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Properties

Label 18.0.18825084986...3072.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{45}\cdot 7^{15}$
Root discriminant $223.15$
Ramified primes $2, 3, 7$
Class number $11285508$ (GRH)
Class group $[2, 5642754]$ (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![54965524992, 0, 119538913536, 0, 56923292160, 0, 10435936896, 0, 958402368, 0, 49441392, 0, 1498224, 0, 26460, 0, 252, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 54965524992)
 
gp: K = bnfinit(x^18 + 252*x^16 + 26460*x^14 + 1498224*x^12 + 49441392*x^10 + 958402368*x^8 + 10435936896*x^6 + 56923292160*x^4 + 119538913536*x^2 + 54965524992, 1)
 

Normalized defining polynomial

\( x^{18} + 252 x^{16} + 26460 x^{14} + 1498224 x^{12} + 49441392 x^{10} + 958402368 x^{8} + 10435936896 x^{6} + 56923292160 x^{4} + 119538913536 x^{2} + 54965524992 \)

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:  \(-1882508498699710301766760487236393418883072=-\,2^{27}\cdot 3^{45}\cdot 7^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $223.15$
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:  \(1512=2^{3}\cdot 3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{1512}(1,·)$, $\chi_{1512}(131,·)$, $\chi_{1512}(1033,·)$, $\chi_{1512}(635,·)$, $\chi_{1512}(529,·)$, $\chi_{1512}(1235,·)$, $\chi_{1512}(755,·)$, $\chi_{1512}(25,·)$, $\chi_{1512}(731,·)$, $\chi_{1512}(227,·)$, $\chi_{1512}(1009,·)$, $\chi_{1512}(1129,·)$, $\chi_{1512}(1259,·)$, $\chi_{1512}(625,·)$, $\chi_{1512}(1139,·)$, $\chi_{1512}(121,·)$, $\chi_{1512}(505,·)$, $\chi_{1512}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{56} a^{6}$, $\frac{1}{56} a^{7}$, $\frac{1}{112} a^{8}$, $\frac{1}{36176} a^{9} + \frac{9}{2584} a^{7} - \frac{67}{646} a^{5} - \frac{145}{646} a^{3} - \frac{143}{323} a$, $\frac{1}{28723744} a^{10} + \frac{10399}{14361872} a^{8} - \frac{7627}{897617} a^{6} + \frac{37969}{512924} a^{4} + \frac{13979}{128231} a^{2} + \frac{117}{397}$, $\frac{1}{28723744} a^{11} + \frac{11}{2051696} a^{9} + \frac{14521}{1795234} a^{7} + \frac{2644}{128231} a^{5} - \frac{7062}{128231} a^{3} - \frac{24935}{128231} a$, $\frac{1}{402132416} a^{12} + \frac{3821}{14361872} a^{8} + \frac{40597}{7180936} a^{6} - \frac{16665}{512924} a^{4} + \frac{159}{128231} a^{2} - \frac{96}{397}$, $\frac{1}{402132416} a^{13} - \frac{149}{14361872} a^{9} + \frac{353}{53992} a^{7} + \frac{2391}{512924} a^{5} + \frac{31522}{128231} a^{3} + \frac{23778}{128231} a$, $\frac{1}{804264832} a^{14} + \frac{14407}{3590468} a^{8} + \frac{783}{188972} a^{6} - \frac{1315}{13498} a^{4} - \frac{42103}{256462} a^{2} - \frac{35}{397}$, $\frac{1}{804264832} a^{15} + \frac{9}{2051696} a^{9} - \frac{6373}{7180936} a^{7} - \frac{7530}{128231} a^{5} - \frac{30193}{256462} a^{3} + \frac{13706}{128231} a$, $\frac{1}{1608529664} a^{16} - \frac{20355}{14361872} a^{8} + \frac{1525}{256462} a^{6} + \frac{14149}{512924} a^{4} + \frac{23793}{128231} a^{2} + \frac{172}{397}$, $\frac{1}{1608529664} a^{17} - \frac{27}{3590468} a^{9} + \frac{35951}{7180936} a^{7} - \frac{3049}{256462} a^{5} + \frac{30542}{128231} a^{3} - \frac{1099}{7543} a$

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

Class group and class number

$C_{2}\times C_{5642754}$, which has order $11285508$ (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

\(\Q(\sqrt{-42}) \), \(\Q(\zeta_{9})^+\), 6.0.3456649728.1, 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 $18$ R $18$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
7Data not computed