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Properties

Label 18.0.50869871130...1703.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 23^{9}$
Root discriminant $20.75$
Ramified primes $3, 23$
Class number $3$
Class group $[3]$
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![575, 0, 234, 0, -1977, 0, 506, 0, 1134, 0, 177, 0, -57, 0, -18, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^14 - 57*x^12 + 177*x^10 + 1134*x^8 + 506*x^6 - 1977*x^4 + 234*x^2 + 575)
 
gp: K = bnfinit(x^18 - 18*x^14 - 57*x^12 + 177*x^10 + 1134*x^8 + 506*x^6 - 1977*x^4 + 234*x^2 + 575, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{14} - 57 x^{12} + 177 x^{10} + 1134 x^{8} + 506 x^{6} - 1977 x^{4} + 234 x^{2} + 575 \)

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:  \(-508698711308514601331703=-\,3^{24}\cdot 23^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $20.75$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{10} a^{11} - \frac{1}{5} a^{9} - \frac{1}{2} a^{8} - \frac{3}{10} a^{7} + \frac{2}{5} a^{5} - \frac{1}{2} a^{4} + \frac{1}{5} a^{3} - \frac{1}{2} a^{2} - \frac{1}{5} a$, $\frac{1}{10} a^{12} - \frac{1}{5} a^{10} - \frac{3}{10} a^{8} - \frac{1}{10} a^{6} + \frac{1}{5} a^{4} - \frac{1}{2} a^{3} + \frac{3}{10} a^{2} - \frac{1}{2}$, $\frac{1}{10} a^{13} - \frac{1}{5} a^{9} + \frac{3}{10} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{3}{10} a^{3} - \frac{1}{2} a^{2} + \frac{1}{10} a - \frac{1}{2}$, $\frac{1}{2530} a^{14} - \frac{1}{506} a^{12} + \frac{124}{1265} a^{10} + \frac{14}{1265} a^{8} - \frac{1}{2} a^{7} + \frac{35}{253} a^{6} - \frac{1}{2} a^{5} - \frac{193}{2530} a^{4} + \frac{533}{1265} a^{2} - \frac{1}{2} a + \frac{5}{22}$, $\frac{1}{2530} a^{15} - \frac{1}{506} a^{13} - \frac{1}{506} a^{11} + \frac{267}{1265} a^{9} + \frac{1109}{2530} a^{7} - \frac{1}{2} a^{6} - \frac{241}{506} a^{5} - \frac{1}{2} a^{4} + \frac{56}{253} a^{3} + \frac{47}{110} a$, $\frac{1}{2206364930} a^{16} - \frac{266471}{2206364930} a^{14} + \frac{451372}{157597495} a^{12} + \frac{90313675}{441272986} a^{10} - \frac{235732614}{1103182465} a^{8} + \frac{209987567}{2206364930} a^{6} - \frac{1}{2} a^{5} - \frac{43503043}{100289315} a^{4} + \frac{317828607}{1103182465} a^{2} - \frac{1}{2} a + \frac{1408749}{19185782}$, $\frac{1}{2206364930} a^{17} - \frac{266471}{2206364930} a^{15} + \frac{451372}{157597495} a^{13} + \frac{10295389}{2206364930} a^{11} + \frac{205540372}{1103182465} a^{9} - \frac{134511681}{441272986} a^{7} - \frac{1}{2} a^{6} - \frac{4689036}{20057863} a^{5} - \frac{123444379}{1103182465} a^{3} - \frac{1}{2} a^{2} + \frac{45415309}{95928910} a$

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

Class group and class number

$C_{3}$, which has order $3$

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:  \( 10047.0042383 \)
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, 9.1.148718980881.2 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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ 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
$3$3.9.12.3$x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$$3$$3$$12$$C_9$$[2]^{3}$
3.9.12.3$x^{9} + 6 x^{8} + 3 x^{6} + 9 x^{3} + 135$$3$$3$$12$$C_9$$[2]^{3}$
$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}$