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Properties

Label 18.10.2110600968...2841.8
Degree $18$
Signature $[10, 4]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{3}$
Root discriminant $25.52$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T840

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, -6, 96, -195, -102, 1121, -2361, 2715, -1742, 267, 645, -712, 357, -63, -36, 30, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 30*x^16 - 36*x^15 - 63*x^14 + 357*x^13 - 712*x^12 + 645*x^11 + 267*x^10 - 1742*x^9 + 2715*x^8 - 2361*x^7 + 1121*x^6 - 102*x^5 - 195*x^4 + 96*x^3 - 6*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^18 - 9*x^17 + 30*x^16 - 36*x^15 - 63*x^14 + 357*x^13 - 712*x^12 + 645*x^11 + 267*x^10 - 1742*x^9 + 2715*x^8 - 2361*x^7 + 1121*x^6 - 102*x^5 - 195*x^4 + 96*x^3 - 6*x^2 - 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 30 x^{16} - 36 x^{15} - 63 x^{14} + 357 x^{13} - 712 x^{12} + 645 x^{11} + 267 x^{10} - 1742 x^{9} + 2715 x^{8} - 2361 x^{7} + 1121 x^{6} - 102 x^{5} - 195 x^{4} + 96 x^{3} - 6 x^{2} - 6 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(21106009685505727678262841=3^{24}\cdot 73^{3}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $25.52$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{530209} a^{16} - \frac{8}{530209} a^{15} + \frac{72555}{530209} a^{14} + \frac{22464}{530209} a^{13} - \frac{200318}{530209} a^{12} - \frac{150906}{530209} a^{11} + \frac{30324}{530209} a^{10} + \frac{667}{530209} a^{9} + \frac{184694}{530209} a^{8} - \frac{216765}{530209} a^{7} - \frac{64742}{530209} a^{6} + \frac{134838}{530209} a^{5} - \frac{264623}{530209} a^{4} + \frac{234924}{530209} a^{3} - \frac{99530}{530209} a^{2} - \frac{213784}{530209} a - \frac{97536}{530209}$, $\frac{1}{530209} a^{17} + \frac{72491}{530209} a^{15} + \frac{72695}{530209} a^{14} - \frac{20606}{530209} a^{13} - \frac{162823}{530209} a^{12} - \frac{116506}{530209} a^{11} + \frac{243259}{530209} a^{10} + \frac{190030}{530209} a^{9} + \frac{200369}{530209} a^{8} - \frac{208235}{530209} a^{7} + \frac{147111}{530209} a^{6} - \frac{246337}{530209} a^{5} + \frac{238776}{530209} a^{4} + \frac{189235}{530209} a^{3} + \frac{50394}{530209} a^{2} - \frac{217181}{530209} a - \frac{250079}{530209}$

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:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 1087281.49682 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T840:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 165888
The 180 conjugacy class representatives for t18n840 are not computed
Character table for t18n840 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.2.1.2$x^{2} + 365$$2$$1$$1$$C_2$$[\ ]_{2}$
73.4.2.1$x^{4} + 1533 x^{2} + 644809$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
73.4.0.1$x^{4} - x + 13$$1$$4$$0$$C_4$$[\ ]^{4}$
73.6.0.1$x^{6} - x + 5$$1$$6$$0$$C_6$$[\ ]^{6}$
577Data not computed