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Properties

Label 18.0.39531097362...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 5^{6}$
Root discriminant $18.00$
Ramified primes $2, 3, 5$
Class number $1$
Class group Trivial
Galois group $C_3^3:C_2^2$ (as 18T53)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -27, 117, -315, 567, -675, 444, 63, -486, 495, -171, -99, 132, -72, 48, -33, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 33*x^15 + 48*x^14 - 72*x^13 + 132*x^12 - 99*x^11 - 171*x^10 + 495*x^9 - 486*x^8 + 63*x^7 + 444*x^6 - 675*x^5 + 567*x^4 - 315*x^3 + 117*x^2 - 27*x + 3)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 33*x^15 + 48*x^14 - 72*x^13 + 132*x^12 - 99*x^11 - 171*x^10 + 495*x^9 - 486*x^8 + 63*x^7 + 444*x^6 - 675*x^5 + 567*x^4 - 315*x^3 + 117*x^2 - 27*x + 3, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 33 x^{15} + 48 x^{14} - 72 x^{13} + 132 x^{12} - 99 x^{11} - 171 x^{10} + 495 x^{9} - 486 x^{8} + 63 x^{7} + 444 x^{6} - 675 x^{5} + 567 x^{4} - 315 x^{3} + 117 x^{2} - 27 x + 3 \)

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:  \(-39531097362172608000000=-\,2^{12}\cdot 3^{31}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.00$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{13} a^{16} - \frac{6}{13} a^{15} + \frac{5}{13} a^{14} - \frac{2}{13} a^{13} - \frac{6}{13} a^{12} - \frac{4}{13} a^{11} - \frac{5}{13} a^{10} + \frac{3}{13} a^{9} - \frac{3}{13} a^{8} + \frac{5}{13} a^{6} - \frac{4}{13} a^{5} + \frac{6}{13} a^{4} + \frac{5}{13} a^{3} - \frac{1}{13} a^{2} + \frac{5}{13} a - \frac{3}{13}$, $\frac{1}{2295696845} a^{17} + \frac{9464172}{459139369} a^{16} - \frac{92054018}{2295696845} a^{15} + \frac{6222553}{2295696845} a^{14} - \frac{977800638}{2295696845} a^{13} - \frac{72635317}{176592065} a^{12} - \frac{72366916}{2295696845} a^{11} + \frac{813252598}{2295696845} a^{10} - \frac{571841262}{2295696845} a^{9} - \frac{509717011}{2295696845} a^{8} - \frac{299608889}{2295696845} a^{7} - \frac{388781214}{2295696845} a^{6} - \frac{44268476}{176592065} a^{5} + \frac{1135356}{2295696845} a^{4} - \frac{2577396}{35318413} a^{3} + \frac{174990116}{459139369} a^{2} + \frac{509494997}{2295696845} a - \frac{485706986}{2295696845}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( \frac{7705865226}{13584005} a^{17} - \frac{3594133491}{2716801} a^{16} + \frac{80462613842}{13584005} a^{15} - \frac{200545730832}{13584005} a^{14} + \frac{235892729622}{13584005} a^{13} - \frac{397196252901}{13584005} a^{12} + \frac{751850308854}{13584005} a^{11} - \frac{260551546092}{13584005} a^{10} - \frac{1491962555632}{13584005} a^{9} + \frac{2817625117614}{13584005} a^{8} - \frac{1862305106334}{13584005} a^{7} - \frac{758996697084}{13584005} a^{6} + \frac{2914233331782}{13584005} a^{5} - \frac{3254187818214}{13584005} a^{4} + \frac{438975228936}{2716801} a^{3} - \frac{192184283232}{2716801} a^{2} + \frac{259694452782}{13584005} a - \frac{34628641556}{13584005} \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 28271.4308598 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^3:C_2^2$ (as 18T53):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 15 conjugacy class representatives for $C_3^3:C_2^2$
Character table for $C_3^3:C_2^2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.7873200.2, 6.0.34992.1

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

Sibling fields

Degree 12 sibling: 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 R ${\href{/LocalNumberField/7.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{9}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$