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Properties

Label 18.0.33954867934...9136.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 211^{9}$
Root discriminant $23.06$
Ramified primes $2, 211$
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![81, 513, 1579, -2226, -3036, 2694, 3354, -4074, 513, 1953, -1913, 534, 444, -588, 372, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 372*x^14 - 588*x^13 + 444*x^12 + 534*x^11 - 1913*x^10 + 1953*x^9 + 513*x^8 - 4074*x^7 + 3354*x^6 + 2694*x^5 - 3036*x^4 - 2226*x^3 + 1579*x^2 + 513*x + 81)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 372*x^14 - 588*x^13 + 444*x^12 + 534*x^11 - 1913*x^10 + 1953*x^9 + 513*x^8 - 4074*x^7 + 3354*x^6 + 2694*x^5 - 3036*x^4 - 2226*x^3 + 1579*x^2 + 513*x + 81, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 372 x^{14} - 588 x^{13} + 444 x^{12} + 534 x^{11} - 1913 x^{10} + 1953 x^{9} + 513 x^{8} - 4074 x^{7} + 3354 x^{6} + 2694 x^{5} - 3036 x^{4} - 2226 x^{3} + 1579 x^{2} + 513 x + 81 \)

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:  \(-3395486793483559622619136=-\,2^{12}\cdot 211^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $23.06$
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:  $2, 211$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{5} - \frac{1}{6} a$, $\frac{1}{84} a^{10} - \frac{5}{84} a^{9} + \frac{1}{28} a^{8} + \frac{3}{14} a^{7} - \frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{2}{7} a^{3} - \frac{1}{12} a^{2} + \frac{29}{84} a - \frac{1}{28}$, $\frac{1}{84} a^{11} + \frac{1}{14} a^{9} - \frac{3}{28} a^{8} - \frac{5}{28} a^{7} - \frac{13}{28} a^{4} - \frac{13}{84} a^{3} - \frac{1}{14} a^{2} - \frac{1}{7} a - \frac{5}{28}$, $\frac{1}{84} a^{12} - \frac{1}{12} a^{9} + \frac{3}{28} a^{8} + \frac{3}{14} a^{7} + \frac{1}{28} a^{5} + \frac{29}{84} a^{4} + \frac{3}{14} a^{3} - \frac{1}{7} a^{2} - \frac{5}{12} a - \frac{2}{7}$, $\frac{1}{84} a^{13} + \frac{1}{42} a^{9} - \frac{1}{28} a^{8} - \frac{3}{14} a^{6} - \frac{17}{42} a^{5} + \frac{13}{28} a^{4} - \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{25}{84} a + \frac{1}{4}$, $\frac{1}{1176} a^{14} - \frac{1}{168} a^{13} - \frac{1}{196} a^{12} + \frac{1}{1176} a^{11} - \frac{5}{1176} a^{10} + \frac{47}{588} a^{9} - \frac{1}{56} a^{8} + \frac{51}{392} a^{7} - \frac{97}{1176} a^{6} + \frac{5}{84} a^{5} + \frac{149}{392} a^{4} + \frac{425}{1176} a^{3} - \frac{5}{84} a^{2} + \frac{569}{1176} a + \frac{127}{392}$, $\frac{1}{1176} a^{15} + \frac{1}{1176} a^{13} + \frac{1}{1176} a^{12} + \frac{1}{588} a^{11} + \frac{1}{392} a^{10} - \frac{1}{24} a^{9} + \frac{2}{49} a^{8} + \frac{67}{588} a^{7} + \frac{1}{8} a^{6} + \frac{335}{1176} a^{5} + \frac{61}{147} a^{4} + \frac{67}{168} a^{3} + \frac{185}{392} a^{2} + \frac{257}{588} a - \frac{25}{56}$, $\frac{1}{102312} a^{16} - \frac{1}{12789} a^{15} - \frac{17}{102312} a^{14} + \frac{37}{14616} a^{13} + \frac{74}{12789} a^{12} + \frac{25}{102312} a^{11} - \frac{179}{102312} a^{10} - \frac{367}{51156} a^{9} + \frac{6511}{51156} a^{8} + \frac{6065}{102312} a^{7} - \frac{15139}{102312} a^{6} - \frac{14543}{51156} a^{5} + \frac{11297}{102312} a^{4} - \frac{9727}{102312} a^{3} - \frac{22865}{51156} a^{2} + \frac{47435}{102312} a + \frac{655}{5684}$, $\frac{1}{23224824} a^{17} + \frac{5}{1105944} a^{16} - \frac{835}{3870804} a^{15} - \frac{103}{3870804} a^{14} + \frac{3467}{1935402} a^{13} + \frac{3169}{967701} a^{12} + \frac{1291}{322567} a^{11} + \frac{13283}{3870804} a^{10} - \frac{1436429}{23224824} a^{9} - \frac{1611839}{7741608} a^{8} + \frac{164914}{967701} a^{7} + \frac{261049}{3870804} a^{6} + \frac{109859}{3870804} a^{5} - \frac{496345}{3870804} a^{4} - \frac{477}{5684} a^{3} - \frac{319915}{1935402} a^{2} - \frac{8527697}{23224824} a - \frac{155513}{2580536}$

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:  \( 226324.427557 \)
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

\(\Q(\sqrt{-211}) \), 3.1.211.1 x3, 6.0.9393931.1, 9.1.126855644224.1 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 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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
$2$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
211Data not computed