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Properties

Label 18.0.60436082803...1264.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{37}$
Root discriminant $27.06$
Ramified primes $2, 3$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
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![256, -2304, 8352, -16992, 24408, -26784, 23988, -18360, 13050, -7616, 4473, -1872, 1068, -288, 162, -24, 18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256)
 
gp: K = bnfinit(x^18 + 18*x^16 - 24*x^15 + 162*x^14 - 288*x^13 + 1068*x^12 - 1872*x^11 + 4473*x^10 - 7616*x^9 + 13050*x^8 - 18360*x^7 + 23988*x^6 - 26784*x^5 + 24408*x^4 - 16992*x^3 + 8352*x^2 - 2304*x + 256, 1)
 

Normalized defining polynomial

\( x^{18} + 18 x^{16} - 24 x^{15} + 162 x^{14} - 288 x^{13} + 1068 x^{12} - 1872 x^{11} + 4473 x^{10} - 7616 x^{9} + 13050 x^{8} - 18360 x^{7} + 23988 x^{6} - 26784 x^{5} + 24408 x^{4} - 16992 x^{3} + 8352 x^{2} - 2304 x + 256 \)

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:  \(-60436082803655481715851264=-\,2^{27}\cdot 3^{37}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.06$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} + \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{8} a^{8} - \frac{1}{4} a^{5} + \frac{1}{8} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{10} - \frac{1}{8} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{11} - \frac{1}{8} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{16} a^{12} - \frac{1}{16} a^{10} - \frac{1}{16} a^{8} + \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{13} + \frac{1}{32} a^{11} - \frac{1}{32} a^{9} + \frac{3}{32} a^{7} - \frac{1}{4} a^{5} + \frac{1}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{32} a^{14} - \frac{1}{32} a^{12} + \frac{1}{32} a^{10} + \frac{1}{32} a^{8} - \frac{1}{16} a^{6} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a$, $\frac{1}{1088} a^{15} - \frac{1}{1088} a^{14} + \frac{11}{1088} a^{13} + \frac{31}{1088} a^{12} + \frac{53}{1088} a^{11} - \frac{19}{1088} a^{10} - \frac{55}{1088} a^{9} - \frac{19}{1088} a^{8} + \frac{45}{544} a^{7} - \frac{1}{16} a^{6} - \frac{15}{272} a^{5} + \frac{11}{272} a^{4} - \frac{41}{136} a^{3} + \frac{7}{17} a^{2} - \frac{8}{17} a - \frac{3}{17}$, $\frac{1}{2176} a^{16} - \frac{3}{272} a^{14} + \frac{1}{272} a^{13} - \frac{9}{1088} a^{12} + \frac{7}{544} a^{10} + \frac{3}{68} a^{9} + \frac{37}{2176} a^{8} - \frac{5}{136} a^{7} - \frac{49}{544} a^{6} + \frac{15}{272} a^{5} + \frac{31}{544} a^{4} + \frac{67}{136} a^{3} - \frac{21}{136} a^{2} - \frac{11}{34} a + \frac{7}{17}$, $\frac{1}{10569569224448} a^{17} - \frac{758859609}{5284784612224} a^{16} + \frac{514208499}{2642392306112} a^{15} - \frac{14106836161}{1321196153056} a^{14} + \frac{21416486097}{5284784612224} a^{13} - \frac{5081856671}{203260946624} a^{12} - \frac{23642997}{685623328} a^{11} + \frac{81277697559}{1321196153056} a^{10} - \frac{439851939999}{10569569224448} a^{9} - \frac{88589790557}{5284784612224} a^{8} - \frac{193988823677}{2642392306112} a^{7} + \frac{12464565213}{660598076528} a^{6} - \frac{33790372685}{155434841536} a^{5} - \frac{221685412901}{1321196153056} a^{4} + \frac{113746523867}{660598076528} a^{3} - \frac{1780590869}{25407618328} a^{2} + \frac{11987875502}{41287379783} a - \frac{16729895994}{41287379783}$

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

Class group and class number

$C_{2}$, which has order $2$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 46631737.4813 \) (assuming GRH)
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

\(\Q(\sqrt{-6}) \), 3.1.216.1 x3, 6.0.1119744.1, 9.1.1586874322944.5 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 R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
3Data not computed