{3} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{10} - \frac{1}{3} a^{7} - \frac{1}{9} a^{6} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{11} - \frac{4}{9} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{135} a^{16} + \frac{7}{135} a^{15} - \frac{1}{27} a^{14} - \frac{7}{135} a^{13} + \frac{14}{135} a^{12} + \frac{4}{45} a^{11} + \frac{2}{135} a^{10} + \frac{2}{27} a^{9} - \frac{2}{135} a^{8} + \frac{2}{5} a^{7} + \frac{8}{27} a^{6} - \frac{37}{135} a^{5} + \frac{22}{45} a^{4} - \frac{32}{135} a^{3} + \frac{2}{45} a^{2} - \frac{1}{5} a$, $\frac{1}{28135232390252197391557272435} a^{17} - \frac{1762453202159123081268481}{686225180250053594916031035} a^{16} - \frac{358045601947720602537483011}{28135232390252197391557272435} a^{15} + \frac{124616669363201987079081763}{2557748399113836126505206585} a^{14} + \frac{11956875192439571456104465}{5627046478050439478311454487} a^{13} - \frac{97623011896933067836299556}{625227386450048830923494943} a^{12} - \frac{177796237258541733707754904}{28135232390252197391557272435} a^{11} + \frac{3290043559302530902350810214}{28135232390252197391557272435} a^{10} + \frac{10679393535336337023308032}{1480801704750115652187224865} a^{9} + \frac{208028936924095493668136687}{1875682159350146492770484829} a^{8} + \frac{6432642640983780812319835798}{28135232390252197391557272435} a^{7} + \frac{8585567184801624016863806003}{28135232390252197391557272435} a^{6} - \frac{2780923387379958604936834046}{9378410796750732463852424145} a^{5} - \frac{2166888677532476072251973395}{5627046478050439478311454487} a^{4} + \frac{262776290553145766688166372}{1339772970964390351978917735} a^{3} + \frac{440345571215731792111758767}{1875682159350146492770484829} a^{2} + \frac{44521536669816732995136652}{183890407779426126742204395} a - \frac{10688187101714734732639555}{208409128816682943641164981}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 943839.99321 \)
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

\(\Q(\sqrt{-823}) \), 3.1.823.1 x3, 6.0.557441767.1, 9.1.458774574241.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
823Data not computed