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Properties

Label 18.0.29765082070...9888.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{31}\cdot 7^{6}$
Root discriminant $20.14$
Ramified primes $2, 3, 7$
Class number $1$
Class group Trivial
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3, -27, 117, -333, 729, -1269, 1608, -1215, 162, 711, -819, 369, 126, -324, 264, -129, 42, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3)
 
gp: K = bnfinit(x^18 - 9*x^17 + 42*x^16 - 129*x^15 + 264*x^14 - 324*x^13 + 126*x^12 + 369*x^11 - 819*x^10 + 711*x^9 + 162*x^8 - 1215*x^7 + 1608*x^6 - 1269*x^5 + 729*x^4 - 333*x^3 + 117*x^2 - 27*x + 3, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 42 x^{16} - 129 x^{15} + 264 x^{14} - 324 x^{13} + 126 x^{12} + 369 x^{11} - 819 x^{10} + 711 x^{9} + 162 x^{8} - 1215 x^{7} + 1608 x^{6} - 1269 x^{5} + 729 x^{4} - 333 x^{3} + 117 x^{2} - 27 x + 3 \)

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:  \(-297650820707983690149888=-\,2^{12}\cdot 3^{31}\cdot 7^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.14$
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:  $2, 3, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{7} a^{15} + \frac{2}{7} a^{14} - \frac{2}{7} a^{13} + \frac{3}{7} a^{12} - \frac{2}{7} a^{10} + \frac{3}{7} a^{9} + \frac{2}{7} a^{8} + \frac{1}{7} a^{7} - \frac{1}{7} a^{6} - \frac{1}{7} a^{5} + \frac{1}{7} a^{4} - \frac{1}{7} a^{3} - \frac{1}{7} a^{2} - \frac{1}{7} a + \frac{3}{7}$, $\frac{1}{7} a^{16} + \frac{1}{7} a^{14} + \frac{1}{7} a^{12} - \frac{2}{7} a^{11} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{3}{7} a^{7} + \frac{1}{7} a^{6} + \frac{3}{7} a^{5} - \frac{3}{7} a^{4} + \frac{1}{7} a^{3} + \frac{1}{7} a^{2} - \frac{2}{7} a + \frac{1}{7}$, $\frac{1}{80802735730283} a^{17} - \frac{4535132339027}{80802735730283} a^{16} - \frac{1793526400509}{80802735730283} a^{15} + \frac{16511630724905}{80802735730283} a^{14} + \frac{35329446934030}{80802735730283} a^{13} + \frac{16534233289659}{80802735730283} a^{12} + \frac{27898233651924}{80802735730283} a^{11} - \frac{11456376131096}{80802735730283} a^{10} - \frac{6305423980162}{80802735730283} a^{9} - \frac{36571844611397}{80802735730283} a^{8} + \frac{12048473729864}{80802735730283} a^{7} + \frac{18216096698679}{80802735730283} a^{6} - \frac{5565846668190}{11543247961469} a^{5} + \frac{37678358602995}{80802735730283} a^{4} - \frac{35618967041375}{80802735730283} a^{3} - \frac{366900269653}{11543247961469} a^{2} - \frac{526136218134}{80802735730283} a + \frac{33305999809806}{80802735730283}$

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:  \( -\frac{223828226532}{24612469001} a^{17} + \frac{1848054751407}{24612469001} a^{16} - \frac{8004176785874}{24612469001} a^{15} + \frac{22744707214086}{24612469001} a^{14} - \frac{41439062414148}{24612469001} a^{13} + \frac{39693396171237}{24612469001} a^{12} + \frac{4733183177958}{24612469001} a^{11} - \frac{81795320579322}{24612469001} a^{10} + \frac{121021578911702}{24612469001} a^{9} - \frac{61498762820730}{24612469001} a^{8} - \frac{91284537460350}{24612469001} a^{7} + \frac{206828092379994}{24612469001} a^{6} - \frac{195981226422180}{24612469001} a^{5} + \frac{121386158887752}{24612469001} a^{4} - \frac{59494279599486}{24612469001} a^{3} + \frac{22963470724908}{24612469001} a^{2} - \frac{5727667099788}{24612469001} a + \frac{661411624648}{24612469001} \) (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:  \( 145692.568555 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.3.756.1, 6.0.1714608.1, 6.0.314928.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$