↑

Properties

Label 18.0.79423921004...3667.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{27}\cdot 109^{6}\cdot 199^{6}$
Root discriminant $144.91$
Ramified primes $3, 109, 199$
Class number $243$ (GRH)
Class group $[3, 3, 3, 9]$ (GRH)
Galois group $C_3^2\times S_3$ (as 18T17)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![537367797, 0, 0, -106906158, 0, 0, 9295021, 0, 0, -441378, 0, 0, 11935, 0, 0, -171, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 171*x^15 + 11935*x^12 - 441378*x^9 + 9295021*x^6 - 106906158*x^3 + 537367797)
 
gp: K = bnfinit(x^18 - 171*x^15 + 11935*x^12 - 441378*x^9 + 9295021*x^6 - 106906158*x^3 + 537367797, 1)
 

Normalized defining polynomial

\( x^{18} - 171 x^{15} + 11935 x^{12} - 441378 x^{9} + 9295021 x^{6} - 106906158 x^{3} + 537367797 \)

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:  \(-794239210048451650839200748292120763667=-\,3^{27}\cdot 109^{6}\cdot 199^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $144.91$
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:  $3, 109, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{8} + \frac{1}{3} a^{2}$, $\frac{1}{305979963600243} a^{15} - \frac{34745074444672}{101993321200081} a^{12} + \frac{108009923629930}{305979963600243} a^{9} + \frac{19008277071591}{101993321200081} a^{6} - \frac{14232314223332}{305979963600243} a^{3} + \frac{43208840823819}{101993321200081}$, $\frac{1}{248761710406997559} a^{16} + \frac{7365935208657635}{27640190045221951} a^{13} - \frac{106984977336455120}{248761710406997559} a^{10} - \frac{10498975991251146}{27640190045221951} a^{7} + \frac{61487740369425511}{248761710406997559} a^{4} + \frac{10417721709349535}{27640190045221951} a$, $\frac{1}{67414423520296338489} a^{17} + \frac{864211826610538116}{7490491502255148721} a^{14} - \frac{25978202859664201256}{67414423520296338489} a^{11} + \frac{3555085539842380533}{7490491502255148721} a^{8} + \frac{5783007079730369368}{67414423520296338489} a^{5} - \frac{3665727554305169948}{7490491502255148721} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $243$ (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:  \( \frac{69797934131}{248761710406997559} a^{16} - \frac{1255766051088}{27640190045221951} a^{13} + \frac{706090030776140}{248761710406997559} a^{10} - \frac{2377764821085930}{27640190045221951} a^{7} + \frac{317395251357221354}{248761710406997559} a^{4} - \frac{205108352675598483}{27640190045221951} a \) (order $18$)
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:  \( 63469931610.31867 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3^2\times S_3$ (as 18T17):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_3^2\times S_3$
Character table for $C_3^2\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 6.0.12703485987.3, 6.0.9260841284523.1, \(\Q(\zeta_{9})\), Deg 6

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
3.6.9.10$x^{6} + 6 x^{4} + 3$$6$$1$$9$$C_6$$[2]_{2}$
$109$$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
$\Q_{109}$$x + 6$$1$$1$$0$Trivial$[\ ]$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
109.3.2.1$x^{3} - 109$$3$$1$$2$$C_3$$[\ ]_{3}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$
199.3.2.1$x^{3} - 199$$3$$1$$2$$C_3$$[\ ]_{3}$