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Properties

Label 18.0.14730304268...7343.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{9}\cdot 37^{17}$
Root discriminant $131.96$
Ramified primes $19, 37$
Class number $1154174$ (GRH)
Class group $[1154174]$ (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![3387073609, -551296124, 910645155, -942303485, 309774585, -333953027, 129371239, -46920758, 31097213, -4301040, 4450991, -300446, 308310, -10650, 10480, -169, 168, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + 168*x^16 - 169*x^15 + 10480*x^14 - 10650*x^13 + 308310*x^12 - 300446*x^11 + 4450991*x^10 - 4301040*x^9 + 31097213*x^8 - 46920758*x^7 + 129371239*x^6 - 333953027*x^5 + 309774585*x^4 - 942303485*x^3 + 910645155*x^2 - 551296124*x + 3387073609)
 
gp: K = bnfinit(x^18 - x^17 + 168*x^16 - 169*x^15 + 10480*x^14 - 10650*x^13 + 308310*x^12 - 300446*x^11 + 4450991*x^10 - 4301040*x^9 + 31097213*x^8 - 46920758*x^7 + 129371239*x^6 - 333953027*x^5 + 309774585*x^4 - 942303485*x^3 + 910645155*x^2 - 551296124*x + 3387073609, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} + 168 x^{16} - 169 x^{15} + 10480 x^{14} - 10650 x^{13} + 308310 x^{12} - 300446 x^{11} + 4450991 x^{10} - 4301040 x^{9} + 31097213 x^{8} - 46920758 x^{7} + 129371239 x^{6} - 333953027 x^{5} + 309774585 x^{4} - 942303485 x^{3} + 910645155 x^{2} - 551296124 x + 3387073609 \)

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:  \(-147303042689029667841882658956210597343=-\,19^{9}\cdot 37^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $131.96$
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:  $19, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(703=19\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{703}(1,·)$, $\chi_{703}(514,·)$, $\chi_{703}(132,·)$, $\chi_{703}(455,·)$, $\chi_{703}(151,·)$, $\chi_{703}(398,·)$, $\chi_{703}(343,·)$, $\chi_{703}(474,·)$, $\chi_{703}(284,·)$, $\chi_{703}(419,·)$, $\chi_{703}(229,·)$, $\chi_{703}(305,·)$, $\chi_{703}(360,·)$, $\chi_{703}(552,·)$, $\chi_{703}(248,·)$, $\chi_{703}(571,·)$, $\chi_{703}(189,·)$, $\chi_{703}(702,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{337187} a^{16} + \frac{119214}{337187} a^{15} + \frac{74850}{337187} a^{14} - \frac{47312}{337187} a^{13} + \frac{34212}{337187} a^{12} + \frac{72344}{337187} a^{11} + \frac{100284}{337187} a^{10} - \frac{77712}{337187} a^{9} + \frac{146281}{337187} a^{8} - \frac{87488}{337187} a^{7} - \frac{45808}{337187} a^{6} - \frac{54573}{337187} a^{5} - \frac{48181}{337187} a^{4} + \frac{67558}{337187} a^{3} - \frac{87560}{337187} a^{2} - \frac{52402}{337187} a - \frac{1551}{10877}$, $\frac{1}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{17} + \frac{33060210665921508864692246503550832205241509132466234317537320149226635}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{16} - \frac{362997701397264998404063623277816383188754281764260814352959275093234083721}{1143218057835462401019587728073672783923994248378569369486730082316166967153} a^{15} + \frac{10080543555665956295397839339546659314386944083709754160831096482026602711405}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{14} + \frac{13789527680728525761925036700510432660739323222268667281864880350983334314943}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{13} + \frac{6028526839710190004600160582241225014944098565816619359578011099254593566452}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{12} + \frac{1542219455454366679837289247235053885755652711921089645432547182456234838240}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{11} + \frac{12281250057023756310692997672253393460267185825613312241408765619110791256980}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{10} - \frac{16977624660153843432227893393166925027566601923606917110695388098797463612266}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{9} - \frac{4153728912104497749975440509385733652288726928617118454036298815940095233411}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{8} - \frac{2991095152294822741095062854898816830715627184927932702442152819077488788126}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{7} + \frac{4436895566621111356114297998311459895459745598205277801034400007166696068197}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{6} + \frac{14039499286979555774843451085736814984280981527550226805801103284986411639561}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{5} + \frac{11558601236617305071745621561026409868403551073054528363762103306959073943943}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{4} - \frac{16541634121743331470578388920995492034995656478185154211605757794187577254237}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{3} - \frac{10205139415474954053306759760161667568710126209445403770367415819957955274492}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a^{2} + \frac{12234407364267713844423530114196297422202604685135214228280446048142746578997}{35439759792899334431607219570283856301643821699735650454088632551801175981743} a + \frac{399535389319233800801587347710965125090411867684732811862347045855100014526}{1143218057835462401019587728073672783923994248378569369486730082316166967153}$

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

Class group and class number

$C_{1154174}$, which has order $1154174$ (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:  \( 409151.3102125697 \) (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

\(\Q(\sqrt{-703}) \), 3.3.1369.1, 6.0.475630201063.1, 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ $18$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
19Data not computed
37Data not computed