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Properties

Label 18.6.81166666365...4481.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 113^{2}\cdot 2143^{2}$
Root discriminant $14.51$
Ramified primes $7, 113, 2143$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T703

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 2, -11, -1, 15, -64, -63, 76, 62, -63, -32, 40, 3, -16, 8, 5, -5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 5*x^16 + 5*x^15 + 8*x^14 - 16*x^13 + 3*x^12 + 40*x^11 - 32*x^10 - 63*x^9 + 62*x^8 + 76*x^7 - 63*x^6 - 64*x^5 + 15*x^4 - x^3 - 11*x^2 + 2*x + 1)
 
gp: K = bnfinit(x^18 - x^17 - 5*x^16 + 5*x^15 + 8*x^14 - 16*x^13 + 3*x^12 + 40*x^11 - 32*x^10 - 63*x^9 + 62*x^8 + 76*x^7 - 63*x^6 - 64*x^5 + 15*x^4 - x^3 - 11*x^2 + 2*x + 1, 1)
 

Normalized defining polynomial

\( x^{18} - x^{17} - 5 x^{16} + 5 x^{15} + 8 x^{14} - 16 x^{13} + 3 x^{12} + 40 x^{11} - 32 x^{10} - 63 x^{9} + 62 x^{8} + 76 x^{7} - 63 x^{6} - 64 x^{5} + 15 x^{4} - x^{3} - 11 x^{2} + 2 x + 1 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(811666663658785884481=7^{12}\cdot 113^{2}\cdot 2143^{2}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $14.51$
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:  $7, 113, 2143$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{1299296733253} a^{17} + \frac{33521752109}{1299296733253} a^{16} + \frac{436479268642}{1299296733253} a^{15} + \frac{562426137400}{1299296733253} a^{14} + \frac{248834323520}{1299296733253} a^{13} + \frac{497481452072}{1299296733253} a^{12} - \frac{625499523959}{1299296733253} a^{11} + \frac{202045890070}{1299296733253} a^{10} + \frac{460992765112}{1299296733253} a^{9} - \frac{500467628483}{1299296733253} a^{8} + \frac{259769783403}{1299296733253} a^{7} + \frac{459732286065}{1299296733253} a^{6} - \frac{370609195126}{1299296733253} a^{5} - \frac{391639059631}{1299296733253} a^{4} + \frac{550171922382}{1299296733253} a^{3} - \frac{133647538659}{1299296733253} a^{2} - \frac{183380852981}{1299296733253} a + \frac{211168412244}{1299296733253}$

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:  \( 2267.33209131 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T703:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 41472
The 55 conjugacy class representatives for t18n703 are not computed
Character table for t18n703 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.3.252121807.1

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

Sibling fields

Degree 12 sibling: data not computed
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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.6.0.1}{6} }$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.6.0.1}{6} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
$7$7.6.4.3$x^{6} + 56 x^{3} + 1323$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
$113$$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
$\Q_{113}$$x + 3$$1$$1$$0$Trivial$[\ ]$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.2.0.1$x^{2} - x + 10$$1$$2$$0$$C_2$$[\ ]^{2}$
113.4.2.1$x^{4} + 2147 x^{2} + 1276900$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
2143Data not computed