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Properties

Label 18.0.13266257117...2288.4
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 17^{12}$
Root discriminant $78.65$
Ramified primes $2, 3, 17$
Class number $81$ (GRH)
Class group $[3, 3, 9]$ (GRH)
Galois group $He_3:C_2$ (as 18T21)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![34012224, 0, 0, 2099520, 0, 0, 409536, 0, 0, -28944, 0, 0, 1944, 0, 0, -48, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 48*x^15 + 1944*x^12 - 28944*x^9 + 409536*x^6 + 2099520*x^3 + 34012224)
 
gp: K = bnfinit(x^18 - 48*x^15 + 1944*x^12 - 28944*x^9 + 409536*x^6 + 2099520*x^3 + 34012224, 1)
 

Normalized defining polynomial

\( x^{18} - 48 x^{15} + 1944 x^{12} - 28944 x^{9} + 409536 x^{6} + 2099520 x^{3} + 34012224 \)

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:  \(-13266257117930788904129800273932288=-\,2^{12}\cdot 3^{33}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.65$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3}$, $\frac{1}{6} a^{4}$, $\frac{1}{18} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{36} a^{6}$, $\frac{1}{108} a^{7} - \frac{1}{18} a^{4} + \frac{1}{3} a$, $\frac{1}{108} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{1296} a^{9} + \frac{1}{108} a^{6} - \frac{1}{18} a^{3} - \frac{1}{2}$, $\frac{1}{1296} a^{10} + \frac{1}{6} a$, $\frac{1}{3888} a^{11} + \frac{1}{18} a^{2}$, $\frac{1}{69984} a^{12} + \frac{1}{324} a^{3}$, $\frac{1}{69984} a^{13} + \frac{1}{324} a^{4}$, $\frac{1}{209952} a^{14} + \frac{1}{972} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{37371456} a^{15} - \frac{29}{6228576} a^{12} + \frac{11}{28836} a^{9} + \frac{361}{173016} a^{6} + \frac{745}{28836} a^{3} + \frac{7}{89}$, $\frac{1}{112114368} a^{16} - \frac{29}{18685728} a^{13} - \frac{5}{38448} a^{10} - \frac{1241}{519048} a^{7} + \frac{7153}{86508} a^{4} - \frac{25}{178} a$, $\frac{1}{1009029312} a^{17} - \frac{37}{21021444} a^{14} - \frac{5}{346032} a^{11} + \frac{8371}{4671432} a^{8} + \frac{2923}{194643} a^{5} + \frac{281}{4806} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}$, which has order $81$ (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:  \( \frac{5}{37371456} a^{15} - \frac{7}{778572} a^{12} + \frac{7}{19224} a^{9} - \frac{1399}{173016} a^{6} + \frac{553}{7209} a^{3} + \frac{35}{89} \) (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:  \( 898516250.6729627 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 18T21):

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 $He_3:C_2$
Character table for $He_3:C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.2255067.2, 9.1.66498764695119936.21 x3

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.14$x^{6} + 12 x^{3} + 18 x^{2} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.7$x^{6} + 21$$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}$