magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^9 + 1)
gp: K = bnfinit(x^18 - x^9 + 1, 1)
Normalized defining polynomial
\( x^{18} - x^{9} + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 9]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-2954312706550833698643=-\,3^{45}\) | magma: Discriminant(K);
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.59$ | magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3$ | magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(27=3^{3}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{27}(1,·)$, $\chi_{27}(2,·)$, $\chi_{27}(4,·)$, $\chi_{27}(5,·)$, $\chi_{27}(7,·)$, $\chi_{27}(8,·)$, $\chi_{27}(10,·)$, $\chi_{27}(11,·)$, $\chi_{27}(13,·)$, $\chi_{27}(14,·)$, $\chi_{27}(16,·)$, $\chi_{27}(17,·)$, $\chi_{27}(19,·)$, $\chi_{27}(20,·)$, $\chi_{27}(22,·)$, $\chi_{27}(23,·)$, $\chi_{27}(25,·)$, $\chi_{27}(26,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
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: | $8$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( a \) (order $54$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| Fundamental units: | \( a^{13} + a \), \( a^{15} - a^{9} + 1 \), \( a^{14} + a^{9} - a^{5} \), \( a^{13} + a^{9} \), \( a^{8} - a^{7} \), \( a^{16} + a^{9} - a^{7} \), \( a^{4} + a^{2} \), \( a^{10} + a^{9} - a \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 40934.0329443 \) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
| A cyclic group of order 18 |
| The 18 conjugacy class representatives for $C_{18}$ |
| Character table for $C_{18}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{9})\), \(\Q(\zeta_{27})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
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 | $18$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/LocalNumberField/31.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ | $18$ | ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ | $18$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
| 3 | Data not computed | ||||||