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Properties

Label 18.8.70530723208...7031.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{24}\cdot 7^{12}\cdot 71^{5}$
Root discriminant $51.74$
Ramified primes $3, 7, 71$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T459

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-881, -12381, 7236, 293774, 225774, -258648, 77273, 134298, -145626, 66689, -3681, -12102, 8755, -3504, 960, -220, 36, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 36*x^16 - 220*x^15 + 960*x^14 - 3504*x^13 + 8755*x^12 - 12102*x^11 - 3681*x^10 + 66689*x^9 - 145626*x^8 + 134298*x^7 + 77273*x^6 - 258648*x^5 + 225774*x^4 + 293774*x^3 + 7236*x^2 - 12381*x - 881)
 
gp: K = bnfinit(x^18 - 6*x^17 + 36*x^16 - 220*x^15 + 960*x^14 - 3504*x^13 + 8755*x^12 - 12102*x^11 - 3681*x^10 + 66689*x^9 - 145626*x^8 + 134298*x^7 + 77273*x^6 - 258648*x^5 + 225774*x^4 + 293774*x^3 + 7236*x^2 - 12381*x - 881, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 36 x^{16} - 220 x^{15} + 960 x^{14} - 3504 x^{13} + 8755 x^{12} - 12102 x^{11} - 3681 x^{10} + 66689 x^{9} - 145626 x^{8} + 134298 x^{7} + 77273 x^{6} - 258648 x^{5} + 225774 x^{4} + 293774 x^{3} + 7236 x^{2} - 12381 x - 881 \)

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:  \(-7053072320828130442597556717031=-\,3^{24}\cdot 7^{12}\cdot 71^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $51.74$
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:  $3, 7, 71$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} + \frac{1}{3} a^{8} + \frac{1}{3} a^{5} + \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{3} a^{14} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} - \frac{1}{3} a^{9} + \frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{15} + \frac{1}{9} a^{13} - \frac{2}{9} a^{11} + \frac{1}{9} a^{10} + \frac{2}{9} a^{9} + \frac{2}{9} a^{8} + \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{3} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a^{2} + \frac{4}{9} a - \frac{4}{9}$, $\frac{1}{639} a^{16} - \frac{5}{213} a^{15} - \frac{80}{639} a^{14} - \frac{22}{213} a^{13} - \frac{92}{639} a^{12} - \frac{26}{639} a^{11} - \frac{82}{639} a^{10} - \frac{10}{639} a^{9} + \frac{31}{213} a^{8} + \frac{20}{213} a^{7} + \frac{110}{639} a^{6} + \frac{82}{213} a^{5} - \frac{263}{639} a^{4} - \frac{160}{639} a^{3} + \frac{139}{639} a^{2} + \frac{119}{639} a - \frac{37}{213}$, $\frac{1}{6082305325079479847700447419802958895710955343} a^{17} - \frac{1287768864488843943766246703965380027793343}{2027435108359826615900149139934319631903651781} a^{16} - \frac{174772824311368852981823444703204909686704897}{6082305325079479847700447419802958895710955343} a^{15} - \frac{173018006062495890391977171089264763725437520}{2027435108359826615900149139934319631903651781} a^{14} - \frac{423859810848654838573956542909724938641098319}{6082305325079479847700447419802958895710955343} a^{13} - \frac{445696993308600062271361854882778099538828627}{6082305325079479847700447419802958895710955343} a^{12} - \frac{534283839254864749509457951658596406636603113}{2027435108359826615900149139934319631903651781} a^{11} + \frac{430228224151164255175147259920852462306675826}{2027435108359826615900149139934319631903651781} a^{10} + \frac{2471794109028999649058167247403626489111817254}{6082305325079479847700447419802958895710955343} a^{9} - \frac{356726715755108223743825256759483522987054547}{6082305325079479847700447419802958895710955343} a^{8} + \frac{1851642687044049501116754333523399534567217705}{6082305325079479847700447419802958895710955343} a^{7} - \frac{8013688125197701100596806937394931719662429}{2027435108359826615900149139934319631903651781} a^{6} + \frac{288225774306377835360103704071115768324691868}{2027435108359826615900149139934319631903651781} a^{5} + \frac{187050917740312920328229916979052492135102051}{6082305325079479847700447419802958895710955343} a^{4} + \frac{2877236554921607141300375990738311962769147658}{6082305325079479847700447419802958895710955343} a^{3} + \frac{786630961999934143980434690463723338241677545}{6082305325079479847700447419802958895710955343} a^{2} + \frac{2664092374424995607457136722217235579393654587}{6082305325079479847700447419802958895710955343} a - \frac{1158752265642307710394489630148919306092059087}{6082305325079479847700447419802958895710955343}$

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:  $12$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 415677511.077 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T459:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 96 conjugacy class representatives for t18n459 are not computed
Character table for t18n459 is not computed

Intermediate fields

3.3.3969.2, 3.3.3969.1, \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 9.9.62523502209.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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}$
7Data not computed
71Data not computed