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Properties

Label 18.0.36767744115...6256.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{45}\cdot 7^{15}$
Root discriminant $157.79$
Ramified primes $2, 3, 7$
Class number $1208172$ (GRH)
Class group $[2, 604086]$ (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![107354541, 0, 466948881, 0, 444713220, 0, 163061514, 0, 29950074, 0, 3090087, 0, 187278, 0, 6615, 0, 126, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 126*x^16 + 6615*x^14 + 187278*x^12 + 3090087*x^10 + 29950074*x^8 + 163061514*x^6 + 444713220*x^4 + 466948881*x^2 + 107354541)
 
gp: K = bnfinit(x^18 + 126*x^16 + 6615*x^14 + 187278*x^12 + 3090087*x^10 + 29950074*x^8 + 163061514*x^6 + 444713220*x^4 + 466948881*x^2 + 107354541, 1)
 

Normalized defining polynomial

\( x^{18} + 126 x^{16} + 6615 x^{14} + 187278 x^{12} + 3090087 x^{10} + 29950074 x^{8} + 163061514 x^{6} + 444713220 x^{4} + 466948881 x^{2} + 107354541 \)

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:  \(-3676774411522871683138204076633580896256=-\,2^{18}\cdot 3^{45}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $157.79$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(756=2^{2}\cdot 3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{756}(1,·)$, $\chi_{756}(131,·)$, $\chi_{756}(503,·)$, $\chi_{756}(529,·)$, $\chi_{756}(277,·)$, $\chi_{756}(505,·)$, $\chi_{756}(25,·)$, $\chi_{756}(731,·)$, $\chi_{756}(479,·)$, $\chi_{756}(227,·)$, $\chi_{756}(625,·)$, $\chi_{756}(755,·)$, $\chi_{756}(373,·)$, $\chi_{756}(251,·)$, $\chi_{756}(121,·)$, $\chi_{756}(635,·)$, $\chi_{756}(253,·)$, $\chi_{756}(383,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6}$, $\frac{1}{7} a^{7}$, $\frac{1}{7} a^{8}$, $\frac{1}{2261} a^{9} + \frac{9}{323} a^{7} - \frac{134}{323} a^{5} - \frac{145}{323} a^{3} - \frac{143}{323} a$, $\frac{1}{897617} a^{10} + \frac{10399}{897617} a^{8} - \frac{61016}{897617} a^{6} + \frac{37969}{128231} a^{4} + \frac{27958}{128231} a^{2} + \frac{117}{397}$, $\frac{1}{897617} a^{11} + \frac{11}{128231} a^{9} + \frac{58084}{897617} a^{7} + \frac{10576}{128231} a^{5} - \frac{14124}{128231} a^{3} - \frac{24935}{128231} a$, $\frac{1}{6283319} a^{12} + \frac{3821}{897617} a^{8} + \frac{40597}{897617} a^{6} - \frac{16665}{128231} a^{4} + \frac{318}{128231} a^{2} - \frac{96}{397}$, $\frac{1}{6283319} a^{13} - \frac{149}{897617} a^{9} + \frac{353}{6749} a^{7} + \frac{2391}{128231} a^{5} + \frac{63044}{128231} a^{3} + \frac{23778}{128231} a$, $\frac{1}{6283319} a^{14} + \frac{57628}{897617} a^{8} + \frac{1566}{47243} a^{6} - \frac{2630}{6749} a^{4} - \frac{42103}{128231} a^{2} - \frac{35}{397}$, $\frac{1}{6283319} a^{15} + \frac{9}{128231} a^{9} - \frac{6373}{897617} a^{7} - \frac{30120}{128231} a^{5} - \frac{30193}{128231} a^{3} + \frac{13706}{128231} a$, $\frac{1}{6283319} a^{16} - \frac{20355}{897617} a^{8} + \frac{6100}{128231} a^{6} + \frac{14149}{128231} a^{4} + \frac{47586}{128231} a^{2} + \frac{172}{397}$, $\frac{1}{6283319} a^{17} - \frac{108}{897617} a^{9} + \frac{35951}{897617} a^{7} - \frac{6098}{128231} a^{5} + \frac{61084}{128231} a^{3} - \frac{1099}{7543} a$

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

Class group and class number

$C_{2}\times C_{604086}$, which has order $1208172$ (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:  \( 10392888.21418944 \) (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

\(\Q(\sqrt{-21}) \), \(\Q(\zeta_{9})^+\), 6.0.432081216.1, 9.9.3691950281939241.2

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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ $18$

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
7Data not computed