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Properties

Label 18.0.32517009212...1552.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{27}\cdot 43^{15}$
Root discriminant $337.64$
Ramified primes $2, 3, 43$
Class number $115788512$ (GRH)
Class group $[2, 2, 146, 198268]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4501933129728, 0, 3376449847296, 0, 914455166976, 0, 119555334144, 0, 8434990080, 0, 336547584, 0, 7580040, 0, 91332, 0, 516, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 516*x^16 + 91332*x^14 + 7580040*x^12 + 336547584*x^10 + 8434990080*x^8 + 119555334144*x^6 + 914455166976*x^4 + 3376449847296*x^2 + 4501933129728)
 
gp: K = bnfinit(x^18 + 516*x^16 + 91332*x^14 + 7580040*x^12 + 336547584*x^10 + 8434990080*x^8 + 119555334144*x^6 + 914455166976*x^4 + 3376449847296*x^2 + 4501933129728, 1)
 

Normalized defining polynomial

\( x^{18} + 516 x^{16} + 91332 x^{14} + 7580040 x^{12} + 336547584 x^{10} + 8434990080 x^{8} + 119555334144 x^{6} + 914455166976 x^{4} + 3376449847296 x^{2} + 4501933129728 \)

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:  \(-3251700921275639876819745944740786500431511552=-\,2^{27}\cdot 3^{27}\cdot 43^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $337.64$
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:  $2, 3, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(3096=2^{3}\cdot 3^{2}\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{3096}(1,·)$, $\chi_{3096}(515,·)$, $\chi_{3096}(2113,·)$, $\chi_{3096}(1033,·)$, $\chi_{3096}(1547,·)$, $\chi_{3096}(2579,·)$, $\chi_{3096}(337,·)$, $\chi_{3096}(467,·)$, $\chi_{3096}(1369,·)$, $\chi_{3096}(1499,·)$, $\chi_{3096}(2401,·)$, $\chi_{3096}(2531,·)$, $\chi_{3096}(2243,·)$, $\chi_{3096}(2065,·)$, $\chi_{3096}(49,·)$, $\chi_{3096}(179,·)$, $\chi_{3096}(1081,·)$, $\chi_{3096}(1211,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{1032} a^{6}$, $\frac{1}{2064} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8256} a^{8} - \frac{1}{2064} a^{6} + \frac{1}{16} a^{4} + \frac{1}{8} a^{2}$, $\frac{1}{16512} a^{9} - \frac{1}{4128} a^{7} + \frac{1}{32} a^{5} - \frac{3}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{66048} a^{10} - \frac{1}{16512} a^{8} - \frac{7}{16512} a^{6} - \frac{3}{64} a^{4} - \frac{1}{8} a^{2}$, $\frac{1}{132096} a^{11} - \frac{1}{33024} a^{9} - \frac{7}{33024} a^{7} + \frac{13}{128} a^{5} - \frac{1}{16} a^{3} - \frac{1}{2} a$, $\frac{1}{1090584576} a^{12} - \frac{1}{704512} a^{10} - \frac{47}{2113536} a^{8} - \frac{439}{1056768} a^{6} - \frac{61}{512} a^{4} + \frac{21}{128} a^{2} - \frac{7}{16}$, $\frac{1}{2181169152} a^{13} - \frac{1}{1409024} a^{11} - \frac{47}{4227072} a^{9} - \frac{439}{2113536} a^{7} + \frac{67}{1024} a^{5} + \frac{21}{256} a^{3} - \frac{7}{32} a$, $\frac{1}{61072736256} a^{14} + \frac{5}{15268184064} a^{12} - \frac{19}{5636096} a^{10} + \frac{89}{59179008} a^{8} + \frac{725}{3698688} a^{6} + \frac{27}{1024} a^{4} + \frac{141}{896} a^{2} - \frac{11}{28}$, $\frac{1}{122145472512} a^{15} + \frac{5}{30536368128} a^{13} - \frac{19}{11272192} a^{11} + \frac{89}{118358016} a^{9} + \frac{725}{7397376} a^{7} + \frac{27}{2048} a^{5} + \frac{141}{1792} a^{3} - \frac{11}{56} a$, $\frac{1}{86435510750281728} a^{16} + \frac{147641}{21608877687570432} a^{14} - \frac{6999103}{21608877687570432} a^{12} - \frac{139488575}{83755339874304} a^{10} + \frac{34244643}{872451457024} a^{8} + \frac{6765223}{30434353152} a^{6} + \frac{30634423}{1268098048} a^{4} + \frac{15069401}{79256128} a^{2} + \frac{2118825}{4953508}$, $\frac{1}{172871021500563456} a^{17} + \frac{147641}{43217755375140864} a^{15} - \frac{6999103}{43217755375140864} a^{13} - \frac{139488575}{167510679748608} a^{11} + \frac{34244643}{1744902914048} a^{9} + \frac{6765223}{60868706304} a^{7} - \frac{286390089}{2536196096} a^{5} + \frac{15069401}{158512256} a^{3} + \frac{2118825}{9907016} a$

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

Class group and class number

$C_{2}\times C_{2}\times C_{146}\times C_{198268}$, which has order $115788512$ (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:  \( 10847494.59839338 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-258}) \), \(\Q(\zeta_{9})^+\), 3.3.149769.1, 3.3.149769.2, 3.3.1849.1, 6.0.801247375872.6, 6.0.1481506397987328.2, 6.0.1481506397987328.1, 6.0.2032244716032.2, 9.9.3359431500123609.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.5$x^{6} - 4 x^{4} + 4 x^{2} + 8$$2$$3$$9$$C_6$$[3]^{3}$
3Data not computed
43Data not computed