magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -9, 45, 120, -330, -462, 924, 792, -1287, -715, 1001, 364, -455, -105, 120, 16, -17, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1)
gp: K = bnfinit(x^18 - x^17 - 17*x^16 + 16*x^15 + 120*x^14 - 105*x^13 - 455*x^12 + 364*x^11 + 1001*x^10 - 715*x^9 - 1287*x^8 + 792*x^7 + 924*x^6 - 462*x^5 - 330*x^4 + 120*x^3 + 45*x^2 - 9*x - 1, 1)
Normalized defining polynomial
\( x^{18} - x^{17} - 17 x^{16} + 16 x^{15} + 120 x^{14} - 105 x^{13} - 455 x^{12} + 364 x^{11} + 1001 x^{10} - 715 x^{9} - 1287 x^{8} + 792 x^{7} + 924 x^{6} - 462 x^{5} - 330 x^{4} + 120 x^{3} + 45 x^{2} - 9 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $18$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[18, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(456487940826035155404146917=37^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $30.27$ | 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: | $37$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(37\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{37}(1,·)$, $\chi_{37}(3,·)$, $\chi_{37}(4,·)$, $\chi_{37}(7,·)$, $\chi_{37}(9,·)$, $\chi_{37}(10,·)$, $\chi_{37}(11,·)$, $\chi_{37}(12,·)$, $\chi_{37}(16,·)$, $\chi_{37}(21,·)$, $\chi_{37}(25,·)$, $\chi_{37}(26,·)$, $\chi_{37}(27,·)$, $\chi_{37}(28,·)$, $\chi_{37}(30,·)$, $\chi_{37}(33,·)$, $\chi_{37}(34,·)$, $\chi_{37}(36,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
Trivial group, which has order $1$ (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: | $17$ | 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: | \( 24199468.4506 \) (assuming GRH) | 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
| \(\Q(\sqrt{37}) \), 3.3.1369.1, 6.6.69343957.1, 9.9.3512479453921.1 |
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 | $18$ | ${\href{/LocalNumberField/3.9.0.1}{9} }^{2}$ | $18$ | ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ | $18$ | $18$ | $18$ | ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{9}$ | R | ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{9}$ | ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ | ${\href{/LocalNumberField/53.9.0.1}{9} }^{2}$ | $18$ |
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 |
|---|---|---|---|---|---|---|---|
| 37 | Data not computed | ||||||