↑

Properties

Label 18.0.73311456037...4827.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 19^{12}\cdot 433^{12}$
Root discriminant $2117.59$
Ramified primes $3, 19, 433$
Class number $45935634276$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 21, 63, 126, 378]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![23917744283328, 0, 0, 0, 0, 0, 13431802185, 0, 0, 0, 0, 0, 530694, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 530694*x^12 + 13431802185*x^6 + 23917744283328)
 
gp: K = bnfinit(x^18 + 530694*x^12 + 13431802185*x^6 + 23917744283328, 1)
 

Normalized defining polynomial

\( x^{18} + 530694 x^{12} + 13431802185 x^{6} + 23917744283328 \)

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:  \(-733114560372703860308311985400170653539593347139329480594827=-\,3^{27}\cdot 19^{12}\cdot 433^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $2117.59$
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:  $3, 19, 433$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{18} a^{6} - \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{882} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2} - \frac{143}{294} a$, $\frac{1}{1764} a^{8} - \frac{143}{588} a^{2}$, $\frac{1}{7056} a^{9} + \frac{1033}{2352} a^{3} - \frac{1}{2}$, $\frac{1}{14112} a^{10} - \frac{1}{1764} a^{7} - \frac{1}{6} a^{5} - \frac{143}{4704} a^{4} - \frac{1}{2} a^{2} - \frac{151}{588} a$, $\frac{1}{84672} a^{11} - \frac{1}{3528} a^{8} + \frac{6913}{28224} a^{5} + \frac{1}{3} a^{3} - \frac{445}{1176} a^{2}$, $\frac{1}{73815525504} a^{12} + \frac{1}{42336} a^{10} - \frac{1}{1764} a^{7} + \frac{99805409}{24605175168} a^{6} - \frac{1}{6} a^{5} + \frac{2209}{14112} a^{4} - \frac{1}{6} a^{2} + \frac{143}{588} a - \frac{705749}{2615346}$, $\frac{1}{73815525504} a^{13} - \frac{240463}{502146432} a^{7} - \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{41555957}{128151954} a$, $\frac{1}{3616960749696} a^{14} + \frac{211393505}{1205653583232} a^{8} + \frac{1}{6} a^{4} - \frac{2462494571}{6279445746} a^{2} - \frac{1}{2} a$, $\frac{1}{354462153470208} a^{15} - \frac{1}{147631051008} a^{12} + \frac{1}{42336} a^{10} - \frac{813822127}{118154051156736} a^{9} - \frac{1}{1764} a^{7} - \frac{99805409}{49210350336} a^{6} + \frac{1}{6} a^{5} + \frac{2209}{14112} a^{4} + \frac{978906422803}{2461542732432} a^{3} - \frac{1}{6} a^{2} + \frac{143}{588} a + \frac{705749}{5230692}$, $\frac{1}{52105936560120576} a^{16} - \frac{1}{147631051008} a^{13} - \frac{402698349871}{17368645520040192} a^{10} - \frac{1}{5292} a^{8} + \frac{240463}{1004292864} a^{7} - \frac{1}{6} a^{5} - \frac{7736964272645}{361846781667504} a^{4} - \frac{1}{3} a^{3} - \frac{739}{1764} a^{2} + \frac{41555957}{256303908} a + \frac{1}{3}$, $\frac{1}{2553190891445908224} a^{17} - \frac{1}{7233921499392} a^{14} + \frac{3905001431885}{851063630481969408} a^{11} - \frac{1}{21168} a^{9} + \frac{472083583}{2411307166464} a^{8} - \frac{7783730539305941}{70921969206830784} a^{5} + \frac{1}{6} a^{4} - \frac{3385}{7056} a^{3} - \frac{10688190587}{25117782984} a^{2} - \frac{1}{6} a - \frac{1}{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{21}\times C_{63}\times C_{126}\times C_{378}$, which has order $45935634276$ (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:  \( \frac{1}{820514244144} a^{15} + \frac{270149}{410257122072} a^{9} + \frac{5741797263}{273504748048} a^{3} + \frac{1}{2} \) (order $6$)
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:  \( 81466624347417.27 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.203050587.2 x3, 3.3.5482365849.1, 6.0.123688622643133707.1, 6.0.1332214901307.1 x2, 6.0.90169005906844472403.1, 9.3.494339478621963410856630165147.1 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
433Data not computed