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Properties

Label 18.0.76200670127...0544.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{9}\cdot 19^{16}$
Root discriminant $67.11$
Ramified primes $2, 3, 19$
Class number $18506$ (GRH)
Class group $[18506]$ (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![103115431, -20623090, 76910381, -14144668, 28239050, -4749676, 6677144, -1017454, 1120553, -152462, 138889, -16470, 12807, -1276, 860, -66, 39, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2*x^17 + 39*x^16 - 66*x^15 + 860*x^14 - 1276*x^13 + 12807*x^12 - 16470*x^11 + 138889*x^10 - 152462*x^9 + 1120553*x^8 - 1017454*x^7 + 6677144*x^6 - 4749676*x^5 + 28239050*x^4 - 14144668*x^3 + 76910381*x^2 - 20623090*x + 103115431)
 
gp: K = bnfinit(x^18 - 2*x^17 + 39*x^16 - 66*x^15 + 860*x^14 - 1276*x^13 + 12807*x^12 - 16470*x^11 + 138889*x^10 - 152462*x^9 + 1120553*x^8 - 1017454*x^7 + 6677144*x^6 - 4749676*x^5 + 28239050*x^4 - 14144668*x^3 + 76910381*x^2 - 20623090*x + 103115431, 1)
 

Normalized defining polynomial

\( x^{18} - 2 x^{17} + 39 x^{16} - 66 x^{15} + 860 x^{14} - 1276 x^{13} + 12807 x^{12} - 16470 x^{11} + 138889 x^{10} - 152462 x^{9} + 1120553 x^{8} - 1017454 x^{7} + 6677144 x^{6} - 4749676 x^{5} + 28239050 x^{4} - 14144668 x^{3} + 76910381 x^{2} - 20623090 x + 103115431 \)

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:  \(-762006701275810777282417256300544=-\,2^{27}\cdot 3^{9}\cdot 19^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $67.11$
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:  $2, 3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(456=2^{3}\cdot 3\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{456}(1,·)$, $\chi_{456}(197,·)$, $\chi_{456}(385,·)$, $\chi_{456}(73,·)$, $\chi_{456}(77,·)$, $\chi_{456}(149,·)$, $\chi_{456}(121,·)$, $\chi_{456}(25,·)$, $\chi_{456}(5,·)$, $\chi_{456}(289,·)$, $\chi_{456}(101,·)$, $\chi_{456}(389,·)$, $\chi_{456}(169,·)$, $\chi_{456}(365,·)$, $\chi_{456}(49,·)$, $\chi_{456}(245,·)$, $\chi_{456}(313,·)$, $\chi_{456}(125,·)$$\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}{11397902101497910952248547269128046773158497715010193} a^{17} - \frac{4453014023645486683893960454492367546132742519418952}{11397902101497910952248547269128046773158497715010193} a^{16} - \frac{1364864153031884256603374070839441273498594439377407}{11397902101497910952248547269128046773158497715010193} a^{15} + \frac{1423741677668014749449734359490430577209164081256556}{11397902101497910952248547269128046773158497715010193} a^{14} + \frac{2811571502576202302610390689504648591584198929036197}{11397902101497910952248547269128046773158497715010193} a^{13} + \frac{1697865407113942381635786223243801959677909063788097}{11397902101497910952248547269128046773158497715010193} a^{12} - \frac{4622783572484723910880868933931592866917151576117116}{11397902101497910952248547269128046773158497715010193} a^{11} - \frac{3137646039042559954547230071650667206662845897507002}{11397902101497910952248547269128046773158497715010193} a^{10} - \frac{1844291215837052166007847369410402668384508412109208}{11397902101497910952248547269128046773158497715010193} a^{9} - \frac{4730804161440837544624356305116491681732703004018741}{11397902101497910952248547269128046773158497715010193} a^{8} - \frac{2968135861399743970995642578558019009398780690339820}{11397902101497910952248547269128046773158497715010193} a^{7} + \frac{3400047423942085109297283754105781086255392104535727}{11397902101497910952248547269128046773158497715010193} a^{6} - \frac{3874642653144424197269676616835443312537000288079231}{11397902101497910952248547269128046773158497715010193} a^{5} + \frac{233874806041657713396886064385678197825444517074866}{11397902101497910952248547269128046773158497715010193} a^{4} - \frac{4279319990509687307061911220376595738353969784916641}{11397902101497910952248547269128046773158497715010193} a^{3} - \frac{5680718670543692451533133227995455275519951353695096}{11397902101497910952248547269128046773158497715010193} a^{2} + \frac{3581290796299036060785301439357412050483857431130504}{11397902101497910952248547269128046773158497715010193} a - \frac{561212618259320843158764617057537221276102245026838}{11397902101497910952248547269128046773158497715010193}$

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

Class group and class number

$C_{18506}$, which has order $18506$ (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:  \( 22305.8950792 \) (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

\(\Q(\sqrt{-6}) \), 3.3.361.1, 6.0.1801557504.1, \(\Q(\zeta_{19})^+\)

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ $18$ R $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ $18$ $18$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
3Data not computed
19Data not computed