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Properties

Label 18.6.19109609009...4193.1
Degree $18$
Signature $[6, 6]$
Discriminant $7^{12}\cdot 673^{5}$
Root discriminant $22.33$
Ramified primes $7, 673$
Class number $1$
Class group Trivial
Galois group 18T188

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -15, 0, 217, -242, -500, 1061, -327, -596, 355, 358, -446, 80, 123, -73, -7, 21, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 8*x^17 + 21*x^16 - 7*x^15 - 73*x^14 + 123*x^13 + 80*x^12 - 446*x^11 + 358*x^10 + 355*x^9 - 596*x^8 - 327*x^7 + 1061*x^6 - 500*x^5 - 242*x^4 + 217*x^3 - 15*x - 1)
 
gp: K = bnfinit(x^18 - 8*x^17 + 21*x^16 - 7*x^15 - 73*x^14 + 123*x^13 + 80*x^12 - 446*x^11 + 358*x^10 + 355*x^9 - 596*x^8 - 327*x^7 + 1061*x^6 - 500*x^5 - 242*x^4 + 217*x^3 - 15*x - 1, 1)
 

Normalized defining polynomial

\( x^{18} - 8 x^{17} + 21 x^{16} - 7 x^{15} - 73 x^{14} + 123 x^{13} + 80 x^{12} - 446 x^{11} + 358 x^{10} + 355 x^{9} - 596 x^{8} - 327 x^{7} + 1061 x^{6} - 500 x^{5} - 242 x^{4} + 217 x^{3} - 15 x - 1 \)

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:  \(1910960900904599329104193=7^{12}\cdot 673^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.33$
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:  $7, 673$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{8015230320946787} a^{17} - \frac{765895450117990}{8015230320946787} a^{16} - \frac{1558970287650453}{8015230320946787} a^{15} + \frac{2491773779091233}{8015230320946787} a^{14} - \frac{3455968604858131}{8015230320946787} a^{13} + \frac{143085369465550}{8015230320946787} a^{12} + \frac{796180848404152}{8015230320946787} a^{11} - \frac{2434573518676336}{8015230320946787} a^{10} - \frac{2762843316835891}{8015230320946787} a^{9} + \frac{2737066827134848}{8015230320946787} a^{8} + \frac{1470740661369989}{8015230320946787} a^{7} + \frac{2299957097971810}{8015230320946787} a^{6} - \frac{93460105307053}{8015230320946787} a^{5} + \frac{1116251475620353}{8015230320946787} a^{4} - \frac{3675797975858210}{8015230320946787} a^{3} + \frac{2747112378351382}{8015230320946787} a^{2} + \frac{231214536626331}{8015230320946787} a - \frac{2199197448661229}{8015230320946787}$

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:  $11$
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
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 84423.3379433 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T188:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 648
The 88 conjugacy class representatives for t18n188 are not computed
Character table for t18n188 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 6.2.1615873.1, 9.9.53286643921.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ $18$ R $18$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ $18$ $18$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ $18$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{4}$ $18$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ $18$

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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
673Data not computed