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Properties

Label 18.0.31634955213...5952.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{9}\cdot 19^{10}$
Root discriminant $17.78$
Ramified primes $2, 3, 19$
Class number $1$ (GRH)
Class group Trivial (GRH)
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![223, -1479, 4555, -8627, 11285, -10621, 7497, -4388, 2430, -1107, 137, 234, -145, 37, -29, 20, 0, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 20*x^15 - 29*x^14 + 37*x^13 - 145*x^12 + 234*x^11 + 137*x^10 - 1107*x^9 + 2430*x^8 - 4388*x^7 + 7497*x^6 - 10621*x^5 + 11285*x^4 - 8627*x^3 + 4555*x^2 - 1479*x + 223)
 
gp: K = bnfinit(x^18 - 4*x^17 + 20*x^15 - 29*x^14 + 37*x^13 - 145*x^12 + 234*x^11 + 137*x^10 - 1107*x^9 + 2430*x^8 - 4388*x^7 + 7497*x^6 - 10621*x^5 + 11285*x^4 - 8627*x^3 + 4555*x^2 - 1479*x + 223, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 20 x^{15} - 29 x^{14} + 37 x^{13} - 145 x^{12} + 234 x^{11} + 137 x^{10} - 1107 x^{9} + 2430 x^{8} - 4388 x^{7} + 7497 x^{6} - 10621 x^{5} + 11285 x^{4} - 8627 x^{3} + 4555 x^{2} - 1479 x + 223 \)

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:  \(-31634955213811766525952=-\,2^{18}\cdot 3^{9}\cdot 19^{10}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $17.78$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2669706912519605506852337} a^{17} - \frac{219761845213571043618281}{2669706912519605506852337} a^{16} + \frac{449305631840929459210072}{2669706912519605506852337} a^{15} + \frac{272411010319379609307323}{2669706912519605506852337} a^{14} + \frac{1242172874246443284393100}{2669706912519605506852337} a^{13} + \frac{839349566732754035792762}{2669706912519605506852337} a^{12} - \frac{539128712035708114788218}{2669706912519605506852337} a^{11} + \frac{76046664646417514896360}{2669706912519605506852337} a^{10} - \frac{544916082402384821819329}{2669706912519605506852337} a^{9} + \frac{922012451916870318722852}{2669706912519605506852337} a^{8} + \frac{858841162698082772045247}{2669706912519605506852337} a^{7} + \frac{55368051420084758724825}{140510890132610816150123} a^{6} + \frac{513973164105446201521165}{2669706912519605506852337} a^{5} - \frac{194440980125309142912983}{2669706912519605506852337} a^{4} - \frac{258354282029907200722848}{2669706912519605506852337} a^{3} - \frac{426292187745219882256149}{2669706912519605506852337} a^{2} - \frac{1327507362567014367254649}{2669706912519605506852337} a - \frac{619888728546747070896008}{2669706912519605506852337}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  \( \frac{9455791585982}{369426739661573} a^{17} - \frac{31857759406780}{369426739661573} a^{16} - \frac{18010708887785}{369426739661573} a^{15} + \frac{171826311154272}{369426739661573} a^{14} - \frac{174803572627643}{369426739661573} a^{13} + \frac{276117927195058}{369426739661573} a^{12} - \frac{1206299074704695}{369426739661573} a^{11} + \frac{1486264963574563}{369426739661573} a^{10} + \frac{1981265005231463}{369426739661573} a^{9} - \frac{9064821264038926}{369426739661573} a^{8} + \frac{17955475749712549}{369426739661573} a^{7} - \frac{1674547090061821}{19443512613767} a^{6} + \frac{53338308409156442}{369426739661573} a^{5} - \frac{71436584997953467}{369426739661573} a^{4} + \frac{69408190540920984}{369426739661573} a^{3} - \frac{46301607649058811}{369426739661573} a^{2} + \frac{19725812668744023}{369426739661573} a - \frac{3799219294104115}{369426739661573} \) (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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 17725.017305991136 \) (assuming GRH)
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

\(\Q(\sqrt{-3}) \), 3.1.152.1, 6.0.9747.1, 6.0.623808.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.2.1.1$x^{2} - 19$$2$$1$$1$$C_2$$[\ ]_{2}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$