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Properties

Label 18.0.25815178338...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{44}\cdot 5^{9}$
Root discriminant $92.75$
Ramified primes $2, 3, 5$
Class number $256298$ (GRH)
Class group $[256298]$ (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![4843960811, -334818, 2631789081, 236460, 688088160, -29034, 113290266, 738, 12937608, -2, 1064547, 0, 63384, 0, 2655, 0, 72, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 72*x^16 + 2655*x^14 + 63384*x^12 + 1064547*x^10 - 2*x^9 + 12937608*x^8 + 738*x^7 + 113290266*x^6 - 29034*x^5 + 688088160*x^4 + 236460*x^3 + 2631789081*x^2 - 334818*x + 4843960811)
 
gp: K = bnfinit(x^18 + 72*x^16 + 2655*x^14 + 63384*x^12 + 1064547*x^10 - 2*x^9 + 12937608*x^8 + 738*x^7 + 113290266*x^6 - 29034*x^5 + 688088160*x^4 + 236460*x^3 + 2631789081*x^2 - 334818*x + 4843960811, 1)
 

Normalized defining polynomial

\( x^{18} + 72 x^{16} + 2655 x^{14} + 63384 x^{12} + 1064547 x^{10} - 2 x^{9} + 12937608 x^{8} + 738 x^{7} + 113290266 x^{6} - 29034 x^{5} + 688088160 x^{4} + 236460 x^{3} + 2631789081 x^{2} - 334818 x + 4843960811 \)

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:  \(-258151783382020583032356864000000000=-\,2^{27}\cdot 3^{44}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.75$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1080=2^{3}\cdot 3^{3}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{1080}(1,·)$, $\chi_{1080}(259,·)$, $\chi_{1080}(961,·)$, $\chi_{1080}(841,·)$, $\chi_{1080}(139,·)$, $\chi_{1080}(721,·)$, $\chi_{1080}(19,·)$, $\chi_{1080}(601,·)$, $\chi_{1080}(859,·)$, $\chi_{1080}(979,·)$, $\chi_{1080}(481,·)$, $\chi_{1080}(739,·)$, $\chi_{1080}(361,·)$, $\chi_{1080}(619,·)$, $\chi_{1080}(241,·)$, $\chi_{1080}(499,·)$, $\chi_{1080}(121,·)$, $\chi_{1080}(379,·)$$\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}{184125042691347889429331094901843541102786395051327075976809} a^{17} - \frac{51323050845853779546199870663092445934563941269658622654607}{184125042691347889429331094901843541102786395051327075976809} a^{16} - \frac{17316497980587398510982517736124799071135102418567233901031}{184125042691347889429331094901843541102786395051327075976809} a^{15} - \frac{26304136847665600776728382941120544031538630263376443356973}{184125042691347889429331094901843541102786395051327075976809} a^{14} - \frac{10448524213724061836741577162561304685249082027725495349773}{184125042691347889429331094901843541102786395051327075976809} a^{13} + \frac{14667930758271240508021631200063944997368404563006589587353}{184125042691347889429331094901843541102786395051327075976809} a^{12} + \frac{17249691063328248638836916091271903788146244746563579166105}{184125042691347889429331094901843541102786395051327075976809} a^{11} - \frac{47237348716651605756503966197659552325937848827930699781194}{184125042691347889429331094901843541102786395051327075976809} a^{10} + \frac{61650379371522435769502439973708747250455646900801018551945}{184125042691347889429331094901843541102786395051327075976809} a^{9} + \frac{78940648108920398953440810036981431884611664528368599942161}{184125042691347889429331094901843541102786395051327075976809} a^{8} - \frac{42203643286385616733151476290182110274589156673434719105827}{184125042691347889429331094901843541102786395051327075976809} a^{7} - \frac{61958491214153839458321585477072053198767151445251085016038}{184125042691347889429331094901843541102786395051327075976809} a^{6} - \frac{50548866930024237413422407176153603988413425410456964261749}{184125042691347889429331094901843541102786395051327075976809} a^{5} + \frac{24659446821746584443232780441522604928999461553337809045684}{184125042691347889429331094901843541102786395051327075976809} a^{4} - \frac{31612836608758053125068706965163565712724969500833983364612}{184125042691347889429331094901843541102786395051327075976809} a^{3} - \frac{29684681514207251719686418409076899930036662072382634970647}{184125042691347889429331094901843541102786395051327075976809} a^{2} - \frac{85338805100421211180857381461751809711780649191497834021201}{184125042691347889429331094901843541102786395051327075976809} a - \frac{1672391528826620245932681159515252711618292184943135906506}{3474057409270714894893039526449878134014837642477869358053}$

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

Class group and class number

$C_{256298}$, which has order $256298$ (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:  \( 40934.0329443 \) (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

\(\Q(\sqrt{-10}) \), \(\Q(\zeta_{9})^+\), 6.0.419904000.3, \(\Q(\zeta_{27})^+\)

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 R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
2Data not computed
3Data not computed
5Data not computed