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Properties

Label 18.0.49786310714...7287.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,7^{9}\cdot 37^{16}$
Root discriminant $65.54$
Ramified primes $7, 37$
Class number $1539$ (GRH)
Class group $[9, 171]$ (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![477047, -538288, 758771, -449504, 331388, -153409, 125132, -73223, 41381, -13245, 4032, -2109, 1582, -453, -87, 62, 6, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 6*x^16 + 62*x^15 - 87*x^14 - 453*x^13 + 1582*x^12 - 2109*x^11 + 4032*x^10 - 13245*x^9 + 41381*x^8 - 73223*x^7 + 125132*x^6 - 153409*x^5 + 331388*x^4 - 449504*x^3 + 758771*x^2 - 538288*x + 477047)
 
gp: K = bnfinit(x^18 - 7*x^17 + 6*x^16 + 62*x^15 - 87*x^14 - 453*x^13 + 1582*x^12 - 2109*x^11 + 4032*x^10 - 13245*x^9 + 41381*x^8 - 73223*x^7 + 125132*x^6 - 153409*x^5 + 331388*x^4 - 449504*x^3 + 758771*x^2 - 538288*x + 477047, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 6 x^{16} + 62 x^{15} - 87 x^{14} - 453 x^{13} + 1582 x^{12} - 2109 x^{11} + 4032 x^{10} - 13245 x^{9} + 41381 x^{8} - 73223 x^{7} + 125132 x^{6} - 153409 x^{5} + 331388 x^{4} - 449504 x^{3} + 758771 x^{2} - 538288 x + 477047 \)

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:  \(-497863107144137244036834347537287=-\,7^{9}\cdot 37^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $65.54$
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:  $7, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(259=7\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{259}(1,·)$, $\chi_{259}(195,·)$, $\chi_{259}(197,·)$, $\chi_{259}(71,·)$, $\chi_{259}(211,·)$, $\chi_{259}(90,·)$, $\chi_{259}(155,·)$, $\chi_{259}(218,·)$, $\chi_{259}(223,·)$, $\chi_{259}(160,·)$, $\chi_{259}(34,·)$, $\chi_{259}(232,·)$, $\chi_{259}(174,·)$, $\chi_{259}(83,·)$, $\chi_{259}(181,·)$, $\chi_{259}(118,·)$, $\chi_{259}(120,·)$, $\chi_{259}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{43} a^{15} + \frac{20}{43} a^{14} + \frac{11}{43} a^{13} - \frac{18}{43} a^{12} + \frac{9}{43} a^{11} - \frac{7}{43} a^{10} + \frac{7}{43} a^{9} + \frac{3}{43} a^{8} + \frac{3}{43} a^{7} + \frac{1}{43} a^{6} - \frac{6}{43} a^{5} + \frac{6}{43} a^{4} - \frac{20}{43} a^{3} - \frac{12}{43} a^{2} + \frac{18}{43} a - \frac{16}{43}$, $\frac{1}{1333} a^{16} + \frac{1}{1333} a^{15} + \frac{190}{1333} a^{14} + \frac{332}{1333} a^{13} - \frac{509}{1333} a^{12} - \frac{436}{1333} a^{11} - \frac{505}{1333} a^{10} + \frac{257}{1333} a^{9} + \frac{505}{1333} a^{8} - \frac{615}{1333} a^{7} - \frac{240}{1333} a^{6} + \frac{292}{1333} a^{5} - \frac{177}{1333} a^{4} - \frac{234}{1333} a^{3} + \frac{547}{1333} a^{2} - \frac{573}{1333} a - \frac{212}{1333}$, $\frac{1}{632519033990960283054653664357437260793759} a^{17} + \frac{120365597013295846605852216801418293853}{632519033990960283054653664357437260793759} a^{16} - \frac{1318733163818801730438844545979899989638}{632519033990960283054653664357437260793759} a^{15} - \frac{170608305710051102217458277266578318455994}{632519033990960283054653664357437260793759} a^{14} + \frac{178012607849045392492878044122091318749416}{632519033990960283054653664357437260793759} a^{13} - \frac{186747306111030567737586746910766105979111}{632519033990960283054653664357437260793759} a^{12} - \frac{252186704286770927491871703463257459249108}{632519033990960283054653664357437260793759} a^{11} - \frac{296033149773464269425390030306155541701604}{632519033990960283054653664357437260793759} a^{10} - \frac{73111506785636410555861081705204541884784}{632519033990960283054653664357437260793759} a^{9} + \frac{51669907827374730235652754798179311105085}{632519033990960283054653664357437260793759} a^{8} + \frac{114112552158767549971170999417126588816980}{632519033990960283054653664357437260793759} a^{7} - \frac{252352813926166220595449806744720523769982}{632519033990960283054653664357437260793759} a^{6} + \frac{275007460699447248425770138764668774706394}{632519033990960283054653664357437260793759} a^{5} + \frac{182112343634792554315687979707773492952077}{632519033990960283054653664357437260793759} a^{4} - \frac{100260499604386824226225464835089752810381}{632519033990960283054653664357437260793759} a^{3} + \frac{68265049603129613360015258668490986072001}{632519033990960283054653664357437260793759} a^{2} + \frac{5310918978990789736011889415302192082720}{20403839806160009130795279495401201961089} a - \frac{122596474230901334025408793775416968154509}{632519033990960283054653664357437260793759}$

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

Class group and class number

$C_{9}\times C_{171}$, which has order $1539$ (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:  \( 409151.310213 \) (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

\(\Q(\sqrt{-7}) \), 3.3.1369.1, 6.0.642837223.1, 9.9.3512479453921.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ $18$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R $18$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
7Data not computed
$37$37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$
37.9.8.1$x^{9} - 37$$9$$1$$8$$C_9$$[\ ]_{9}$