Properties

Label 18.0.54803868577...5939.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,19^{17}$
Root discriminant $16.13$
Ramified prime $19$
Class number $1$
Class group Trivial
Galois Group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1)
gp: K = bnfinit(x^18 - x^17 + x^16 - x^15 + x^14 - x^13 + x^12 - x^11 + x^10 - x^9 + x^8 - x^7 + x^6 - x^5 + x^4 - x^3 + x^2 - x + 1, 1)

Normalized defining polynomial

\(x^{18} \) \(\mathstrut -\mathstrut x^{17} \) \(\mathstrut +\mathstrut x^{16} \) \(\mathstrut -\mathstrut x^{15} \) \(\mathstrut +\mathstrut x^{14} \) \(\mathstrut -\mathstrut x^{13} \) \(\mathstrut +\mathstrut x^{12} \) \(\mathstrut -\mathstrut x^{11} \) \(\mathstrut +\mathstrut x^{10} \) \(\mathstrut -\mathstrut x^{9} \) \(\mathstrut +\mathstrut x^{8} \) \(\mathstrut -\mathstrut x^{7} \) \(\mathstrut +\mathstrut x^{6} \) \(\mathstrut -\mathstrut x^{5} \) \(\mathstrut +\mathstrut x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut +\mathstrut x^{2} \) \(\mathstrut -\mathstrut x \) \(\mathstrut +\mathstrut 1 \)

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:  \(-5480386857784802185939=-\,19^{17}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $16.13$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $19$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(19\)
Dirichlet character group:    $\lbrace$$\chi_{19}(1,·)$, $\chi_{19}(2,·)$, $\chi_{19}(3,·)$, $\chi_{19}(4,·)$, $\chi_{19}(5,·)$, $\chi_{19}(6,·)$, $\chi_{19}(7,·)$, $\chi_{19}(8,·)$, $\chi_{19}(9,·)$, $\chi_{19}(10,·)$, $\chi_{19}(11,·)$, $\chi_{19}(12,·)$, $\chi_{19}(13,·)$, $\chi_{19}(14,·)$, $\chi_{19}(15,·)$, $\chi_{19}(16,·)$, $\chi_{19}(17,·)$, $\chi_{19}(18,·)$$\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 $38$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Fundamental units:  \( a^{12} - a^{11} \),  \( a^{12} + a^{2} \),  \( a^{7} - a^{2} \),  \( a^{16} - a^{9} + a^{4} \),  \( a^{13} - a^{12} + a^{11} - a^{10} + a^{9} - a^{8} + a^{7} - a^{6} + a^{5} - a^{4} + a^{3} - a^{2} + a \),  \( a^{9} - a^{8} + a^{7} \),  \( a^{17} - a^{16} + a^{15} - a^{14} - a^{12} + a^{11} + a^{9} - a^{8} + a^{7} - a^{6} - a^{4} + a^{3} + a - 1 \),  \( a^{17} - 1 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 22305.8950792 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$C_{18}$ (as 18T1):

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

\(\Q(\sqrt{-19}) \), 3.3.361.1, 6.0.2476099.1, \(\Q(\zeta_{19})^+\)

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 $18$ $18$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ $18$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ $18$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{2}$ $18$ $18$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed