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Properties

Label 18.0.49493857988...4683.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 43^{9}$
Root discriminant $96.17$
Ramified primes $3, 43$
Class number $566293$ (GRH)
Class group $[7, 7, 11557]$ (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![9587714579, -4966452585, 6197190291, -2651158680, 1822549050, -655191549, 320905905, -97660890, 37297827, -9587546, 2965176, -634806, 160752, -27720, 5688, -732, 117, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 117*x^16 - 732*x^15 + 5688*x^14 - 27720*x^13 + 160752*x^12 - 634806*x^11 + 2965176*x^10 - 9587546*x^9 + 37297827*x^8 - 97660890*x^7 + 320905905*x^6 - 655191549*x^5 + 1822549050*x^4 - 2651158680*x^3 + 6197190291*x^2 - 4966452585*x + 9587714579)
 
gp: K = bnfinit(x^18 - 9*x^17 + 117*x^16 - 732*x^15 + 5688*x^14 - 27720*x^13 + 160752*x^12 - 634806*x^11 + 2965176*x^10 - 9587546*x^9 + 37297827*x^8 - 97660890*x^7 + 320905905*x^6 - 655191549*x^5 + 1822549050*x^4 - 2651158680*x^3 + 6197190291*x^2 - 4966452585*x + 9587714579, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 117 x^{16} - 732 x^{15} + 5688 x^{14} - 27720 x^{13} + 160752 x^{12} - 634806 x^{11} + 2965176 x^{10} - 9587546 x^{9} + 37297827 x^{8} - 97660890 x^{7} + 320905905 x^{6} - 655191549 x^{5} + 1822549050 x^{4} - 2651158680 x^{3} + 6197190291 x^{2} - 4966452585 x + 9587714579 \)

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:  \(-494938579887862497256991740536934683=-\,3^{44}\cdot 43^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $96.17$
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:  $3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1161=3^{3}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{1161}(1,·)$, $\chi_{1161}(130,·)$, $\chi_{1161}(259,·)$, $\chi_{1161}(388,·)$, $\chi_{1161}(517,·)$, $\chi_{1161}(646,·)$, $\chi_{1161}(775,·)$, $\chi_{1161}(904,·)$, $\chi_{1161}(1033,·)$, $\chi_{1161}(85,·)$, $\chi_{1161}(214,·)$, $\chi_{1161}(343,·)$, $\chi_{1161}(472,·)$, $\chi_{1161}(601,·)$, $\chi_{1161}(730,·)$, $\chi_{1161}(859,·)$, $\chi_{1161}(988,·)$, $\chi_{1161}(1117,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{53} a^{15} + \frac{10}{53} a^{14} + \frac{13}{53} a^{13} - \frac{22}{53} a^{12} + \frac{21}{53} a^{11} + \frac{13}{53} a^{10} - \frac{18}{53} a^{9} + \frac{17}{53} a^{8} + \frac{19}{53} a^{7} - \frac{17}{53} a^{6} - \frac{20}{53} a^{5} - \frac{12}{53} a^{4} - \frac{26}{53} a^{3} + \frac{21}{53} a^{2} + \frac{10}{53} a - \frac{10}{53}$, $\frac{1}{53} a^{16} + \frac{19}{53} a^{14} + \frac{7}{53} a^{13} - \frac{24}{53} a^{12} + \frac{15}{53} a^{11} + \frac{11}{53} a^{10} - \frac{15}{53} a^{9} + \frac{8}{53} a^{8} + \frac{5}{53} a^{7} - \frac{9}{53} a^{6} - \frac{24}{53} a^{5} - \frac{12}{53} a^{4} + \frac{16}{53} a^{3} + \frac{12}{53} a^{2} - \frac{4}{53} a - \frac{6}{53}$, $\frac{1}{700916985026129819774124824138029223771535099612222696937} a^{17} - \frac{77766310472984657783010790807900340780856073594829426}{700916985026129819774124824138029223771535099612222696937} a^{16} + \frac{3284484117751900212463699184740989524160183441071326990}{700916985026129819774124824138029223771535099612222696937} a^{15} + \frac{29895692531049783434431232218704903586032991654624215412}{700916985026129819774124824138029223771535099612222696937} a^{14} + \frac{182570987552581745829641401570359056313207952961247054736}{700916985026129819774124824138029223771535099612222696937} a^{13} + \frac{323028591462153801125179572440698069879624675068462278441}{700916985026129819774124824138029223771535099612222696937} a^{12} - \frac{179093450692480344893430368576172173786924147679593767422}{700916985026129819774124824138029223771535099612222696937} a^{11} + \frac{294323770387873591094977530963818472236910312382185859155}{700916985026129819774124824138029223771535099612222696937} a^{10} - \frac{79649151564129851576443337300103317307764530853893243574}{700916985026129819774124824138029223771535099612222696937} a^{9} + \frac{226877431109971619925955100663072866333304403029370352657}{700916985026129819774124824138029223771535099612222696937} a^{8} - \frac{59002607487077327264965206342826351922381419296976827418}{700916985026129819774124824138029223771535099612222696937} a^{7} - \frac{95739632018699421569389455434542663706468482150410832764}{700916985026129819774124824138029223771535099612222696937} a^{6} + \frac{159433890256940661204362404641459593052305432236651772671}{700916985026129819774124824138029223771535099612222696937} a^{5} + \frac{163859049010281125907387795016891492130906044168093471369}{700916985026129819774124824138029223771535099612222696937} a^{4} - \frac{188278115286974558331689645061090804427508539217338819940}{700916985026129819774124824138029223771535099612222696937} a^{3} + \frac{77241149368360601950237232306600087146294710221931736543}{700916985026129819774124824138029223771535099612222696937} a^{2} - \frac{216902500733884508058130742693749084160459649232735255101}{700916985026129819774124824138029223771535099612222696937} a - \frac{155208285586682069771334322240251056936199520262042726369}{700916985026129819774124824138029223771535099612222696937}$

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

Class group and class number

$C_{7}\times C_{7}\times C_{11557}$, which has order $566293$ (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-43}) \), \(\Q(\zeta_{9})^+\), 6.0.521645427.2, \(\Q(\zeta_{27})^+\)

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 $18$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
43Data not computed