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Properties

Label 18.0.12565821915...0192.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 1123^{12}$
Root discriminant $891.16$
Ramified primes $2, 3, 1123$
Class number $79767385425$ (GRH)
Class group $[301, 4515, 58695]$ (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![1728, 0, 0, 0, 0, 0, 14701076640, 0, 0, 0, 0, 0, 242604, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 242604*x^12 + 14701076640*x^6 + 1728)
 
gp: K = bnfinit(x^18 + 242604*x^12 + 14701076640*x^6 + 1728, 1)
 

Normalized defining polynomial

\( x^{18} + 242604 x^{12} + 14701076640 x^{6} + 1728 \)

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:  \(-125658219152626752999993187563351622531489799076360192=-\,2^{12}\cdot 3^{27}\cdot 1123^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $891.16$
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:  $2, 3, 1123$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{36} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3}$, $\frac{1}{36} a^{7} + \frac{1}{6} a^{5} + \frac{1}{3} a$, $\frac{1}{36} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{144} a^{9} - \frac{1}{72} a^{6} + \frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{6}$, $\frac{1}{144} a^{10} - \frac{1}{72} a^{7} + \frac{1}{6} a^{5} - \frac{1}{6} a^{4} - \frac{1}{6} a$, $\frac{1}{432} a^{11} - \frac{1}{72} a^{8} - \frac{2}{9} a^{5} - \frac{1}{6} a^{3} - \frac{1}{6} a^{2}$, $\frac{1}{5264276832} a^{12} - \frac{1}{432} a^{10} - \frac{1}{72} a^{7} - \frac{5203723}{438689736} a^{6} + \frac{1}{6} a^{5} + \frac{1}{18} a^{4} - \frac{1}{4} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{10427675}{36557478}$, $\frac{1}{5264276832} a^{13} - \frac{5203723}{438689736} a^{7} - \frac{1}{6} a^{5} - \frac{1}{4} a^{4} + \frac{1}{6} a^{3} - \frac{10427675}{36557478} a$, $\frac{1}{5264276832} a^{14} - \frac{5203723}{438689736} a^{8} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} - \frac{10427675}{36557478} a^{2}$, $\frac{1}{10528553664} a^{15} + \frac{444595}{438689736} a^{9} - \frac{1}{6} a^{5} + \frac{13943977}{73114956} a^{3} - \frac{1}{2}$, $\frac{1}{31585660992} a^{16} + \frac{6982103}{2632138416} a^{10} - \frac{1}{108} a^{8} - \frac{1}{72} a^{7} + \frac{1}{6} a^{5} + \frac{38315629}{219344868} a^{4} - \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{31585660992} a^{17} + \frac{444595}{1316069208} a^{11} - \frac{1}{432} a^{9} - \frac{1}{72} a^{6} - \frac{22613501}{219344868} a^{5} + \frac{1}{6} a^{4} - \frac{1}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{1}{6}$

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

Class group and class number

$C_{301}\times C_{4515}\times C_{58695}$, which has order $79767385425$ (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:  \( -\frac{1}{5820096} a^{15} - \frac{15163}{363756} a^{9} - \frac{306282543}{121252} a^{3} + \frac{1}{2} \) (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:  \( 1065094301.6985265 \) (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

\(\Q(\sqrt{-3}) \), 3.1.15133548.1 x3, 3.3.102151449.1, 6.0.687072825204912.1, 6.0.31304755598398803.2, 6.0.397164833712.2 x2, Deg 9 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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
$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}$
1123Data not computed