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Properties

Label 18.0.31354186474...9487.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,17^{9}\cdot 31^{9}$
Root discriminant $22.96$
Ramified primes $17, 31$
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![25, 80, 246, 30, -925, -303, 1657, -1219, 519, -464, 736, -451, 201, -140, 44, -20, 8, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 8*x^16 - 20*x^15 + 44*x^14 - 140*x^13 + 201*x^12 - 451*x^11 + 736*x^10 - 464*x^9 + 519*x^8 - 1219*x^7 + 1657*x^6 - 303*x^5 - 925*x^4 + 30*x^3 + 246*x^2 + 80*x + 25)
 
gp: K = bnfinit(x^18 + 8*x^16 - 20*x^15 + 44*x^14 - 140*x^13 + 201*x^12 - 451*x^11 + 736*x^10 - 464*x^9 + 519*x^8 - 1219*x^7 + 1657*x^6 - 303*x^5 - 925*x^4 + 30*x^3 + 246*x^2 + 80*x + 25, 1)
 

Normalized defining polynomial

\( x^{18} + 8 x^{16} - 20 x^{15} + 44 x^{14} - 140 x^{13} + 201 x^{12} - 451 x^{11} + 736 x^{10} - 464 x^{9} + 519 x^{8} - 1219 x^{7} + 1657 x^{6} - 303 x^{5} - 925 x^{4} + 30 x^{3} + 246 x^{2} + 80 x + 25 \)

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:  \(-3135418647416996838649487=-\,17^{9}\cdot 31^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.96$
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:  $17, 31$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{35} a^{13} + \frac{12}{35} a^{12} + \frac{4}{35} a^{11} + \frac{2}{35} a^{10} + \frac{3}{35} a^{9} - \frac{6}{35} a^{8} + \frac{3}{35} a^{7} - \frac{3}{35} a^{6} - \frac{13}{35} a^{5} + \frac{12}{35} a^{4} - \frac{12}{35} a^{3} - \frac{1}{7} a^{2} - \frac{3}{35} a - \frac{2}{7}$, $\frac{1}{35} a^{14} - \frac{11}{35} a^{11} + \frac{2}{5} a^{10} - \frac{1}{5} a^{9} + \frac{1}{7} a^{8} - \frac{4}{35} a^{7} - \frac{12}{35} a^{6} - \frac{1}{5} a^{5} - \frac{16}{35} a^{4} - \frac{1}{35} a^{3} - \frac{13}{35} a^{2} - \frac{9}{35} a + \frac{3}{7}$, $\frac{1}{595} a^{15} + \frac{1}{595} a^{14} - \frac{2}{595} a^{13} - \frac{7}{17} a^{12} + \frac{2}{7} a^{11} + \frac{38}{595} a^{10} - \frac{113}{595} a^{9} + \frac{293}{595} a^{8} + \frac{13}{595} a^{7} - \frac{118}{595} a^{6} + \frac{73}{595} a^{5} + \frac{99}{595} a^{4} + \frac{2}{119} a^{3} + \frac{23}{595} a^{2} + \frac{117}{595} a + \frac{1}{17}$, $\frac{1}{1103725} a^{16} + \frac{764}{1103725} a^{15} + \frac{416}{44149} a^{14} + \frac{15654}{1103725} a^{13} + \frac{10104}{44149} a^{12} + \frac{352244}{1103725} a^{11} + \frac{355162}{1103725} a^{10} - \frac{274864}{1103725} a^{9} - \frac{295693}{1103725} a^{8} - \frac{1181}{31535} a^{7} + \frac{276406}{1103725} a^{6} - \frac{14201}{220745} a^{5} - \frac{224197}{1103725} a^{4} - \frac{281296}{1103725} a^{3} + \frac{539464}{1103725} a^{2} + \frac{40274}{220745} a + \frac{6855}{44149}$, $\frac{1}{77547465620046625} a^{17} + \frac{21536644627}{77547465620046625} a^{16} + \frac{23767810724362}{77547465620046625} a^{15} + \frac{17545340446629}{77547465620046625} a^{14} - \frac{463987741848248}{77547465620046625} a^{13} + \frac{21552355061781639}{77547465620046625} a^{12} + \frac{856873130173329}{77547465620046625} a^{11} + \frac{1449476896255757}{77547465620046625} a^{10} + \frac{1468084314304603}{3101898624801865} a^{9} + \frac{1388202506743698}{11078209374292375} a^{8} + \frac{21102597838515141}{77547465620046625} a^{7} - \frac{4201244862935537}{77547465620046625} a^{6} + \frac{32415742491849233}{77547465620046625} a^{5} - \frac{26284968307021012}{77547465620046625} a^{4} + \frac{21117363034785376}{77547465620046625} a^{3} - \frac{16408196477508093}{77547465620046625} a^{2} + \frac{398281929119831}{912323124941725} a - \frac{2092783743538}{8360912735315}$

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:  \( 48092.8610332 \)
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

\(\Q(\sqrt{-527}) \), 3.1.527.1 x3, 6.0.146363183.2, 9.1.77133397441.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$17$17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
$31$31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} - 31$$2$$1$$1$$C_2$$[\ ]_{2}$