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Properties

Label 18.8.78063323494...8859.1
Degree $18$
Signature $[8, 5]$
Discriminant $-\,3^{27}\cdot 73^{2}\cdot 577^{3}$
Root discriminant $24.15$
Ramified primes $3, 73, 577$
Class number $1$
Class group Trivial
Galois group 18T879

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-233, -420, -69, 1155, 1161, -1530, -431, 330, -195, 401, -213, 12, 22, 0, 21, -8, 0, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 8*x^15 + 21*x^14 + 22*x^12 + 12*x^11 - 213*x^10 + 401*x^9 - 195*x^8 + 330*x^7 - 431*x^6 - 1530*x^5 + 1161*x^4 + 1155*x^3 - 69*x^2 - 420*x - 233)
 
gp: K = bnfinit(x^18 - 3*x^17 - 8*x^15 + 21*x^14 + 22*x^12 + 12*x^11 - 213*x^10 + 401*x^9 - 195*x^8 + 330*x^7 - 431*x^6 - 1530*x^5 + 1161*x^4 + 1155*x^3 - 69*x^2 - 420*x - 233, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} - 8 x^{15} + 21 x^{14} + 22 x^{12} + 12 x^{11} - 213 x^{10} + 401 x^{9} - 195 x^{8} + 330 x^{7} - 431 x^{6} - 1530 x^{5} + 1161 x^{4} + 1155 x^{3} - 69 x^{2} - 420 x - 233 \)

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:  \(-7806332349433625305658859=-\,3^{27}\cdot 73^{2}\cdot 577^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $24.15$
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:  $3, 73, 577$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{97248596570002128332659590413} a^{17} - \frac{34853493596469430154808157740}{97248596570002128332659590413} a^{16} - \frac{33379827541235703908055378722}{97248596570002128332659590413} a^{15} - \frac{33697296789973720460351513516}{97248596570002128332659590413} a^{14} - \frac{20636689208969471648747097184}{97248596570002128332659590413} a^{13} + \frac{35486555316094121897596367942}{97248596570002128332659590413} a^{12} + \frac{6799305450231094014294343975}{97248596570002128332659590413} a^{11} - \frac{42137267709594961670365490207}{97248596570002128332659590413} a^{10} + \frac{20655281452595749551880443692}{97248596570002128332659590413} a^{9} - \frac{143258706799453631650140362}{97248596570002128332659590413} a^{8} - \frac{15732643696018810216803122571}{97248596570002128332659590413} a^{7} - \frac{45854133671360268308608357152}{97248596570002128332659590413} a^{6} - \frac{24135205327466523548826248803}{97248596570002128332659590413} a^{5} - \frac{48203177330211819496316854753}{97248596570002128332659590413} a^{4} - \frac{38658262988267129008789979591}{97248596570002128332659590413} a^{3} - \frac{3694102315773599203385910628}{97248596570002128332659590413} a^{2} - \frac{43863224351949757116454096019}{97248596570002128332659590413} a + \frac{20813095834419006074689623813}{97248596570002128332659590413}$

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:  \( 389761.820311 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T879:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 331776
The 360 conjugacy class representatives for t18n879 are not computed
Character table for t18n879 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.22384826361.1

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 $18$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ $18$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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
3Data not computed
73Data not computed
577Data not computed