Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^9 + 1)
gp: K = bnfinit(x^18 - x^9 + 1, 1)
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]);
\( x^{18} - x^{9} + 1 \)
sage: K.defining_polynomial()
gp: K.pol
magma: DefiningPolynomial(K);
Invariants
| Degree: | $18$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
| |
| Signature: | $[0, 9]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
| |
| Discriminant: |
\(-2954312706550833698643\)
\(\medspace = -\,3^{45}\)
| sage: K.disc()
gp: K.disc
magma: Discriminant(Integers(K));
| |
| Root discriminant: | \(15.59\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
| |
| Ramified primes: |
\(3\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(Integers(K)));
| |
| $\card{ \Gal(K/\Q) }$: | $18$ | ||
| 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. | |||
| Reflex fields: | unavailable$^{256}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
| Monogenic: | Yes | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
sage: K.class_group().invariants()
gp: K.clgp
magma: ClassGroup(K);
Unit group
sage: UK = K.unit_group()
magma: UK, f := UnitGroup(K);
| Rank: | $8$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
| |
| Torsion generator: |
\( a \)
(order $54$)
| sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
| |
| Fundamental units: |
$a^{13}+a$, $a^{15}-a^{9}+1$, $a^{14}+a^{9}-a^{5}$, $a^{13}+a^{9}$, $a^{8}-a^{7}$, $a^{16}+a^{9}-a^{7}$, $a^{4}+a^{2}$, $a^{10}+a^{9}-a$
| sage: UK.fundamental_units()
gp: K.fu
magma: [K!f(g): g in Generators(UK)];
| |
| Regulator: | \( 40934.0329443 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
|
Class number formula
Galois group
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
magma: GaloisGroup(K);
| 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 | $18$ | ${\href{/padicField/7.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/13.9.0.1}{9} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | $18$ | $18$ | ${\href{/padicField/31.9.0.1}{9} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | $18$ | ${\href{/padicField/43.9.0.1}{9} }^{2}$ | $18$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | $18$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
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]])
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];
Local algebras for ramified primes
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $45$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| * | 1.3.2t1.a.a | $1$ | $ 3 $ | \(\Q(\sqrt{-3}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.27.9t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.a | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
| * | 1.27.9t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.b | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
| * | 1.9.3t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.27.9t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.c | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
| * | 1.27.9t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.d | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 1.27.9t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.e | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
| * | 1.27.9t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})^+\) | $C_9$ (as 9T1) | $0$ | $1$ |
| * | 1.27.18t1.a.f | $1$ | $ 3^{3}$ | \(\Q(\zeta_{27})\) | $C_{18}$ (as 18T1) | $0$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.