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Properties

Label 18.0.47429540431...9375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 5^{9}\cdot 37^{16}$
Root discriminant $95.94$
Ramified primes $3, 5, 37$
Class number $37058$ (GRH)
Class group $[37058]$ (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![24056839, -16951280, 23891131, -11979744, 9673910, -4112929, 2517118, -952935, 445351, -139961, 52234, -14541, 4672, -1033, 241, -50, 24, -7, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 7*x^17 + 24*x^16 - 50*x^15 + 241*x^14 - 1033*x^13 + 4672*x^12 - 14541*x^11 + 52234*x^10 - 139961*x^9 + 445351*x^8 - 952935*x^7 + 2517118*x^6 - 4112929*x^5 + 9673910*x^4 - 11979744*x^3 + 23891131*x^2 - 16951280*x + 24056839)
 
gp: K = bnfinit(x^18 - 7*x^17 + 24*x^16 - 50*x^15 + 241*x^14 - 1033*x^13 + 4672*x^12 - 14541*x^11 + 52234*x^10 - 139961*x^9 + 445351*x^8 - 952935*x^7 + 2517118*x^6 - 4112929*x^5 + 9673910*x^4 - 11979744*x^3 + 23891131*x^2 - 16951280*x + 24056839, 1)
 

Normalized defining polynomial

\( x^{18} - 7 x^{17} + 24 x^{16} - 50 x^{15} + 241 x^{14} - 1033 x^{13} + 4672 x^{12} - 14541 x^{11} + 52234 x^{10} - 139961 x^{9} + 445351 x^{8} - 952935 x^{7} + 2517118 x^{6} - 4112929 x^{5} + 9673910 x^{4} - 11979744 x^{3} + 23891131 x^{2} - 16951280 x + 24056839 \)

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:  \(-474295404311594698259070089068359375=-\,3^{9}\cdot 5^{9}\cdot 37^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $95.94$
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:  $3, 5, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(555=3\cdot 5\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{555}(256,·)$, $\chi_{555}(1,·)$, $\chi_{555}(194,·)$, $\chi_{555}(451,·)$, $\chi_{555}(329,·)$, $\chi_{555}(269,·)$, $\chi_{555}(271,·)$, $\chi_{555}(16,·)$, $\chi_{555}(211,·)$, $\chi_{555}(404,·)$, $\chi_{555}(149,·)$, $\chi_{555}(419,·)$, $\chi_{555}(164,·)$, $\chi_{555}(359,·)$, $\chi_{555}(44,·)$, $\chi_{555}(46,·)$, $\chi_{555}(181,·)$, $\chi_{555}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{31} a^{15} - \frac{11}{31} a^{13} - \frac{15}{31} a^{12} - \frac{6}{31} a^{11} - \frac{15}{31} a^{10} - \frac{3}{31} a^{9} - \frac{15}{31} a^{8} - \frac{7}{31} a^{7} - \frac{11}{31} a^{6} + \frac{12}{31} a^{5} + \frac{2}{31} a^{3} - \frac{10}{31} a^{2} + \frac{3}{31} a - \frac{5}{31}$, $\frac{1}{1333} a^{16} + \frac{11}{1333} a^{15} + \frac{578}{1333} a^{14} + \frac{639}{1333} a^{13} + \frac{232}{1333} a^{12} + \frac{663}{1333} a^{11} - \frac{261}{1333} a^{10} - \frac{17}{1333} a^{9} - \frac{482}{1333} a^{8} - \frac{491}{1333} a^{7} + \frac{170}{1333} a^{6} - \frac{178}{1333} a^{5} - \frac{277}{1333} a^{4} - \frac{608}{1333} a^{3} + \frac{79}{1333} a^{2} + \frac{245}{1333} a + \frac{658}{1333}$, $\frac{1}{64971458031073856764107624306495112618547511907147} a^{17} + \frac{10855352648754751264624883363113784346061469617}{64971458031073856764107624306495112618547511907147} a^{16} + \frac{678505292192008662079851671582867901810651054038}{64971458031073856764107624306495112618547511907147} a^{15} - \frac{948952357634944857828333237457170904056609310187}{64971458031073856764107624306495112618547511907147} a^{14} + \frac{31979620323865523677828331676428956663951243843848}{64971458031073856764107624306495112618547511907147} a^{13} - \frac{7594911627955378585121678144157143899281454035796}{64971458031073856764107624306495112618547511907147} a^{12} + \frac{25641147891166306433759233013701402992777340551873}{64971458031073856764107624306495112618547511907147} a^{11} + \frac{3788672312029927438809268487024063407226028078035}{64971458031073856764107624306495112618547511907147} a^{10} - \frac{14048572067672628311839041052333127720775297105916}{64971458031073856764107624306495112618547511907147} a^{9} + \frac{497859119894942174811061351097876181453916323154}{2095853484873350218197020138919197181243468126037} a^{8} - \frac{3993868551973815096516418981935365286720872698003}{64971458031073856764107624306495112618547511907147} a^{7} - \frac{32322532740403421661562564663323247356246910132330}{64971458031073856764107624306495112618547511907147} a^{6} - \frac{24481921868684232512978419506737579246751386669431}{64971458031073856764107624306495112618547511907147} a^{5} + \frac{21493951171703839757412446415840764842601434042180}{64971458031073856764107624306495112618547511907147} a^{4} - \frac{18981304189441199146027291165647090398163590979468}{64971458031073856764107624306495112618547511907147} a^{3} + \frac{15819099291892933330882451732303331230076482609344}{64971458031073856764107624306495112618547511907147} a^{2} - \frac{19088097017670198806704950873161495470661807024764}{64971458031073856764107624306495112618547511907147} a - \frac{10035722839041652311612604191363721359007297505219}{64971458031073856764107624306495112618547511907147}$

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

Class group and class number

$C_{37058}$, which has order $37058$ (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

\(\Q(\sqrt{-15}) \), 3.3.1369.1, 6.0.6325293375.1, 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 R $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ R $18$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ $18$

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
3Data not computed
5Data not computed
37Data not computed