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Properties

Label 18.18.1379190349...0625.1
Degree $18$
Signature $[18, 0]$
Discriminant $3^{32}\cdot 5^{15}\cdot 29^{3}$
Root discriminant $47.25$
Ramified primes $3, 5, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3\wr C_2$ (as 18T93)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![125, 0, -2250, -1875, 12375, 19125, -13725, -38475, -5400, 25925, 10800, -7200, -4275, 810, 675, -30, -45, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 45*x^16 - 30*x^15 + 675*x^14 + 810*x^13 - 4275*x^12 - 7200*x^11 + 10800*x^10 + 25925*x^9 - 5400*x^8 - 38475*x^7 - 13725*x^6 + 19125*x^5 + 12375*x^4 - 1875*x^3 - 2250*x^2 + 125)
 
gp: K = bnfinit(x^18 - 45*x^16 - 30*x^15 + 675*x^14 + 810*x^13 - 4275*x^12 - 7200*x^11 + 10800*x^10 + 25925*x^9 - 5400*x^8 - 38475*x^7 - 13725*x^6 + 19125*x^5 + 12375*x^4 - 1875*x^3 - 2250*x^2 + 125, 1)
 

Normalized defining polynomial

\( x^{18} - 45 x^{16} - 30 x^{15} + 675 x^{14} + 810 x^{13} - 4275 x^{12} - 7200 x^{11} + 10800 x^{10} + 25925 x^{9} - 5400 x^{8} - 38475 x^{7} - 13725 x^{6} + 19125 x^{5} + 12375 x^{4} - 1875 x^{3} - 2250 x^{2} + 125 \)

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:  \(1379190349911729435699462890625=3^{32}\cdot 5^{15}\cdot 29^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $47.25$
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:  $3, 5, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{5} a^{8}$, $\frac{1}{5} a^{9}$, $\frac{1}{15} a^{10} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{15} a^{11} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{15} a^{7} + \frac{1}{15} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{75} a^{12} - \frac{1}{15} a^{9} + \frac{1}{15} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{75} a^{13} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} - \frac{1}{15} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{75} a^{14} + \frac{1}{15} a^{9} - \frac{1}{15} a^{7} + \frac{1}{15} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{75} a^{15} - \frac{1}{15} a^{9} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{75} a^{16} + \frac{1}{15} a^{9} - \frac{1}{15} a^{8} + \frac{1}{15} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{8953079925} a^{17} + \frac{1760369}{8953079925} a^{16} + \frac{2359699}{358123197} a^{15} - \frac{8088563}{1790615985} a^{14} + \frac{3378709}{1790615985} a^{13} - \frac{35483162}{8953079925} a^{12} + \frac{6297298}{1790615985} a^{11} - \frac{5335738}{596871995} a^{10} - \frac{12555686}{596871995} a^{9} - \frac{32725426}{596871995} a^{8} + \frac{77938169}{1790615985} a^{7} - \frac{163888408}{1790615985} a^{6} - \frac{57401526}{119374399} a^{5} - \frac{73843952}{358123197} a^{4} + \frac{49284419}{119374399} a^{3} - \frac{42940235}{119374399} a^{2} - \frac{11091905}{358123197} a - \frac{44463904}{119374399}$

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:  \( 1736468069.05 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3\wr C_2$ (as 18T93):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 27 conjugacy class representatives for $C_3\times S_3\wr C_2$
Character table for $C_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\zeta_{9})^+\), 6.6.820125.1, 6.6.594590625.1

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.6.0.1}{6} }$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
5Data not computed
$29$29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.3.0.1$x^{3} - x + 3$$1$$3$$0$$C_3$$[\ ]^{3}$
29.6.3.2$x^{6} - 841 x^{2} + 73167$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
29.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$