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Properties

Label 18.0.13445405384...0192.6
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{43}$
Root discriminant $21.90$
Ramified primes $2, 3$
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![13, -36, 45, -3, -27, -9, 159, -423, 837, -1330, 1674, -1665, 1311, -819, 405, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 405*x^14 - 819*x^13 + 1311*x^12 - 1665*x^11 + 1674*x^10 - 1330*x^9 + 837*x^8 - 423*x^7 + 159*x^6 - 9*x^5 - 27*x^4 - 3*x^3 + 45*x^2 - 36*x + 13)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 405*x^14 - 819*x^13 + 1311*x^12 - 1665*x^11 + 1674*x^10 - 1330*x^9 + 837*x^8 - 423*x^7 + 159*x^6 - 9*x^5 - 27*x^4 - 3*x^3 + 45*x^2 - 36*x + 13, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 405 x^{14} - 819 x^{13} + 1311 x^{12} - 1665 x^{11} + 1674 x^{10} - 1330 x^{9} + 837 x^{8} - 423 x^{7} + 159 x^{6} - 9 x^{5} - 27 x^{4} - 3 x^{3} + 45 x^{2} - 36 x + 13 \)

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:  \(-1344540538448023869960192=-\,2^{12}\cdot 3^{43}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $21.90$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{191843} a^{16} - \frac{8}{191843} a^{15} - \frac{91309}{191843} a^{14} + \frac{63774}{191843} a^{13} + \frac{17982}{191843} a^{12} + \frac{21897}{191843} a^{11} - \frac{42}{10097} a^{10} - \frac{50707}{191843} a^{9} - \frac{55265}{191843} a^{8} - \frac{32365}{191843} a^{7} + \frac{48460}{191843} a^{6} - \frac{31146}{191843} a^{5} - \frac{72765}{191843} a^{4} - \frac{41948}{191843} a^{3} - \frac{34706}{191843} a^{2} + \frac{67060}{191843} a - \frac{76037}{191843}$, $\frac{1}{124506107} a^{17} + \frac{316}{124506107} a^{16} - \frac{17743457}{124506107} a^{15} - \frac{21079250}{124506107} a^{14} - \frac{19989958}{124506107} a^{13} - \frac{35781866}{124506107} a^{12} + \frac{16302294}{124506107} a^{11} - \frac{830367}{11318737} a^{10} - \frac{57346892}{124506107} a^{9} + \frac{55153958}{124506107} a^{8} + \frac{53637762}{124506107} a^{7} + \frac{9914604}{124506107} a^{6} - \frac{1181159}{6552953} a^{5} - \frac{29756784}{124506107} a^{4} + \frac{20714039}{124506107} a^{3} - \frac{34007001}{124506107} a^{2} - \frac{18827470}{124506107} a - \frac{31734179}{124506107}$

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:  \( -\frac{1234866}{6552953} a^{17} + \frac{10496361}{6552953} a^{16} - \frac{49821384}{6552953} a^{15} + \frac{163733160}{6552953} a^{14} - \frac{400088640}{6552953} a^{13} + \frac{754069134}{6552953} a^{12} - \frac{1108051608}{6552953} a^{11} + \frac{114676998}{595723} a^{10} - \frac{47871176}{284911} a^{9} + \frac{731973021}{6552953} a^{8} - \frac{371320368}{6552953} a^{7} + \frac{146252064}{6552953} a^{6} - \frac{23788266}{6552953} a^{5} - \frac{44424291}{6552953} a^{4} + \frac{21118320}{6552953} a^{3} + \frac{21545538}{6552953} a^{2} - \frac{38049846}{6552953} a + \frac{16867347}{6552953} \) (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:  \( 205747.43076295865 \)
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

\(\Q(\sqrt{-3}) \), 3.1.972.1 x3, 6.0.2834352.1, 9.3.669462604992.3

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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