magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -27, 0, 819, 0, -7371, 0, 30888, 0, -72930, 0, 107406, 0, -104652, 0, 69768, 0, -32319, 0, 10395, 0, -2277, 0, 324, 0, -27, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 27*x^25 + 324*x^23 - 2277*x^21 + 10395*x^19 - 32319*x^17 + 69768*x^15 - 104652*x^13 + 107406*x^11 - 72930*x^9 + 30888*x^7 - 7371*x^5 + 819*x^3 - 27*x - 1)
gp: K = bnfinit(x^27 - 27*x^25 + 324*x^23 - 2277*x^21 + 10395*x^19 - 32319*x^17 + 69768*x^15 - 104652*x^13 + 107406*x^11 - 72930*x^9 + 30888*x^7 - 7371*x^5 + 819*x^3 - 27*x - 1, 1)
Normalized defining polynomial
\( x^{27} - 27 x^{25} + 324 x^{23} - 2277 x^{21} + 10395 x^{19} - 32319 x^{17} + 69768 x^{15} - 104652 x^{13} + 107406 x^{11} - 72930 x^{9} + 30888 x^{7} - 7371 x^{5} + 819 x^{3} - 27 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $27$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[27, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(706965049015104706497203195837614914543357369=3^{94}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $45.82$ | 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: | $3$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(81=3^{4}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{81}(64,·)$, $\chi_{81}(1,·)$, $\chi_{81}(67,·)$, $\chi_{81}(4,·)$, $\chi_{81}(70,·)$, $\chi_{81}(7,·)$, $\chi_{81}(73,·)$, $\chi_{81}(10,·)$, $\chi_{81}(76,·)$, $\chi_{81}(13,·)$, $\chi_{81}(79,·)$, $\chi_{81}(16,·)$, $\chi_{81}(19,·)$, $\chi_{81}(22,·)$, $\chi_{81}(25,·)$, $\chi_{81}(28,·)$, $\chi_{81}(31,·)$, $\chi_{81}(34,·)$, $\chi_{81}(37,·)$, $\chi_{81}(40,·)$, $\chi_{81}(43,·)$, $\chi_{81}(46,·)$, $\chi_{81}(49,·)$, $\chi_{81}(52,·)$, $\chi_{81}(55,·)$, $\chi_{81}(58,·)$, $\chi_{81}(61,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$
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: | $26$ | 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: | \( 68981171346332.37 \) (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 27 |
| The 27 conjugacy class representatives for $C_{27}$ |
| Character table for $C_{27}$ is not computed |
Intermediate fields
| \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\) |
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 | $27$ | R | $27$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ | ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ | $27$ | $27$ | $27$ | ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ | $27$ |
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 | ||||||