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Properties

Label 18.0.92546510514...0000.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 5^{12}$
Root discriminant $35.78$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![51200, 0, -38400, 0, -13920, 0, -256, 0, 5274, 0, 3321, 0, -996, 0, 54, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200)
 
gp: K = bnfinit(x^18 - 6*x^16 + 54*x^14 - 996*x^12 + 3321*x^10 + 5274*x^8 - 256*x^6 - 13920*x^4 - 38400*x^2 + 51200, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{16} + 54 x^{14} - 996 x^{12} + 3321 x^{10} + 5274 x^{8} - 256 x^{6} - 13920 x^{4} - 38400 x^{2} + 51200 \)

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:  \(-9254651051409408000000000000=-\,2^{27}\cdot 3^{24}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.78$
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 over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{11} - \frac{1}{16} a^{9} + \frac{1}{16} a^{7} + \frac{1}{16} a^{5} - \frac{1}{8} a^{3}$, $\frac{1}{112} a^{12} - \frac{1}{16} a^{10} - \frac{1}{112} a^{8} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{3}{28} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{1}{7}$, $\frac{1}{224} a^{13} - \frac{1}{224} a^{12} - \frac{1}{32} a^{11} - \frac{1}{32} a^{10} - \frac{1}{224} a^{9} - \frac{13}{224} a^{8} - \frac{3}{32} a^{7} + \frac{1}{32} a^{6} - \frac{5}{28} a^{5} + \frac{27}{112} a^{4} - \frac{1}{8} a^{3} - \frac{1}{4} a^{2} - \frac{1}{14} a - \frac{3}{7}$, $\frac{1}{896} a^{14} + \frac{1}{448} a^{12} - \frac{1}{32} a^{11} + \frac{17}{448} a^{10} + \frac{1}{32} a^{9} - \frac{1}{56} a^{8} - \frac{1}{32} a^{7} + \frac{37}{896} a^{6} + \frac{7}{32} a^{5} - \frac{33}{448} a^{4} + \frac{5}{16} a^{3} + \frac{33}{112} a^{2} + \frac{3}{14}$, $\frac{1}{8960} a^{15} + \frac{1}{640} a^{13} - \frac{1}{224} a^{12} - \frac{53}{4480} a^{11} + \frac{1}{32} a^{10} + \frac{13}{320} a^{9} + \frac{1}{224} a^{8} - \frac{719}{8960} a^{7} + \frac{3}{32} a^{6} + \frac{61}{640} a^{5} - \frac{1}{14} a^{4} - \frac{107}{1120} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a - \frac{3}{7}$, $\frac{1}{3381903042560} a^{16} + \frac{129664951}{241564503040} a^{14} - \frac{287145611}{73519631360} a^{12} + \frac{5462833333}{120782251520} a^{10} + \frac{122025720281}{3381903042560} a^{8} - \frac{1}{8} a^{7} - \frac{325688303}{10502804480} a^{6} + \frac{3303534029}{211368940160} a^{4} - \frac{1}{8} a^{3} - \frac{1163019049}{3019556288} a^{2} - \frac{1}{4} a + \frac{826731057}{2642111752}$, $\frac{1}{13527612170240} a^{17} - \frac{32096993}{966258012160} a^{15} - \frac{63993375}{58815705088} a^{13} - \frac{1722091159}{96625801216} a^{11} - \frac{1}{16} a^{10} - \frac{732508709223}{13527612170240} a^{9} - \frac{1}{16} a^{8} - \frac{3802397911}{42011217920} a^{7} - \frac{1}{16} a^{6} - \frac{6907056141}{169095152128} a^{5} + \frac{1}{16} a^{4} + \frac{944802017}{8627303680} a^{3} - \frac{3}{8} a^{2} - \frac{3513881107}{10568447008} a - \frac{1}{2}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( 366199805.276 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-2}) \), 3.1.16200.1 x3, 3.1.648.1 x3, 3.1.200.1 x3, 3.1.16200.2 x3, 6.0.2099520000.2, 6.0.3359232.4, 6.0.320000.1, 6.0.2099520000.1, 9.1.34012224000000.6 x9

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 9 sibling: data not computed

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 ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$