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Properties

Label 18.0.43973453648...4375.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,5^{12}\cdot 23^{9}$
Root discriminant $14.02$
Ramified primes $5, 23$
Class number $1$
Class group Trivial
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![1, -10, 41, -84, 96, -74, -1, 148, -88, 127, 34, 39, 68, -32, 50, -18, 12, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 12*x^16 - 18*x^15 + 50*x^14 - 32*x^13 + 68*x^12 + 39*x^11 + 34*x^10 + 127*x^9 - 88*x^8 + 148*x^7 - x^6 - 74*x^5 + 96*x^4 - 84*x^3 + 41*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 12*x^16 - 18*x^15 + 50*x^14 - 32*x^13 + 68*x^12 + 39*x^11 + 34*x^10 + 127*x^9 - 88*x^8 + 148*x^7 - x^6 - 74*x^5 + 96*x^4 - 84*x^3 + 41*x^2 - 10*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 12 x^{16} - 18 x^{15} + 50 x^{14} - 32 x^{13} + 68 x^{12} + 39 x^{11} + 34 x^{10} + 127 x^{9} - 88 x^{8} + 148 x^{7} - x^{6} - 74 x^{5} + 96 x^{4} - 84 x^{3} + 41 x^{2} - 10 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-439734536489990234375=-\,5^{12}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.02$
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:  $5, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{35} a^{15} + \frac{3}{35} a^{14} - \frac{1}{5} a^{13} + \frac{17}{35} a^{12} - \frac{1}{7} a^{11} - \frac{12}{35} a^{10} + \frac{17}{35} a^{9} - \frac{1}{7} a^{8} - \frac{11}{35} a^{7} + \frac{12}{35} a^{6} + \frac{11}{35} a^{5} + \frac{12}{35} a^{4} - \frac{1}{7} a^{3} + \frac{3}{7} a^{2} + \frac{2}{35} a + \frac{8}{35}$, $\frac{1}{805} a^{16} - \frac{226}{805} a^{14} + \frac{178}{805} a^{13} - \frac{1}{5} a^{12} + \frac{108}{805} a^{11} - \frac{367}{805} a^{10} + \frac{37}{115} a^{9} + \frac{389}{805} a^{8} + \frac{79}{161} a^{7} - \frac{12}{161} a^{6} + \frac{27}{115} a^{5} + \frac{379}{805} a^{4} + \frac{62}{161} a^{3} + \frac{202}{805} a^{2} - \frac{348}{805} a + \frac{256}{805}$, $\frac{1}{20487978636895} a^{17} + \frac{11408352853}{20487978636895} a^{16} + \frac{117531612511}{20487978636895} a^{15} + \frac{5217227014691}{20487978636895} a^{14} - \frac{2696112146281}{20487978636895} a^{13} + \frac{7928697267144}{20487978636895} a^{12} - \frac{128678456811}{890781679865} a^{11} - \frac{8008372844716}{20487978636895} a^{10} - \frac{38969292511}{4097595727379} a^{9} + \frac{117662078001}{1205175213935} a^{8} + \frac{8459759982088}{20487978636895} a^{7} + \frac{5958019009368}{20487978636895} a^{6} - \frac{6094146051937}{20487978636895} a^{5} + \frac{9334120508241}{20487978636895} a^{4} + \frac{5044544428227}{20487978636895} a^{3} + \frac{23800987217}{172167887705} a^{2} - \frac{7934486794}{169322137495} a + \frac{9525912999989}{20487978636895}$

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

Class group and class number

Trivial group, which has order $1$

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:  \( 635.399863828 \)
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

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, 6.0.12167.1, 9.1.4372515625.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$