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Properties

Label 18.6.53894585314...1989.2
Degree $18$
Signature $[6, 6]$
Discriminant $3^{18}\cdot 7^{14}\cdot 29^{5}$
Root discriminant $34.72$
Ramified primes $3, 7, 29$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T766

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-29, 0, -753, 0, 1617, 0, 1919, 0, -642, 0, -1053, 0, -258, 0, 18, 0, 12, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 12*x^16 + 18*x^14 - 258*x^12 - 1053*x^10 - 642*x^8 + 1919*x^6 + 1617*x^4 - 753*x^2 - 29)
 
gp: K = bnfinit(x^18 + 12*x^16 + 18*x^14 - 258*x^12 - 1053*x^10 - 642*x^8 + 1919*x^6 + 1617*x^4 - 753*x^2 - 29, 1)
 

Normalized defining polynomial

\( x^{18} + 12 x^{16} + 18 x^{14} - 258 x^{12} - 1053 x^{10} - 642 x^{8} + 1919 x^{6} + 1617 x^{4} - 753 x^{2} - 29 \)

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:  \(5389458531481507431123541989=3^{18}\cdot 7^{14}\cdot 29^{5}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.72$
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:  $3, 7, 29$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{52} a^{14} - \frac{1}{13} a^{12} + \frac{1}{52} a^{10} - \frac{1}{26} a^{8} - \frac{5}{26} a^{6} - \frac{1}{2} a^{5} + \frac{9}{26} a^{4} - \frac{1}{2} a^{3} + \frac{23}{52} a^{2} - \frac{1}{2} a - \frac{1}{52}$, $\frac{1}{104} a^{15} - \frac{1}{104} a^{14} + \frac{11}{52} a^{13} - \frac{11}{52} a^{12} + \frac{1}{104} a^{11} - \frac{1}{104} a^{10} - \frac{1}{52} a^{9} + \frac{1}{52} a^{8} - \frac{5}{52} a^{7} - \frac{21}{52} a^{6} - \frac{1}{13} a^{5} + \frac{1}{13} a^{4} + \frac{49}{104} a^{3} + \frac{3}{104} a^{2} - \frac{1}{104} a + \frac{1}{104}$, $\frac{1}{3384641104} a^{16} - \frac{32048331}{3384641104} a^{14} - \frac{645850585}{3384641104} a^{12} + \frac{618860605}{3384641104} a^{10} - \frac{6152895}{423080138} a^{8} - \frac{505887417}{1692320552} a^{6} - \frac{1}{2} a^{5} + \frac{283991021}{3384641104} a^{4} - \frac{108618369}{1692320552} a^{2} - \frac{1}{2} a - \frac{252406155}{3384641104}$, $\frac{1}{6769282208} a^{17} - \frac{1}{6769282208} a^{16} - \frac{32048331}{6769282208} a^{15} + \frac{32048331}{6769282208} a^{14} + \frac{1046469967}{6769282208} a^{13} - \frac{1046469967}{6769282208} a^{12} - \frac{1073459947}{6769282208} a^{11} + \frac{1073459947}{6769282208} a^{10} + \frac{102693587}{423080138} a^{9} - \frac{102693587}{423080138} a^{8} - \frac{505887417}{3384641104} a^{7} - \frac{1186433135}{3384641104} a^{6} - \frac{3100650083}{6769282208} a^{5} + \frac{3100650083}{6769282208} a^{4} - \frac{954778645}{3384641104} a^{3} + \frac{954778645}{3384641104} a^{2} - \frac{252406155}{6769282208} a - \frac{3132234949}{6769282208}$

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:  \( 14769656.1486 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T766:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 144 conjugacy class representatives for t18n766 are not computed
Character table for t18n766 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 9.9.13632439166829.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.3.0.1}{3} }^{2}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ $18$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ R ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ $18$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$29$29.2.1.1$x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$
29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.4.0.1$x^{4} - x + 19$$1$$4$$0$$C_4$$[\ ]^{4}$
29.6.4.1$x^{6} + 232 x^{3} + 22707$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$