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Properties

Label 18.6.53894585314...1989.1
Degree $18$
Signature $[6, 6]$
Discriminant $3^{18}\cdot 7^{14}\cdot 29^{5}$
Root discriminant $34.72$
Ramified primes $3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T282

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-18647, 64206, -53244, -54520, 109446, -30537, -43488, 33867, -4437, -5300, 2070, -414, 536, -72, -36, -9, -3, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^16 - 9*x^15 - 36*x^14 - 72*x^13 + 536*x^12 - 414*x^11 + 2070*x^10 - 5300*x^9 - 4437*x^8 + 33867*x^7 - 43488*x^6 - 30537*x^5 + 109446*x^4 - 54520*x^3 - 53244*x^2 + 64206*x - 18647)
 
gp: K = bnfinit(x^18 - 3*x^16 - 9*x^15 - 36*x^14 - 72*x^13 + 536*x^12 - 414*x^11 + 2070*x^10 - 5300*x^9 - 4437*x^8 + 33867*x^7 - 43488*x^6 - 30537*x^5 + 109446*x^4 - 54520*x^3 - 53244*x^2 + 64206*x - 18647, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{16} - 9 x^{15} - 36 x^{14} - 72 x^{13} + 536 x^{12} - 414 x^{11} + 2070 x^{10} - 5300 x^{9} - 4437 x^{8} + 33867 x^{7} - 43488 x^{6} - 30537 x^{5} + 109446 x^{4} - 54520 x^{3} - 53244 x^{2} + 64206 x - 18647 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5389458531481507431123541989=3^{18}\cdot 7^{14}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.72$
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:  $3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{80515068330588964776145125434119480213534} a^{17} + \frac{1359812116168475187171820348086564072775}{80515068330588964776145125434119480213534} a^{16} - \frac{3944718095000669424408841493239171419335}{40257534165294482388072562717059740106767} a^{15} - \frac{14573531698209097699584505390757870678087}{40257534165294482388072562717059740106767} a^{14} - \frac{38700146334692866089702273072372029925877}{80515068330588964776145125434119480213534} a^{13} + \frac{10077504215733436313473982853605004553631}{80515068330588964776145125434119480213534} a^{12} - \frac{5999386116182719026773748596032820147361}{80515068330588964776145125434119480213534} a^{11} - \frac{1397589451420938381023200570318922026273}{40257534165294482388072562717059740106767} a^{10} - \frac{11852699896405046185685931797251477186093}{80515068330588964776145125434119480213534} a^{9} - \frac{16142841800002240381389763903244789039953}{40257534165294482388072562717059740106767} a^{8} - \frac{30712147617528307460931453645595872591071}{80515068330588964776145125434119480213534} a^{7} - \frac{2173985209052332532696313675572829313443}{80515068330588964776145125434119480213534} a^{6} - \frac{25575429882125855839813580475731544272793}{80515068330588964776145125434119480213534} a^{5} + \frac{34912202955808366488613151717692874126437}{80515068330588964776145125434119480213534} a^{4} + \frac{19913598251330778637733382095404985495378}{40257534165294482388072562717059740106767} a^{3} + \frac{35010939030927601167973886176811292688401}{80515068330588964776145125434119480213534} a^{2} - \frac{16043899865915400687454097116062489224727}{40257534165294482388072562717059740106767} a - \frac{14501618187719469039667208856582191413757}{40257534165294482388072562717059740106767}$

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:  $11$
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:  \( 13284133.1042 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T282:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 1296
The 56 conjugacy class representatives for t18n282 are not computed
Character table for t18n282 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.69629.1, 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{29}$$x + 2$$1$$1$$0$Trivial$[\ ]$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.6.5.2$x^{6} + 58$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$