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Properties

Label 18.0.15174987087...8407.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{21}\cdot 29^{9}$
Root discriminant $19.40$
Ramified primes $3, 29$
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![87, 0, 168, 102, 63, 129, 136, -489, -237, -40, 345, 60, -63, -60, 9, 20, -3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 3*x^16 + 20*x^15 + 9*x^14 - 60*x^13 - 63*x^12 + 60*x^11 + 345*x^10 - 40*x^9 - 237*x^8 - 489*x^7 + 136*x^6 + 129*x^5 + 63*x^4 + 102*x^3 + 168*x^2 + 87)
 
gp: K = bnfinit(x^18 - 3*x^17 - 3*x^16 + 20*x^15 + 9*x^14 - 60*x^13 - 63*x^12 + 60*x^11 + 345*x^10 - 40*x^9 - 237*x^8 - 489*x^7 + 136*x^6 + 129*x^5 + 63*x^4 + 102*x^3 + 168*x^2 + 87, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 3 x^{16} + 20 x^{15} + 9 x^{14} - 60 x^{13} - 63 x^{12} + 60 x^{11} + 345 x^{10} - 40 x^{9} - 237 x^{8} - 489 x^{7} + 136 x^{6} + 129 x^{5} + 63 x^{4} + 102 x^{3} + 168 x^{2} + 87 \)

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:  \(-151749870875069854858407=-\,3^{21}\cdot 29^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $19.40$
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:  $3, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5} a^{11} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} - \frac{2}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} + \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{12} + \frac{1}{5} a^{10} + \frac{4}{15} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{4}{15} a^{6} - \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{1}{5} a^{3} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{15} a^{13} - \frac{2}{15} a^{10} + \frac{2}{5} a^{9} - \frac{1}{3} a^{7} - \frac{2}{5} a^{6} + \frac{2}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{1305} a^{14} + \frac{5}{261} a^{13} + \frac{28}{1305} a^{12} + \frac{97}{1305} a^{11} - \frac{632}{1305} a^{10} + \frac{113}{261} a^{9} + \frac{343}{1305} a^{8} - \frac{62}{1305} a^{7} - \frac{362}{1305} a^{6} - \frac{122}{435} a^{5} + \frac{22}{435} a^{4} - \frac{25}{87} a^{3} + \frac{214}{435} a^{2} - \frac{2}{15} a + \frac{7}{15}$, $\frac{1}{1305} a^{15} + \frac{4}{435} a^{13} + \frac{2}{435} a^{12} + \frac{5}{87} a^{11} - \frac{28}{145} a^{10} - \frac{41}{435} a^{9} + \frac{79}{435} a^{8} - \frac{2}{87} a^{7} - \frac{364}{1305} a^{6} - \frac{49}{145} a^{5} + \frac{13}{29} a^{4} + \frac{8}{29} a^{3} - \frac{101}{435} a^{2} + \frac{1}{5} a - \frac{7}{15}$, $\frac{1}{1066185} a^{16} - \frac{371}{1066185} a^{15} + \frac{193}{1066185} a^{14} - \frac{15929}{1066185} a^{13} - \frac{18659}{1066185} a^{12} + \frac{19820}{213237} a^{11} + \frac{73817}{213237} a^{10} + \frac{152398}{1066185} a^{9} + \frac{31024}{1066185} a^{8} + \frac{30118}{71079} a^{7} - \frac{17713}{71079} a^{6} + \frac{86851}{355395} a^{5} + \frac{33575}{71079} a^{4} + \frac{62906}{355395} a^{3} - \frac{146689}{355395} a^{2} - \frac{1967}{4085} a + \frac{1606}{4085}$, $\frac{1}{198213387165} a^{17} - \frac{59282}{198213387165} a^{16} + \frac{15928022}{198213387165} a^{15} + \frac{58948961}{198213387165} a^{14} - \frac{54789766}{10432283535} a^{13} + \frac{3726355988}{198213387165} a^{12} - \frac{5840332768}{198213387165} a^{11} - \frac{74976762616}{198213387165} a^{10} + \frac{53588138297}{198213387165} a^{9} + \frac{38173571158}{198213387165} a^{8} + \frac{726296420}{2086456707} a^{7} + \frac{8346817051}{66071129055} a^{6} + \frac{16143320291}{66071129055} a^{5} + \frac{584851211}{22023709685} a^{4} + \frac{9269398984}{22023709685} a^{3} - \frac{18153170524}{66071129055} a^{2} - \frac{173563253}{2278314795} a + \frac{44412117}{151887653}$

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:  \( 19560.8039608 \)
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

\(\Q(\sqrt{-87}) \), 3.1.87.1 x3, 6.0.658503.1, 9.1.41764235769.1 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
3.6.7.1$x^{6} + 6 x^{2} + 6$$6$$1$$7$$S_3$$[3/2]_{2}$
$29$29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.1.2$x^{2} + 58$$2$$1$$1$$C_2$$[\ ]_{2}$