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Properties

Label 18.0.89850521392...7311.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 37^{17}$
Root discriminant $52.44$
Ramified primes $3, 37$
Class number $152$ (GRH)
Class group $[152]$ (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![22717, 620, 21875, 13403, -3327, 981, 9767, -6238, 785, -1196, -405, -302, 174, -142, 120, -21, 20, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717)
 
gp: K = bnfinit(x^18 - x^17 + 20*x^16 - 21*x^15 + 120*x^14 - 142*x^13 + 174*x^12 - 302*x^11 - 405*x^10 - 1196*x^9 + 785*x^8 - 6238*x^7 + 9767*x^6 + 981*x^5 - 3327*x^4 + 13403*x^3 + 21875*x^2 + 620*x + 22717, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 20 x^{16} - 21 x^{15} + 120 x^{14} - 142 x^{13} + 174 x^{12} - 302 x^{11} - 405 x^{10} - 1196 x^{9} + 785 x^{8} - 6238 x^{7} + 9767 x^{6} + 981 x^{5} - 3327 x^{4} + 13403 x^{3} + 21875 x^{2} + 620 x + 22717 \)

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:  \(-8985052139278849963819823767311=-\,3^{9}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.44$
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:  $3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(111=3\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{111}(1,·)$, $\chi_{111}(70,·)$, $\chi_{111}(65,·)$, $\chi_{111}(10,·)$, $\chi_{111}(11,·)$, $\chi_{111}(77,·)$, $\chi_{111}(16,·)$, $\chi_{111}(46,·)$, $\chi_{111}(95,·)$, $\chi_{111}(34,·)$, $\chi_{111}(100,·)$, $\chi_{111}(101,·)$, $\chi_{111}(104,·)$, $\chi_{111}(41,·)$, $\chi_{111}(7,·)$, $\chi_{111}(110,·)$, $\chi_{111}(49,·)$, $\chi_{111}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7415767366721339600983887549343294947109} a^{17} - \frac{2462785049595615436919553479502327921311}{7415767366721339600983887549343294947109} a^{16} + \frac{774669895576947310736113670471801776335}{7415767366721339600983887549343294947109} a^{15} - \frac{1548545534084382552266979114771018098170}{7415767366721339600983887549343294947109} a^{14} + \frac{2356502744364577316694440055670972813355}{7415767366721339600983887549343294947109} a^{13} + \frac{454294533849196871405307581760709637290}{7415767366721339600983887549343294947109} a^{12} - \frac{3633924172154913711809080067885534031551}{7415767366721339600983887549343294947109} a^{11} + \frac{2013977797504593992879965315893322970533}{7415767366721339600983887549343294947109} a^{10} + \frac{193684202940283594169091672536397854431}{7415767366721339600983887549343294947109} a^{9} - \frac{1204929247890478317956721624745554396963}{7415767366721339600983887549343294947109} a^{8} + \frac{1212576010650177974592570695267056941992}{7415767366721339600983887549343294947109} a^{7} - \frac{483638872561937492501210904768618346046}{7415767366721339600983887549343294947109} a^{6} + \frac{2653194181267985353348771171461603565276}{7415767366721339600983887549343294947109} a^{5} - \frac{657224678249696879707869552084479205614}{7415767366721339600983887549343294947109} a^{4} + \frac{3061476643534552802712483448167331465620}{7415767366721339600983887549343294947109} a^{3} - \frac{569199575377776564275914979126162765414}{7415767366721339600983887549343294947109} a^{2} - \frac{3603261108065949328197138435933705130279}{7415767366721339600983887549343294947109} a - \frac{2492648156736596658877137399967543322686}{7415767366721339600983887549343294947109}$

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

Class group and class number

$C_{152}$, which has order $152$ (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

\(\Q(\sqrt{-111}) \), 3.3.1369.1, 6.0.1872286839.1, 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ R $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
37Data not computed