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Properties

Label 18.0.49364918521...3123.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{25}\cdot 17^{12}$
Root discriminant $30.41$
Ramified primes $3, 17$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![27, 0, 0, -153, 0, 0, 588, 0, 0, -561, 0, 0, 196, 0, 0, -17, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 17*x^15 + 196*x^12 - 561*x^9 + 588*x^6 - 153*x^3 + 27)
 
gp: K = bnfinit(x^18 - 17*x^15 + 196*x^12 - 561*x^9 + 588*x^6 - 153*x^3 + 27, 1)
 

Normalized defining polynomial

\( x^{18} - 17 x^{15} + 196 x^{12} - 561 x^{9} + 588 x^{6} - 153 x^{3} + 27 \)

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:  \(-493649185212973862185233123=-\,3^{25}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.41$
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:  $3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{6} a^{8} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{6} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{4}$, $\frac{1}{6} a^{11} + \frac{1}{6} a^{7} + \frac{1}{6} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36} a^{12} + \frac{1}{18} a^{9} - \frac{1}{4} a^{6} + \frac{1}{6} a^{3} + \frac{1}{4}$, $\frac{1}{108} a^{13} - \frac{1}{108} a^{12} - \frac{1}{27} a^{10} + \frac{1}{27} a^{9} + \frac{5}{36} a^{7} - \frac{5}{36} a^{6} + \frac{7}{18} a^{4} - \frac{7}{18} a^{3} - \frac{5}{12} a + \frac{5}{12}$, $\frac{1}{108} a^{14} - \frac{1}{108} a^{12} - \frac{1}{27} a^{11} + \frac{1}{27} a^{9} - \frac{1}{36} a^{8} - \frac{1}{6} a^{7} + \frac{7}{36} a^{6} - \frac{1}{9} a^{5} - \frac{1}{2} a^{4} - \frac{7}{18} a^{3} + \frac{1}{12} a^{2} - \frac{1}{2} a + \frac{5}{12}$, $\frac{1}{35316} a^{15} - \frac{71}{8829} a^{12} + \frac{2449}{35316} a^{9} - \frac{194}{981} a^{6} + \frac{799}{11772} a^{3} - \frac{124}{981}$, $\frac{1}{35316} a^{16} + \frac{43}{35316} a^{13} - \frac{1}{108} a^{12} + \frac{1141}{35316} a^{10} + \frac{1}{27} a^{9} - \frac{77}{1308} a^{7} - \frac{5}{36} a^{6} + \frac{5377}{11772} a^{4} - \frac{7}{18} a^{3} + \frac{1793}{3924} a + \frac{5}{12}$, $\frac{1}{105948} a^{17} - \frac{71}{26487} a^{14} + \frac{8335}{105948} a^{11} + \frac{133}{2943} a^{8} + \frac{12571}{35316} a^{5} + \frac{733}{5886} a^{2}$

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:  $8$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( \frac{70}{8829} a^{15} - \frac{1241}{8829} a^{12} + \frac{14470}{8829} a^{9} - \frac{5270}{981} a^{6} + \frac{16690}{2943} a^{3} - \frac{385}{981} \) (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:  \( 8709025.293307293 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.2255067.2, 9.3.4275897935643.1

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

Sibling fields

Degree 9 siblings: data not computed
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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.9$x^{6} + 3$$6$$1$$11$$S_3$$[5/2]_{2}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$