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Properties

Label 18.0.50396986113...488.12
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 19^{12}$
Root discriminant $84.71$
Ramified primes $2, 3, 19$
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![1259712, 0, 0, 1073088, 0, 0, 265824, 0, 0, -33120, 0, 0, 1752, 0, 0, -60, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 60*x^15 + 1752*x^12 - 33120*x^9 + 265824*x^6 + 1073088*x^3 + 1259712)
 
gp: K = bnfinit(x^18 - 60*x^15 + 1752*x^12 - 33120*x^9 + 265824*x^6 + 1073088*x^3 + 1259712, 1)
 

Normalized defining polynomial

\( x^{18} - 60 x^{15} + 1752 x^{12} - 33120 x^{9} + 265824 x^{6} + 1073088 x^{3} + 1259712 \)

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:  \(-50396986113842071567939653456703488=-\,2^{12}\cdot 3^{33}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $84.71$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{2} a^{4}$, $\frac{1}{2} a^{5}$, $\frac{1}{12} a^{6}$, $\frac{1}{12} a^{7}$, $\frac{1}{12} a^{8}$, $\frac{1}{144} a^{9} - \frac{1}{6} a^{3} - \frac{1}{2}$, $\frac{1}{144} a^{10} - \frac{1}{6} a^{4} - \frac{1}{2} a$, $\frac{1}{144} a^{11} - \frac{1}{6} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{9792} a^{12} - \frac{11}{4896} a^{9} + \frac{1}{136} a^{6} + \frac{97}{408} a^{3} + \frac{21}{68}$, $\frac{1}{29376} a^{13} - \frac{5}{1632} a^{10} + \frac{37}{1224} a^{7} - \frac{27}{136} a^{4} + \frac{55}{204} a$, $\frac{1}{88128} a^{14} - \frac{49}{14688} a^{11} + \frac{37}{3672} a^{8} - \frac{13}{1224} a^{5} - \frac{251}{612} a^{2}$, $\frac{1}{2138866560} a^{15} - \frac{2683}{356477760} a^{12} - \frac{109273}{44559720} a^{9} + \frac{1165751}{29706480} a^{6} - \frac{3501593}{14853240} a^{3} + \frac{24127}{137530}$, $\frac{1}{6416599680} a^{16} - \frac{2683}{1069433280} a^{13} - \frac{109273}{133679160} a^{10} + \frac{1165751}{89119440} a^{7} - \frac{10928213}{44559720} a^{4} - \frac{37801}{137530} a$, $\frac{1}{19249799040} a^{17} - \frac{2683}{3208299840} a^{14} - \frac{109273}{401037480} a^{11} - \frac{6260869}{267358320} a^{8} - \frac{33208073}{133679160} a^{5} - \frac{37801}{412590} a^{2}$

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:  \( -\frac{143}{62907840} a^{15} + \frac{2833}{20969280} a^{12} - \frac{42863}{10484640} a^{9} + \frac{8614}{109215} a^{6} - \frac{592027}{873720} a^{3} - \frac{11239}{16180} \) (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:  \( 1228939086.8309562 \) (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

\(\Q(\sqrt{-3}) \), 3.1.1083.1 x3, 6.0.3518667.2, 9.1.129610938470796864.28 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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$$[\ ]_{3}^{6}$
$3$3.6.11.8$x^{6} + 12$$6$$1$$11$$S_3$$[5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.18$x^{6} + 21 x^{3} + 12$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$19$19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19.9.6.2$x^{9} + 228 x^{6} + 16967 x^{3} + 438976$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$