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Properties

Label 18.18.7815538955...3712.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{18}\cdot 3^{31}\cdot 13^{6}$
Root discriminant $31.19$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-8, -168, -828, -276, 4440, 5976, -5103, -12804, -1350, 9322, 3831, -2898, -1776, 384, 342, -18, -30, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 30*x^16 - 18*x^15 + 342*x^14 + 384*x^13 - 1776*x^12 - 2898*x^11 + 3831*x^10 + 9322*x^9 - 1350*x^8 - 12804*x^7 - 5103*x^6 + 5976*x^5 + 4440*x^4 - 276*x^3 - 828*x^2 - 168*x - 8)
 
gp: K = bnfinit(x^18 - 30*x^16 - 18*x^15 + 342*x^14 + 384*x^13 - 1776*x^12 - 2898*x^11 + 3831*x^10 + 9322*x^9 - 1350*x^8 - 12804*x^7 - 5103*x^6 + 5976*x^5 + 4440*x^4 - 276*x^3 - 828*x^2 - 168*x - 8, 1)
 

Normalized defining polynomial

\( x^{18} - 30 x^{16} - 18 x^{15} + 342 x^{14} + 384 x^{13} - 1776 x^{12} - 2898 x^{11} + 3831 x^{10} + 9322 x^{9} - 1350 x^{8} - 12804 x^{7} - 5103 x^{6} + 5976 x^{5} + 4440 x^{4} - 276 x^{3} - 828 x^{2} - 168 x - 8 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(781553895537094671760883712=2^{18}\cdot 3^{31}\cdot 13^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.19$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} + \frac{1}{3}$, $\frac{1}{6} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a$, $\frac{1}{6} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{12} a^{13} - \frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{6} + \frac{1}{4} a^{5} + \frac{5}{12} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{36} a^{14} + \frac{1}{36} a^{13} + \frac{1}{36} a^{12} + \frac{1}{18} a^{11} + \frac{1}{18} a^{10} - \frac{1}{36} a^{9} - \frac{1}{6} a^{8} - \frac{5}{12} a^{7} - \frac{1}{6} a^{6} - \frac{1}{9} a^{5} - \frac{13}{36} a^{4} - \frac{1}{9} a^{3} - \frac{1}{18} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{36} a^{15} + \frac{1}{36} a^{12} - \frac{1}{12} a^{10} + \frac{1}{36} a^{9} - \frac{1}{4} a^{8} + \frac{1}{4} a^{7} + \frac{1}{18} a^{6} + \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{4}{9} a^{3} - \frac{1}{2} a^{2} + \frac{1}{3} a - \frac{4}{9}$, $\frac{1}{36} a^{16} + \frac{1}{36} a^{13} - \frac{1}{12} a^{11} + \frac{1}{36} a^{10} - \frac{1}{12} a^{9} - \frac{1}{4} a^{8} + \frac{1}{18} a^{7} + \frac{1}{4} a^{6} - \frac{1}{4} a^{5} + \frac{1}{18} a^{4} - \frac{1}{6} a^{2} - \frac{4}{9} a + \frac{1}{3}$, $\frac{1}{277308} a^{17} - \frac{385}{277308} a^{16} + \frac{919}{138654} a^{15} + \frac{257}{69327} a^{14} + \frac{853}{46218} a^{13} + \frac{1444}{23109} a^{12} - \frac{1472}{23109} a^{11} + \frac{9551}{138654} a^{10} - \frac{17183}{277308} a^{9} - \frac{46021}{277308} a^{8} + \frac{65393}{138654} a^{7} - \frac{23909}{69327} a^{6} - \frac{82613}{277308} a^{5} - \frac{5605}{92436} a^{4} + \frac{4279}{15406} a^{3} + \frac{6147}{15406} a^{2} + \frac{22622}{69327} a - \frac{28231}{69327}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 56996133.1215 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), \(\Q(\zeta_{9})^+\), 6.6.212891328.1, 6.6.212891328.2, 6.6.23654592.1, \(\Q(\zeta_{36})^+\)

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$13$13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$