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Properties

Label 18.0.39739057971...1767.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{44}\cdot 7^{9}$
Root discriminant $38.80$
Ramified primes $3, 7$
Class number $163$ (GRH)
Class group $[163]$ (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![512, 2304, 20736, 9600, 51840, 4752, 61488, 468, 35964, -5, 12465, 0, 2646, 0, 351, 0, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512)
 
gp: K = bnfinit(x^18 + 27*x^16 + 351*x^14 + 2646*x^12 + 12465*x^10 - 5*x^9 + 35964*x^8 + 468*x^7 + 61488*x^6 + 4752*x^5 + 51840*x^4 + 9600*x^3 + 20736*x^2 + 2304*x + 512, 1)
 

Normalized defining polynomial

\( x^{18} + 27 x^{16} + 351 x^{14} + 2646 x^{12} + 12465 x^{10} - 5 x^{9} + 35964 x^{8} + 468 x^{7} + 61488 x^{6} + 4752 x^{5} + 51840 x^{4} + 9600 x^{3} + 20736 x^{2} + 2304 x + 512 \)

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:  \(-39739057971752889532465351767=-\,3^{44}\cdot 7^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.80$
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:  $3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(139,·)$, $\chi_{189}(76,·)$, $\chi_{189}(13,·)$, $\chi_{189}(148,·)$, $\chi_{189}(85,·)$, $\chi_{189}(22,·)$, $\chi_{189}(160,·)$, $\chi_{189}(97,·)$, $\chi_{189}(34,·)$, $\chi_{189}(169,·)$, $\chi_{189}(106,·)$, $\chi_{189}(43,·)$, $\chi_{189}(181,·)$, $\chi_{189}(118,·)$, $\chi_{189}(55,·)$, $\chi_{189}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{5} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{10} + \frac{3}{8} a^{8} + \frac{1}{4} a^{6} + \frac{1}{8} a^{4} + \frac{3}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{11} - \frac{5}{16} a^{9} + \frac{1}{8} a^{7} - \frac{7}{16} a^{5} - \frac{5}{16} a^{4}$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} - \frac{5}{32} a^{10} + \frac{1}{16} a^{8} - \frac{7}{32} a^{6} - \frac{5}{32} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{64} a^{15} - \frac{1}{64} a^{13} - \frac{5}{64} a^{11} - \frac{15}{32} a^{9} - \frac{7}{64} a^{7} - \frac{5}{64} a^{6} - \frac{1}{2} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{2571136} a^{16} - \frac{1177}{321392} a^{15} + \frac{7691}{2571136} a^{14} - \frac{2879}{160696} a^{13} + \frac{71295}{2571136} a^{12} + \frac{7463}{80348} a^{11} + \frac{215979}{1285568} a^{10} + \frac{41113}{321392} a^{9} + \frac{413313}{2571136} a^{8} - \frac{866365}{2571136} a^{7} - \frac{181171}{642784} a^{6} - \frac{223733}{642784} a^{5} + \frac{68235}{321392} a^{4} + \frac{2360}{20087} a^{3} + \frac{2697}{20087} a^{2} - \frac{12337}{40174} a + \frac{5414}{20087}$, $\frac{1}{74232187974706333938944} a^{17} - \frac{831127728464451}{37116093987353166969472} a^{16} + \frac{377271533833022271719}{74232187974706333938944} a^{15} - \frac{278331278352787660949}{37116093987353166969472} a^{14} - \frac{29942816978172247401}{1400607320277477998848} a^{13} + \frac{1955755165701272933671}{37116093987353166969472} a^{12} + \frac{1096579039430598341909}{37116093987353166969472} a^{11} - \frac{961025827654799578423}{18558046993676583484736} a^{10} - \frac{9671586112271963586087}{74232187974706333938944} a^{9} + \frac{11480494090110452449605}{74232187974706333938944} a^{8} - \frac{8132261261468205560909}{37116093987353166969472} a^{7} - \frac{4056772702328006763867}{9279023496838291742368} a^{6} + \frac{710126919121812551673}{4639511748419145871184} a^{5} + \frac{235751092229239154461}{579938968552393233898} a^{4} + \frac{1091383716516322603135}{2319755874209572935592} a^{3} + \frac{326613003505981304781}{1159877937104786467796} a^{2} - \frac{139489603323674068435}{579938968552393233898} a + \frac{141328716160354249477}{289969484276196616949}$

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

Class group and class number

$C_{163}$, which has order $163$ (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

\(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), 6.0.2250423.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.1.0.1}{1} }^{18}$ $18$

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
7Data not computed