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Properties

Label 18.0.67783088755...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{25}\cdot 5^{9}$
Root discriminant $16.32$
Ramified primes $2, 3, 5$
Class number $2$
Class group $[2]$
Galois group $S_3^2$ (as 18T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9, 0, 0, -27, 72, -117, 144, -162, 126, -69, 45, -27, 36, -15, 30, -12, 9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 9*x^16 - 12*x^15 + 30*x^14 - 15*x^13 + 36*x^12 - 27*x^11 + 45*x^10 - 69*x^9 + 126*x^8 - 162*x^7 + 144*x^6 - 117*x^5 + 72*x^4 - 27*x^3 + 9)
 
gp: K = bnfinit(x^18 - 3*x^17 + 9*x^16 - 12*x^15 + 30*x^14 - 15*x^13 + 36*x^12 - 27*x^11 + 45*x^10 - 69*x^9 + 126*x^8 - 162*x^7 + 144*x^6 - 117*x^5 + 72*x^4 - 27*x^3 + 9, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 9 x^{16} - 12 x^{15} + 30 x^{14} - 15 x^{13} + 36 x^{12} - 27 x^{11} + 45 x^{10} - 69 x^{9} + 126 x^{8} - 162 x^{7} + 144 x^{6} - 117 x^{5} + 72 x^{4} - 27 x^{3} + 9 \)

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:  \(-6778308875544000000000=-\,2^{12}\cdot 3^{25}\cdot 5^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $16.32$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{6} a^{12} - \frac{1}{6} a^{11} - \frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{18} a^{13} - \frac{1}{6} a^{11} - \frac{1}{6} a^{10} + \frac{1}{3} a^{8} - \frac{1}{2} a^{7} - \frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{18} a^{14} - \frac{1}{6} a^{9} - \frac{1}{2} a^{7} - \frac{1}{6} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{36} a^{15} - \frac{1}{12} a^{12} - \frac{1}{12} a^{11} + \frac{1}{12} a^{10} - \frac{1}{12} a^{9} - \frac{1}{2} a^{8} + \frac{1}{4} a^{7} - \frac{1}{12} a^{6} - \frac{1}{2} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{7380} a^{16} - \frac{7}{3690} a^{15} + \frac{2}{615} a^{14} - \frac{35}{1476} a^{13} - \frac{187}{2460} a^{12} + \frac{47}{2460} a^{11} + \frac{91}{820} a^{10} - \frac{16}{123} a^{9} + \frac{643}{2460} a^{8} + \frac{689}{2460} a^{7} + \frac{269}{615} a^{6} - \frac{3}{205} a^{5} + \frac{65}{492} a^{4} - \frac{11}{41} a^{3} - \frac{57}{820} a^{2} + \frac{249}{820} a - \frac{73}{410}$, $\frac{1}{159444900} a^{17} - \frac{1987}{53148300} a^{16} - \frac{2120263}{159444900} a^{15} + \frac{265469}{53148300} a^{14} + \frac{27424}{1374525} a^{13} + \frac{831757}{17716100} a^{12} - \frac{2822261}{53148300} a^{11} - \frac{2028179}{13287075} a^{10} + \frac{586672}{13287075} a^{9} - \frac{954171}{4429025} a^{8} + \frac{3597563}{8858050} a^{7} + \frac{5145157}{53148300} a^{6} - \frac{2551981}{17716100} a^{5} - \frac{1047607}{10629660} a^{4} + \frac{1909459}{8858050} a^{3} - \frac{897143}{4429025} a^{2} - \frac{510977}{8858050} a - \frac{632133}{17716100}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 3197.7993361226186 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 18T11):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

\(\Q(\sqrt{-15}) \), 3.1.135.1 x3, 3.1.108.1, 6.0.4374000.1, 6.0.273375.1, 9.1.4251528000.1 x3

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

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 9 sibling: data not computed
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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$2$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$5$5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$C_2$$[\ ]_{2}$
5.2.1.2$x^{2} + 10$$2$$1$$1$$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}$