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Properties

Label 18.0.33552816771...6128.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 23^{8}$
Root discriminant $23.04$
Ramified primes $2, 3, 23$
Class number $1$
Class group Trivial
Galois group $C_3^2:S_3$ (as 18T24)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 54, -187, 483, -849, 1095, -714, 429, 57, 24, 24, 128, -69, 66, -14, 9, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 9*x^16 - 14*x^15 + 66*x^14 - 69*x^13 + 128*x^12 + 24*x^11 + 24*x^10 + 57*x^9 + 429*x^8 - 714*x^7 + 1095*x^6 - 849*x^5 + 483*x^4 - 187*x^3 + 54*x^2 - 9*x + 1)
 
gp: K = bnfinit(x^18 + 9*x^16 - 14*x^15 + 66*x^14 - 69*x^13 + 128*x^12 + 24*x^11 + 24*x^10 + 57*x^9 + 429*x^8 - 714*x^7 + 1095*x^6 - 849*x^5 + 483*x^4 - 187*x^3 + 54*x^2 - 9*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} + 9 x^{16} - 14 x^{15} + 66 x^{14} - 69 x^{13} + 128 x^{12} + 24 x^{11} + 24 x^{10} + 57 x^{9} + 429 x^{8} - 714 x^{7} + 1095 x^{6} - 849 x^{5} + 483 x^{4} - 187 x^{3} + 54 x^{2} - 9 x + 1 \)

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:  \(-3355281677165339463856128=-\,2^{12}\cdot 3^{21}\cdot 23^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.04$
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:  $2, 3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{8} - \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6}$, $\frac{1}{6} a^{13} - \frac{1}{6} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4} + \frac{1}{6} a^{3} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{6} a^{14} - \frac{1}{6} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{2} a^{6} - \frac{1}{3} a^{5} + \frac{1}{6} a^{4} - \frac{1}{6} a^{2} - \frac{1}{3} a$, $\frac{1}{7866} a^{15} + \frac{14}{207} a^{14} - \frac{119}{3933} a^{13} - \frac{23}{342} a^{12} + \frac{47}{7866} a^{11} - \frac{43}{1311} a^{10} - \frac{43}{1311} a^{9} + \frac{899}{7866} a^{8} - \frac{2329}{7866} a^{7} - \frac{1642}{3933} a^{6} - \frac{571}{7866} a^{5} + \frac{2405}{7866} a^{4} + \frac{163}{342} a^{3} - \frac{1207}{2622} a^{2} - \frac{1147}{3933} a - \frac{1795}{7866}$, $\frac{1}{7866} a^{16} - \frac{43}{3933} a^{14} + \frac{77}{2622} a^{13} - \frac{65}{1311} a^{12} + \frac{479}{3933} a^{11} + \frac{109}{1311} a^{10} - \frac{811}{7866} a^{9} + \frac{30}{437} a^{8} + \frac{74}{171} a^{7} + \frac{2887}{7866} a^{6} - \frac{1073}{2622} a^{5} - \frac{455}{1311} a^{4} + \frac{2497}{7866} a^{3} - \frac{1781}{7866} a^{2} + \frac{2005}{7866} a - \frac{41}{414}$, $\frac{1}{28561446} a^{17} - \frac{89}{9520482} a^{16} - \frac{661}{14280723} a^{15} + \frac{44219}{1586747} a^{14} + \frac{46325}{1586747} a^{13} - \frac{2105603}{28561446} a^{12} - \frac{139604}{4760241} a^{11} + \frac{1763575}{14280723} a^{10} + \frac{140675}{4760241} a^{9} - \frac{4661851}{28561446} a^{8} - \frac{9981035}{28561446} a^{7} - \frac{439811}{3173494} a^{6} + \frac{98386}{1586747} a^{5} - \frac{6478325}{28561446} a^{4} - \frac{10374503}{28561446} a^{3} + \frac{1632260}{14280723} a^{2} + \frac{9990907}{28561446} a + \frac{1229710}{4760241}$

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:  \( \frac{1138412}{4760241} a^{17} + \frac{201973}{4760241} a^{16} + \frac{10319149}{4760241} a^{15} - \frac{4695016}{1586747} a^{14} + \frac{24325327}{1586747} a^{13} - \frac{21984243}{1586747} a^{12} + \frac{136212529}{4760241} a^{11} + \frac{16731421}{1586747} a^{10} + \frac{13329031}{1586747} a^{9} + \frac{1079701}{68989} a^{8} + \frac{504491279}{4760241} a^{7} - \frac{721060708}{4760241} a^{6} + \frac{1134348962}{4760241} a^{5} - \frac{260196844}{1586747} a^{4} + \frac{146867956}{1586747} a^{3} - \frac{151688789}{4760241} a^{2} + \frac{49106263}{4760241} a - \frac{3424937}{4760241} \) (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:  \( 112152.79938374869 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2:S_3$ (as 18T24):

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 $C_3^2:S_3$
Character table for $C_3^2:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.108.1 x3, 6.0.34992.1, 9.3.352519065792.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
3Data not computed
$23$23.6.4.2$x^{6} - 23 x^{3} + 3703$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
23.6.0.1$x^{6} - x + 15$$1$$6$$0$$C_6$$[\ ]^{6}$
23.6.4.2$x^{6} - 23 x^{3} + 3703$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$