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Properties

Label 18.2.43614784164...5344.1
Degree $18$
Signature $[2, 8]$
Discriminant $2^{18}\cdot 3^{27}\cdot 1297^{3}$
Root discriminant $34.32$
Ramified primes $2, 3, 1297$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T767

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-35019, 0, -51273, 0, -11610, 0, 9846, 0, 2241, 0, -882, 0, 78, 0, 54, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 15*x^16 + 54*x^14 + 78*x^12 - 882*x^10 + 2241*x^8 + 9846*x^6 - 11610*x^4 - 51273*x^2 - 35019)
 
gp: K = bnfinit(x^18 - 15*x^16 + 54*x^14 + 78*x^12 - 882*x^10 + 2241*x^8 + 9846*x^6 - 11610*x^4 - 51273*x^2 - 35019, 1)
 

Normalized defining polynomial

\( x^{18} - 15 x^{16} + 54 x^{14} + 78 x^{12} - 882 x^{10} + 2241 x^{8} + 9846 x^{6} - 11610 x^{4} - 51273 x^{2} - 35019 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[2, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(4361478416459465406297145344=2^{18}\cdot 3^{27}\cdot 1297^{3}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.32$
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:  $2, 3, 1297$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{3} a^{8}$, $\frac{1}{3} a^{9}$, $\frac{1}{3} a^{10}$, $\frac{1}{3} a^{11}$, $\frac{1}{63} a^{12} + \frac{1}{7} a^{10} - \frac{2}{21} a^{8} + \frac{1}{21} a^{6} + \frac{1}{7} a^{4} - \frac{2}{7} a^{2} - \frac{3}{7}$, $\frac{1}{63} a^{13} + \frac{1}{7} a^{11} - \frac{2}{21} a^{9} + \frac{1}{21} a^{7} + \frac{1}{7} a^{5} - \frac{2}{7} a^{3} - \frac{3}{7} a$, $\frac{1}{693} a^{14} + \frac{2}{693} a^{12} + \frac{19}{231} a^{10} - \frac{20}{231} a^{8} + \frac{31}{231} a^{6} + \frac{19}{77} a^{4} + \frac{4}{77} a^{2} - \frac{1}{11}$, $\frac{1}{693} a^{15} + \frac{2}{693} a^{13} + \frac{19}{231} a^{11} - \frac{20}{231} a^{9} + \frac{31}{231} a^{7} + \frac{19}{77} a^{5} + \frac{4}{77} a^{3} - \frac{1}{11} a$, $\frac{1}{87372790659} a^{16} + \frac{10587950}{87372790659} a^{14} + \frac{6127535}{87372790659} a^{12} - \frac{94406162}{882553441} a^{10} + \frac{112546129}{29124263553} a^{8} + \frac{4136509693}{29124263553} a^{6} - \frac{2389789356}{9708087851} a^{4} + \frac{421049719}{1386869693} a^{2} + \frac{542879395}{1386869693}$, $\frac{1}{87372790659} a^{17} + \frac{10587950}{87372790659} a^{15} + \frac{6127535}{87372790659} a^{13} - \frac{94406162}{882553441} a^{11} + \frac{112546129}{29124263553} a^{9} + \frac{4136509693}{29124263553} a^{7} - \frac{2389789356}{9708087851} a^{5} + \frac{421049719}{1386869693} a^{3} + \frac{542879395}{1386869693} a$

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:  $9$
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:  \( 2758140.27418 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T767:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 82944
The 110 conjugacy class representatives for t18n767 are not computed
Character table for t18n767 is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.5.689278977.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.12.0.1}{12} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.6.0.1}{6} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
2Data not computed
3Data not computed
1297Data not computed