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Properties

Label 18.14.1217816758...9257.1
Degree $18$
Signature $[14, 2]$
Discriminant $3^{24}\cdot 73^{3}\cdot 577^{4}$
Root discriminant $36.33$
Ramified primes $3, 73, 577$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1151, -3066, 1950, 7480, -2106, -2724, 4129, -8442, -3309, 10051, -21, -3570, 425, 360, -39, 17, -9, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 9*x^16 + 17*x^15 - 39*x^14 + 360*x^13 + 425*x^12 - 3570*x^11 - 21*x^10 + 10051*x^9 - 3309*x^8 - 8442*x^7 + 4129*x^6 - 2724*x^5 - 2106*x^4 + 7480*x^3 + 1950*x^2 - 3066*x - 1151)
 
gp: K = bnfinit(x^18 - 3*x^17 - 9*x^16 + 17*x^15 - 39*x^14 + 360*x^13 + 425*x^12 - 3570*x^11 - 21*x^10 + 10051*x^9 - 3309*x^8 - 8442*x^7 + 4129*x^6 - 2724*x^5 - 2106*x^4 + 7480*x^3 + 1950*x^2 - 3066*x - 1151, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 9 x^{16} + 17 x^{15} - 39 x^{14} + 360 x^{13} + 425 x^{12} - 3570 x^{11} - 21 x^{10} + 10051 x^{9} - 3309 x^{8} - 8442 x^{7} + 4129 x^{6} - 2724 x^{5} - 2106 x^{4} + 7480 x^{3} + 1950 x^{2} - 3066 x - 1151 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[14, 2]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12178167588536804870357659257=3^{24}\cdot 73^{3}\cdot 577^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.33$
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:  $3, 73, 577$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{9} a^{12} + \frac{1}{9} a^{9} - \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{2} - \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} - \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{3} - \frac{2}{9} a$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{2}{9} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{81} a^{15} - \frac{1}{27} a^{14} + \frac{1}{27} a^{13} - \frac{2}{27} a^{11} + \frac{4}{27} a^{10} + \frac{2}{81} a^{9} - \frac{10}{27} a^{8} - \frac{4}{27} a^{7} + \frac{1}{9} a^{6} - \frac{10}{27} a^{4} + \frac{7}{81} a^{3} - \frac{2}{27} a^{2} + \frac{1}{3} a + \frac{35}{81}$, $\frac{1}{81} a^{16} + \frac{1}{27} a^{14} + \frac{1}{27} a^{12} + \frac{1}{27} a^{11} + \frac{2}{81} a^{10} + \frac{4}{27} a^{9} + \frac{11}{27} a^{8} + \frac{1}{3} a^{7} + \frac{8}{27} a^{5} - \frac{2}{81} a^{4} - \frac{4}{27} a^{3} - \frac{1}{9} a^{2} - \frac{1}{81} a + \frac{2}{27}$, $\frac{1}{212948673276938559687} a^{17} + \frac{1014121530687227350}{212948673276938559687} a^{16} - \frac{39626102015684896}{212948673276938559687} a^{15} + \frac{3681454150417396376}{70982891092312853229} a^{14} - \frac{606752179001631890}{23660963697437617743} a^{13} - \frac{1567923954902276}{1339299831930431193} a^{12} - \frac{28204706467699377190}{212948673276938559687} a^{11} + \frac{1280791536967526705}{11207824909312555773} a^{10} + \frac{35085874387272979225}{212948673276938559687} a^{9} - \frac{10235559450559688263}{23660963697437617743} a^{8} + \frac{3095428456605200689}{70982891092312853229} a^{7} - \frac{729040971121650976}{3735941636437518591} a^{6} - \frac{92052516229966068089}{212948673276938559687} a^{5} + \frac{67297989813892194160}{212948673276938559687} a^{4} + \frac{86249292847638118058}{212948673276938559687} a^{3} - \frac{635307359277391618}{212948673276938559687} a^{2} + \frac{14548831801991048999}{212948673276938559687} a + \frac{31633931876213479216}{212948673276938559687}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $15$
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:  \( 65556333.5914 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 siblings: 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 $18$ $18$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/41.12.0.1}{12} }{,}\,{\href{/LocalNumberField/41.6.0.1}{6} }$ $18$ $18$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
73Data not computed
577Data not computed