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Properties

Label 18.8.44348448637...4464.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,2^{12}\cdot 3^{27}\cdot 17^{5}$
Root discriminant $18.12$
Ramified primes $2, 3, 17$
Class number $1$
Class group Trivial
Galois group 18T400

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 12, -12, -102, -15, 282, -3, -213, -2, -60, 42, 39, -6, 15, -12, 3, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 3*x^16 - 12*x^15 + 15*x^14 - 6*x^13 + 39*x^12 + 42*x^11 - 60*x^10 - 2*x^9 - 213*x^8 - 3*x^7 + 282*x^6 - 15*x^5 - 102*x^4 - 12*x^3 + 12*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^18 - 3*x^17 + 3*x^16 - 12*x^15 + 15*x^14 - 6*x^13 + 39*x^12 + 42*x^11 - 60*x^10 - 2*x^9 - 213*x^8 - 3*x^7 + 282*x^6 - 15*x^5 - 102*x^4 - 12*x^3 + 12*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 3 x^{16} - 12 x^{15} + 15 x^{14} - 6 x^{13} + 39 x^{12} + 42 x^{11} - 60 x^{10} - 2 x^{9} - 213 x^{8} - 3 x^{7} + 282 x^{6} - 15 x^{5} - 102 x^{4} - 12 x^{3} + 12 x^{2} + 6 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 5]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-44348448637915901374464=-\,2^{12}\cdot 3^{27}\cdot 17^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $18.12$
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:  $2, 3, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{9} - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{2}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{9} - \frac{1}{9} a^{3} + \frac{1}{9}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{10} - \frac{1}{9} a^{4} + \frac{1}{9} a$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} - \frac{1}{9} a^{5} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{9} - \frac{1}{9} a^{6} + \frac{1}{9}$, $\frac{1}{9} a^{16} - \frac{1}{9} a^{10} - \frac{1}{9} a^{7} + \frac{1}{9} a$, $\frac{1}{6272943921} a^{17} + \frac{41317123}{6272943921} a^{16} - \frac{80612353}{2090981307} a^{15} - \frac{174978416}{6272943921} a^{14} + \frac{245463625}{6272943921} a^{13} - \frac{29012434}{696993769} a^{12} + \frac{138382468}{6272943921} a^{11} - \frac{873425948}{6272943921} a^{10} - \frac{309514748}{2090981307} a^{9} + \frac{445785896}{6272943921} a^{8} - \frac{815009833}{6272943921} a^{7} + \frac{60786551}{2090981307} a^{6} + \frac{3024063122}{6272943921} a^{5} + \frac{2259963182}{6272943921} a^{4} - \frac{368148662}{2090981307} a^{3} + \frac{74527097}{6272943921} a^{2} - \frac{526926790}{6272943921} a - \frac{128105770}{696993769}$

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:  $12$
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:  \( 29639.4320916 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T400:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2592
The 56 conjugacy class representatives for t18n400 are not computed
Character table for t18n400 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 6.4.334611.1, 9.5.9829532736.1

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 R R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ $18$

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$2.6.0.1$x^{6} - x + 1$$1$$6$$0$$C_6$$[\ ]^{6}$
2.12.12.11$x^{12} - 6 x^{10} - 73 x^{8} + 140 x^{6} + 79 x^{4} - 6 x^{2} + 57$$2$$6$$12$$A_4 \times C_2$$[2, 2]^{6}$
3Data not computed
$17$$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{17}$$x + 3$$1$$1$$0$Trivial$[\ ]$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
17.6.0.1$x^{6} - x + 12$$1$$6$$0$$C_6$$[\ ]^{6}$
17.6.5.1$x^{6} - 17$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$