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Properties

Label 18.0.18409332383...1043.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 83^{9}$
Root discriminant $133.61$
Ramified primes $3, 83$
Class number $10774323$ (GRH)
Class group $[10774323]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1743009516489, -576856453545, 665351396991, -185230785420, 113757248880, -26836628409, 11455485945, -2289051090, 749248587, -125652506, 32991876, -4545126, 976272, -105840, 18648, -1452, 207, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 207*x^16 - 1452*x^15 + 18648*x^14 - 105840*x^13 + 976272*x^12 - 4545126*x^11 + 32991876*x^10 - 125652506*x^9 + 749248587*x^8 - 2289051090*x^7 + 11455485945*x^6 - 26836628409*x^5 + 113757248880*x^4 - 185230785420*x^3 + 665351396991*x^2 - 576856453545*x + 1743009516489)
 
gp: K = bnfinit(x^18 - 9*x^17 + 207*x^16 - 1452*x^15 + 18648*x^14 - 105840*x^13 + 976272*x^12 - 4545126*x^11 + 32991876*x^10 - 125652506*x^9 + 749248587*x^8 - 2289051090*x^7 + 11455485945*x^6 - 26836628409*x^5 + 113757248880*x^4 - 185230785420*x^3 + 665351396991*x^2 - 576856453545*x + 1743009516489, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 207 x^{16} - 1452 x^{15} + 18648 x^{14} - 105840 x^{13} + 976272 x^{12} - 4545126 x^{11} + 32991876 x^{10} - 125652506 x^{9} + 749248587 x^{8} - 2289051090 x^{7} + 11455485945 x^{6} - 26836628409 x^{5} + 113757248880 x^{4} - 185230785420 x^{3} + 665351396991 x^{2} - 576856453545 x + 1743009516489 \)

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:  \(-184093323834250344728515453895309591043=-\,3^{44}\cdot 83^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $133.61$
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:  $3, 83$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2241=3^{3}\cdot 83\)
Dirichlet character group:    $\lbrace$$\chi_{2241}(1,·)$, $\chi_{2241}(580,·)$, $\chi_{2241}(1993,·)$, $\chi_{2241}(331,·)$, $\chi_{2241}(1744,·)$, $\chi_{2241}(82,·)$, $\chi_{2241}(1495,·)$, $\chi_{2241}(2074,·)$, $\chi_{2241}(1246,·)$, $\chi_{2241}(1825,·)$, $\chi_{2241}(997,·)$, $\chi_{2241}(1576,·)$, $\chi_{2241}(748,·)$, $\chi_{2241}(1327,·)$, $\chi_{2241}(499,·)$, $\chi_{2241}(1078,·)$, $\chi_{2241}(250,·)$, $\chi_{2241}(829,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{17} + \frac{844025025162697703750515595828224182002632807731313071545598019806044}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{16} + \frac{3355913134219336081047626458394976379443173309582782839985576584069580}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{15} - \frac{3754139160504606336067950492963393691477461638539094846244695642825669}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{14} - \frac{3694024615500895837312585305306878488624159622801100366659096814897243}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{13} - \frac{991131744270859779273680969828719950967777025628576488482957594535040}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{12} - \frac{1465042632504398070272019938899714209276665930739900059713317433324913}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{11} + \frac{1061901044099557858271459772680836866521224744204389347873439414462390}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{10} + \frac{598380782842089103505942907536411045638044254453453503119790110707958}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{9} - \frac{2109413736642792047453016564159506880552312637586368651882437178680581}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{8} - \frac{143421357061014064979183601549814243963729558214911242160431519238459}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{7} - \frac{1927565433869611863606853728717085777342363327575448816210310506269344}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{6} + \frac{255663732526406986648015419682695527669297171364057118412245919044091}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{5} - \frac{1985661768863753763525930893103231005264470416980141049049143762911828}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{4} + \frac{2535617618810669601918069203889901405122813743460769809674281867075966}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{3} - \frac{2372997488237660481463464329416651597419764825429681107724776800982294}{7802082235271410044905863798259772147219849742688159040911865549538753} a^{2} - \frac{562733246139416131208530095779292411306515856556917727620309005219386}{7802082235271410044905863798259772147219849742688159040911865549538753} a + \frac{1259093244963763269288826842179173285548003526772012165100627483611620}{7802082235271410044905863798259772147219849742688159040911865549538753}$

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

Class group and class number

$C_{10774323}$, which has order $10774323$ (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:  \( 40934.03294431194 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-83}) \), \(\Q(\zeta_{9})^+\), 6.0.3751494507.2, \(\Q(\zeta_{27})^+\)

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $18$ R $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$3$3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
3.9.22.8$x^{9} + 18 x^{7} + 24 x^{6} + 18 x^{5} + 18 x^{3} + 3$$9$$1$$22$$C_9$$[2, 3]$
83Data not computed