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Properties

Label 18.0.31165870159...4288.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{44}\cdot 11^{9}$
Root discriminant $137.57$
Ramified primes $2, 3, 11$
Class number $8732666$ (GRH)
Class group $[8732666]$ (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![2600743873751, -5692914, 828116450649, 2105724, 122042249064, -130338, 10917716106, 1602, 653310108, -2, 27134019, 0, 783204, 0, 15183, 0, 180, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 180*x^16 + 15183*x^14 + 783204*x^12 + 27134019*x^10 - 2*x^9 + 653310108*x^8 + 1602*x^7 + 10917716106*x^6 - 130338*x^5 + 122042249064*x^4 + 2105724*x^3 + 828116450649*x^2 - 5692914*x + 2600743873751)
 
gp: K = bnfinit(x^18 + 180*x^16 + 15183*x^14 + 783204*x^12 + 27134019*x^10 - 2*x^9 + 653310108*x^8 + 1602*x^7 + 10917716106*x^6 - 130338*x^5 + 122042249064*x^4 + 2105724*x^3 + 828116450649*x^2 - 5692914*x + 2600743873751, 1)
 

Normalized defining polynomial

\( x^{18} + 180 x^{16} + 15183 x^{14} + 783204 x^{12} + 27134019 x^{10} - 2 x^{9} + 653310108 x^{8} + 1602 x^{7} + 10917716106 x^{6} - 130338 x^{5} + 122042249064 x^{4} + 2105724 x^{3} + 828116450649 x^{2} - 5692914 x + 2600743873751 \)

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:  \(-311658701595221813593917258627482124288=-\,2^{27}\cdot 3^{44}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.57$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2376=2^{3}\cdot 3^{3}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2376}(1,·)$, $\chi_{2376}(901,·)$, $\chi_{2376}(2113,·)$, $\chi_{2376}(265,·)$, $\chi_{2376}(1165,·)$, $\chi_{2376}(2221,·)$, $\chi_{2376}(529,·)$, $\chi_{2376}(1429,·)$, $\chi_{2376}(793,·)$, $\chi_{2376}(1693,·)$, $\chi_{2376}(1057,·)$, $\chi_{2376}(1957,·)$, $\chi_{2376}(1321,·)$, $\chi_{2376}(109,·)$, $\chi_{2376}(1585,·)$, $\chi_{2376}(373,·)$, $\chi_{2376}(1849,·)$, $\chi_{2376}(637,·)$$\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}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{17} + \frac{2670248631723261133709773280547542060427556889216004469932612065995949}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{16} - \frac{183370896623985727340143969387268470614436464361367248042418941427832}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{15} + \frac{19993516628032298103089621223176315542430884981850893660187426935172898}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{14} - \frac{12575761256199997523440412684856630092539118659375325875663474952463237}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{13} - \frac{24905901458452459670975420511315444114982464086319567241361194340263171}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{12} - \frac{27693180730431124584917778368155083426396333669309949887495726540267545}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{11} + \frac{3990792282972868788832006940025628457958835995246386638710292012475727}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{10} - \frac{15425767343928265853510047463843374381388310727807099038892521083297170}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{9} + \frac{3144838424393364695365444761359984043441636798799689976163666061154890}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{8} + \frac{11149133079656007497955764476577187672448330423100623532136539110684540}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{7} + \frac{25057244074472836425437225773661512107933338600686914253518755645593551}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{6} - \frac{24267370251784610749287259880205417978045078977425192482520126975767117}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{5} - \frac{9222588936321344788932235604916858712532280441233442125152198712567589}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{4} - \frac{24040484904130898106553118394502238362062084088441055412364573861656831}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{3} - \frac{11697658079481203494390487442576845682404358247991858590625537634861189}{57957434414862235744505032793950471434587912022430723902564898407813193} a^{2} + \frac{14312743701465294289543345716227708710216712208283714437514415740627956}{57957434414862235744505032793950471434587912022430723902564898407813193} a + \frac{22704029158892252616909661740401552739124264112282401563284627103556427}{57957434414862235744505032793950471434587912022430723902564898407813193}$

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

Class group and class number

$C_{8732666}$, which has order $8732666$ (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

\(\Q(\sqrt{-22}) \), \(\Q(\zeta_{9})^+\), 6.0.4471137792.13, \(\Q(\zeta_{27})^+\)

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 R R $18$ $18$ R ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ $18$

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
2Data not computed
3Data not computed
11Data not computed