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Properties

Label 18.0.32669612138...8187.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 1657^{12}$
Root discriminant $727.61$
Ramified primes $3, 1657$
Class number $10376730000$ (GRH)
Class group $[2, 2, 1050, 2470650]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![291038813883, 0, 0, 0, 0, 0, 354417390, 0, 0, 0, 0, 0, 109359, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 109359*x^12 + 354417390*x^6 + 291038813883)
 
gp: K = bnfinit(x^18 + 109359*x^12 + 354417390*x^6 + 291038813883, 1)
 

Normalized defining polynomial

\( x^{18} + 109359 x^{12} + 354417390 x^{6} + 291038813883 \)

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:  \(-3266961213891533812087277880596312351679162976508187=-\,3^{27}\cdot 1657^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $727.61$
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:  $3, 1657$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3}$, $\frac{1}{423} a^{7} + \frac{1}{3} a^{5} + \frac{61}{141} a$, $\frac{1}{423} a^{8} + \frac{61}{141} a^{2}$, $\frac{1}{846} a^{9} - \frac{1}{18} a^{6} - \frac{1}{3} a^{4} - \frac{40}{141} a^{3} - \frac{1}{6}$, $\frac{1}{846} a^{10} - \frac{1}{846} a^{7} + \frac{1}{3} a^{5} - \frac{40}{141} a^{4} - \frac{61}{282} a$, $\frac{1}{2538} a^{11} - \frac{1}{846} a^{8} + \frac{101}{423} a^{5} + \frac{1}{3} a^{3} - \frac{61}{282} a^{2}$, $\frac{1}{206088138} a^{12} + \frac{1}{2538} a^{10} - \frac{1}{846} a^{7} - \frac{2516131}{68696046} a^{6} + \frac{1}{3} a^{5} + \frac{101}{423} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{61}{282} a - \frac{94933}{487206}$, $\frac{1}{206088138} a^{13} - \frac{80101}{68696046} a^{7} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} + \frac{6743887}{22898682} a$, $\frac{1}{9686142486} a^{14} + \frac{3005537}{3228714162} a^{8} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} - \frac{217208471}{1076238054} a^{2}$, $\frac{1}{455248696842} a^{15} - \frac{19893145}{151749565614} a^{9} - \frac{1}{18} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{9148352467}{50583188538} a^{3} + \frac{1}{3}$, $\frac{1}{64190066254722} a^{16} - \frac{8988543595}{21396688751574} a^{10} - \frac{1}{1269} a^{8} - \frac{1}{846} a^{7} + \frac{1}{3} a^{5} + \frac{1333638650923}{7132229583858} a^{4} - \frac{202}{423} a^{2} + \frac{40}{141} a - \frac{1}{3}$, $\frac{1}{3016933113971934} a^{17} + \frac{8151525337}{502822185661989} a^{11} - \frac{1}{2538} a^{9} - \frac{1}{18} a^{6} + \frac{99161525283415}{335214790441326} a^{5} - \frac{1}{3} a^{4} - \frac{101}{423} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{1050}\times C_{2470650}$, which has order $10376730000$ (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:  \( \frac{1}{16861062846} a^{15} + \frac{55784}{8430531423} a^{9} + \frac{180360033}{5620354282} a^{3} + \frac{1}{2} \) (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:  \( 1266915384.7466033 \) (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

\(\Q(\sqrt{-3}) \), 3.1.8236947.1 x3, 3.3.222397569.1, 6.0.203541887642427.1, 6.0.148382036091329283.2, 6.0.54042609267.2 x2, 9.3.32999804109981894517713027.1 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.11$x^{6} + 6 x^{4} + 12$$6$$1$$9$$C_6$$[2]_{2}$
1657Data not computed