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Properties

Label 18.18.7878167217...0000.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{12}\cdot 3^{44}\cdot 5^{9}$
Root discriminant $52.06$
Ramified primes $2, 3, 5$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_9\times S_3$ (as 18T16)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1216, -10368, 20736, 17280, -69120, -7776, 88704, 1296, -57024, -72, 20592, 0, -4368, 0, 540, 0, -36, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 72*x^9 - 57024*x^8 + 1296*x^7 + 88704*x^6 - 7776*x^5 - 69120*x^4 + 17280*x^3 + 20736*x^2 - 10368*x + 1216)
 
gp: K = bnfinit(x^18 - 36*x^16 + 540*x^14 - 4368*x^12 + 20592*x^10 - 72*x^9 - 57024*x^8 + 1296*x^7 + 88704*x^6 - 7776*x^5 - 69120*x^4 + 17280*x^3 + 20736*x^2 - 10368*x + 1216, 1)
 

Normalized defining polynomial

\( x^{18} - 36 x^{16} + 540 x^{14} - 4368 x^{12} + 20592 x^{10} - 72 x^{9} - 57024 x^{8} + 1296 x^{7} + 88704 x^{6} - 7776 x^{5} - 69120 x^{4} + 17280 x^{3} + 20736 x^{2} - 10368 x + 1216 \)

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:  \(7878167217468889863048000000000=2^{12}\cdot 3^{44}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $52.06$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{4} a^{6}$, $\frac{1}{4} a^{7}$, $\frac{1}{4} a^{8}$, $\frac{1}{8} a^{9}$, $\frac{1}{8} a^{10}$, $\frac{1}{8} a^{11}$, $\frac{1}{16} a^{12}$, $\frac{1}{16} a^{13}$, $\frac{1}{16} a^{14}$, $\frac{1}{32} a^{15}$, $\frac{1}{32} a^{16}$, $\frac{1}{32} a^{17}$

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:  \( 5504906840.3 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9\times S_3$ (as 18T16):

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 $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1

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

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 R $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{9}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
5Data not computed