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Properties

Label 18.0.48941546411...1799.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,199^{9}$
Root discriminant $14.11$
Ramified prime $199$
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![9, -15, 62, -179, 393, -709, 1154, -1740, 2362, -2789, 2804, -2364, 1655, -952, 444, -164, 46, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 46*x^16 - 164*x^15 + 444*x^14 - 952*x^13 + 1655*x^12 - 2364*x^11 + 2804*x^10 - 2789*x^9 + 2362*x^8 - 1740*x^7 + 1154*x^6 - 709*x^5 + 393*x^4 - 179*x^3 + 62*x^2 - 15*x + 9)
 
gp: K = bnfinit(x^18 - 9*x^17 + 46*x^16 - 164*x^15 + 444*x^14 - 952*x^13 + 1655*x^12 - 2364*x^11 + 2804*x^10 - 2789*x^9 + 2362*x^8 - 1740*x^7 + 1154*x^6 - 709*x^5 + 393*x^4 - 179*x^3 + 62*x^2 - 15*x + 9, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 46 x^{16} - 164 x^{15} + 444 x^{14} - 952 x^{13} + 1655 x^{12} - 2364 x^{11} + 2804 x^{10} - 2789 x^{9} + 2362 x^{8} - 1740 x^{7} + 1154 x^{6} - 709 x^{5} + 393 x^{4} - 179 x^{3} + 62 x^{2} - 15 x + 9 \)

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:  \(-489415464119070561799=-\,199^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.11$
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:  $199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{1881} a^{16} - \frac{8}{1881} a^{15} + \frac{62}{627} a^{14} + \frac{92}{1881} a^{13} - \frac{151}{1881} a^{12} - \frac{3}{209} a^{11} - \frac{29}{1881} a^{10} - \frac{119}{1881} a^{9} + \frac{274}{1881} a^{8} - \frac{230}{627} a^{7} + \frac{215}{1881} a^{6} - \frac{43}{99} a^{5} - \frac{149}{627} a^{4} + \frac{193}{1881} a^{3} - \frac{362}{1881} a^{2} - \frac{64}{627} a + \frac{24}{209}$, $\frac{1}{182457} a^{17} + \frac{40}{182457} a^{16} - \frac{386}{5529} a^{15} - \frac{1802}{16587} a^{14} - \frac{3259}{182457} a^{13} + \frac{376}{20273} a^{12} + \frac{21874}{182457} a^{11} + \frac{7894}{182457} a^{10} + \frac{26539}{182457} a^{9} + \frac{3736}{60819} a^{8} + \frac{39200}{182457} a^{7} + \frac{50258}{182457} a^{6} + \frac{7888}{60819} a^{5} - \frac{4897}{16587} a^{4} - \frac{1757}{182457} a^{3} - \frac{24457}{60819} a^{2} + \frac{13093}{60819} a + \frac{8049}{20273}$

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:  \( 1021.61263769 \)
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

\(\Q(\sqrt{-199}) \), 3.1.199.1 x3, 6.0.7880599.1, 9.1.1568239201.1 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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$199$199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$