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Properties

Label 18.0.27321605778...2279.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 173^{9}$
Root discriminant $22.78$
Ramified primes $3, 173$
Class number $2$
Class group $[2]$
Galois group $D_9$ (as 18T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![81, -135, 360, -423, 1065, -1326, 1660, -1066, 578, -186, 208, -269, 244, -47, -41, 27, 2, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 2*x^16 + 27*x^15 - 41*x^14 - 47*x^13 + 244*x^12 - 269*x^11 + 208*x^10 - 186*x^9 + 578*x^8 - 1066*x^7 + 1660*x^6 - 1326*x^5 + 1065*x^4 - 423*x^3 + 360*x^2 - 135*x + 81)
 
gp: K = bnfinit(x^18 - 4*x^17 + 2*x^16 + 27*x^15 - 41*x^14 - 47*x^13 + 244*x^12 - 269*x^11 + 208*x^10 - 186*x^9 + 578*x^8 - 1066*x^7 + 1660*x^6 - 1326*x^5 + 1065*x^4 - 423*x^3 + 360*x^2 - 135*x + 81, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 2 x^{16} + 27 x^{15} - 41 x^{14} - 47 x^{13} + 244 x^{12} - 269 x^{11} + 208 x^{10} - 186 x^{9} + 578 x^{8} - 1066 x^{7} + 1660 x^{6} - 1326 x^{5} + 1065 x^{4} - 423 x^{3} + 360 x^{2} - 135 x + 81 \)

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:  \(-2732160577820469872382279=-\,3^{9}\cdot 173^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.78$
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:  $3, 173$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{3} a^{5} - \frac{1}{9} a^{3} + \frac{1}{3} a$, $\frac{1}{9} a^{10} + \frac{2}{9} a^{4} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{11} + \frac{2}{9} a^{5} - \frac{1}{3} a^{3}$, $\frac{1}{18} a^{12} - \frac{1}{18} a^{9} - \frac{1}{6} a^{7} - \frac{1}{18} a^{6} - \frac{1}{3} a^{4} - \frac{1}{9} a^{3} - \frac{1}{6} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{54} a^{13} - \frac{1}{54} a^{12} + \frac{1}{27} a^{11} + \frac{1}{54} a^{10} - \frac{1}{18} a^{9} + \frac{1}{18} a^{8} + \frac{1}{27} a^{7} + \frac{7}{54} a^{6} + \frac{5}{27} a^{5} - \frac{2}{27} a^{4} - \frac{1}{18} a^{3} - \frac{1}{9} a^{2} + \frac{1}{3} a - \frac{1}{2}$, $\frac{1}{54} a^{14} + \frac{1}{54} a^{12} - \frac{1}{18} a^{11} - \frac{1}{27} a^{10} + \frac{5}{54} a^{8} - \frac{1}{6} a^{7} - \frac{1}{54} a^{6} - \frac{4}{9} a^{5} - \frac{25}{54} a^{4} - \frac{1}{6} a^{3} - \frac{1}{9} a^{2} - \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{324} a^{15} - \frac{1}{162} a^{14} - \frac{1}{108} a^{13} - \frac{1}{81} a^{12} - \frac{4}{81} a^{11} - \frac{1}{18} a^{10} + \frac{2}{81} a^{9} + \frac{5}{324} a^{8} + \frac{25}{324} a^{6} + \frac{85}{324} a^{5} + \frac{37}{108} a^{4} + \frac{1}{54} a^{3} - \frac{4}{9} a^{2} - \frac{1}{6} a - \frac{1}{4}$, $\frac{1}{324} a^{16} - \frac{1}{324} a^{14} + \frac{1}{162} a^{13} + \frac{1}{54} a^{12} - \frac{2}{81} a^{11} + \frac{2}{81} a^{10} - \frac{5}{108} a^{9} - \frac{8}{81} a^{8} - \frac{5}{324} a^{7} - \frac{13}{108} a^{6} + \frac{5}{324} a^{5} - \frac{1}{54} a^{4} - \frac{1}{54} a^{3} - \frac{7}{18} a^{2} - \frac{1}{12} a$, $\frac{1}{56577786803748} a^{17} - \frac{616692019}{56577786803748} a^{16} - \frac{2055293819}{4041270485982} a^{15} + \frac{76962337111}{18859262267916} a^{14} + \frac{249417773881}{56577786803748} a^{13} - \frac{227604599245}{28288893401874} a^{12} - \frac{169263018499}{4041270485982} a^{11} - \frac{668033127443}{56577786803748} a^{10} - \frac{1300937200853}{56577786803748} a^{9} - \frac{213689352187}{3143210377986} a^{8} - \frac{426765344159}{4041270485982} a^{7} + \frac{8656179331973}{56577786803748} a^{6} - \frac{13427012317447}{28288893401874} a^{5} + \frac{241850841803}{6286420755972} a^{4} - \frac{4172239267517}{9429631133958} a^{3} + \frac{1307064383495}{6286420755972} a^{2} - \frac{671396791049}{2095473585324} a + \frac{60572237821}{698491195108}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( -1 \) (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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 555844.602335 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9$ (as 18T5):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_9$
Character table for $D_9$

Intermediate fields

\(\Q(\sqrt{-519}) \), 3.1.519.1 x3, 6.0.139798359.1, 9.1.72555348321.1 x9

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

Sibling fields

Degree 9 sibling: 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 ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} - 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$173$173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$
173.2.1.2$x^{2} + 346$$2$$1$$1$$C_2$$[\ ]_{2}$