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Properties

Label 18.0.49429617521...1968.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 19^{10}$
Root discriminant $14.11$
Ramified primes $2, 3, 19$
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![1, -10, 36, -50, -4, 83, -89, 14, 100, -152, 121, -54, 11, 4, 6, -9, 7, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 7*x^16 - 9*x^15 + 6*x^14 + 4*x^13 + 11*x^12 - 54*x^11 + 121*x^10 - 152*x^9 + 100*x^8 + 14*x^7 - 89*x^6 + 83*x^5 - 4*x^4 - 50*x^3 + 36*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 7*x^16 - 9*x^15 + 6*x^14 + 4*x^13 + 11*x^12 - 54*x^11 + 121*x^10 - 152*x^9 + 100*x^8 + 14*x^7 - 89*x^6 + 83*x^5 - 4*x^4 - 50*x^3 + 36*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 7 x^{16} - 9 x^{15} + 6 x^{14} + 4 x^{13} + 11 x^{12} - 54 x^{11} + 121 x^{10} - 152 x^{9} + 100 x^{8} + 14 x^{7} - 89 x^{6} + 83 x^{5} - 4 x^{4} - 50 x^{3} + 36 x^{2} - 10 x + 1 \)

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:  \(-494296175215808851968=-\,2^{12}\cdot 3^{9}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.11$
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:  $2, 3, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{19} a^{16} - \frac{9}{19} a^{15} + \frac{6}{19} a^{14} - \frac{6}{19} a^{13} - \frac{3}{19} a^{12} - \frac{9}{19} a^{11} + \frac{2}{19} a^{10} - \frac{8}{19} a^{9} + \frac{2}{19} a^{8} - \frac{9}{19} a^{7} + \frac{6}{19} a^{6} - \frac{2}{19} a^{5} - \frac{8}{19} a^{4} - \frac{6}{19} a^{3} - \frac{3}{19} a^{2} - \frac{6}{19} a + \frac{9}{19}$, $\frac{1}{12056232317} a^{17} - \frac{78796658}{12056232317} a^{16} - \frac{246362364}{12056232317} a^{15} - \frac{87533867}{12056232317} a^{14} - \frac{6011434583}{12056232317} a^{13} - \frac{4171359091}{12056232317} a^{12} + \frac{2833490284}{12056232317} a^{11} - \frac{4155883975}{12056232317} a^{10} - \frac{2612118717}{12056232317} a^{9} + \frac{4072613870}{12056232317} a^{8} + \frac{1250577336}{12056232317} a^{7} - \frac{4438343464}{12056232317} a^{6} - \frac{1069765727}{12056232317} a^{5} - \frac{4674193152}{12056232317} a^{4} + \frac{4266916301}{12056232317} a^{3} + \frac{1008712334}{12056232317} a^{2} - \frac{5156845313}{12056232317} a + \frac{3487262309}{12056232317}$

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:  \( \frac{359827682}{6584507} a^{17} - \frac{937879538}{6584507} a^{16} + \frac{2152280244}{6584507} a^{15} - \frac{2397084805}{6584507} a^{14} + \frac{1228684907}{6584507} a^{13} + \frac{1911259608}{6584507} a^{12} + \frac{4713857029}{6584507} a^{11} - \frac{17559802382}{6584507} a^{10} + \frac{36668281790}{6584507} a^{9} - \frac{40372356367}{6584507} a^{8} + \frac{20312585200}{6584507} a^{7} + \frac{12835145887}{6584507} a^{6} - \frac{26918060294}{6584507} a^{5} + \frac{19397991622}{6584507} a^{4} + \frac{6043379052}{6584507} a^{3} - \frac{15533126219}{6584507} a^{2} + \frac{6919910909}{6584507} a - \frac{952242575}{6584507} \) (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:  \( 2261.196159404121 \)
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

\(\Q(\sqrt{-3}) \), 3.1.76.1, 6.0.9747.1, 6.0.155952.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.2$x^{2} + 76$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.5$x^{6} + 1216$$6$$1$$5$$C_6$$[\ ]_{6}$