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Properties

Label 18.0.28654508057...6032.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{12}\cdot 67^{12}$
Root discriminant $635.59$
Ramified primes $2, 3, 7, 67$
Class number $2176782336$ (GRH)
Class group $[2, 6, 6, 6, 36, 36, 36, 108]$ (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![421875000000, 0, 0, 0, 0, 0, 455035104, 0, 0, 0, 0, 0, 123804, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 123804*x^12 + 455035104*x^6 + 421875000000)
 
gp: K = bnfinit(x^18 + 123804*x^12 + 455035104*x^6 + 421875000000, 1)
 

Normalized defining polynomial

\( x^{18} + 123804 x^{12} + 455035104 x^{6} + 421875000000 \)

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:  \(-286545080573078051080746349592284832173804522156032=-\,2^{12}\cdot 3^{31}\cdot 7^{12}\cdot 67^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $635.59$
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:  $2, 3, 7, 67$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{36} a^{6} + \frac{1}{6} a^{4} + \frac{1}{3}$, $\frac{1}{900} a^{7} + \frac{1}{6} a^{5} - \frac{29}{75} a$, $\frac{1}{900} a^{8} - \frac{29}{75} a^{2}$, $\frac{1}{3600} a^{9} - \frac{1}{72} a^{6} + \frac{1}{6} a^{4} + \frac{23}{150} a^{3} - \frac{1}{6}$, $\frac{1}{3600} a^{10} - \frac{1}{1800} a^{7} + \frac{1}{6} a^{5} + \frac{23}{150} a^{4} + \frac{29}{150} a$, $\frac{1}{3600} a^{11} - \frac{1}{1800} a^{8} + \frac{23}{150} a^{5} + \frac{29}{150} a^{2}$, $\frac{1}{496562400} a^{12} - \frac{1}{10800} a^{10} - \frac{1}{1800} a^{7} + \frac{391121}{41380200} a^{6} + \frac{1}{6} a^{5} + \frac{26}{225} a^{4} - \frac{1}{4} a^{3} + \frac{1}{3} a^{2} + \frac{29}{150} a + \frac{29755}{137934}$, $\frac{1}{496562400} a^{13} - \frac{1}{10800} a^{11} - \frac{1}{1800} a^{8} - \frac{22681}{41380200} a^{7} + \frac{26}{225} a^{5} - \frac{1}{4} a^{4} - \frac{1}{6} a^{3} + \frac{29}{150} a^{2} - \frac{1049267}{3448350} a$, $\frac{1}{12414060000} a^{14} - \frac{436483}{1034505000} a^{8} - \frac{1}{4} a^{5} + \frac{1}{6} a^{4} + \frac{38537791}{86208750} a^{2}$, $\frac{1}{620703000000} a^{15} - \frac{1585933}{51725250000} a^{9} - \frac{1}{72} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} + \frac{330498091}{4310437500} a^{3} + \frac{1}{3}$, $\frac{1}{15517575000000} a^{16} + \frac{78231721}{646565625000} a^{10} - \frac{16106636909}{107760937500} a^{4} - \frac{1}{2} a$, $\frac{1}{1163818125000000} a^{17} - \frac{5949989683}{96984843750000} a^{11} - \frac{1}{10800} a^{9} - \frac{1}{1800} a^{8} - \frac{1}{72} a^{6} - \frac{1105210511909}{8082070312500} a^{5} + \frac{1}{6} a^{4} - \frac{23}{450} a^{3} - \frac{23}{75} a^{2} + \frac{1}{3} a - \frac{1}{6}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{6}\times C_{6}\times C_{36}\times C_{36}\times C_{36}\times C_{108}$, which has order $2176782336$ (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:  \( -\frac{1}{22989000000} a^{15} - \frac{3947}{718406250} a^{9} - \frac{14461773}{478937500} a^{3} + \frac{1}{2} \) (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:  \( 1305258438.9061162 \) (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

\(\Q(\sqrt{-3}) \), 3.1.23755788.2 x3, 3.3.17816841.1, 6.0.1693012390502832.3, 6.0.952319469657843.1, 6.0.69271877808.3 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$67$67.9.6.1$x^{9} + 3216 x^{6} + 3443063 x^{3} + 1231925248$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
67.9.6.1$x^{9} + 3216 x^{6} + 3443063 x^{3} + 1231925248$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$