magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -3, 12, 1, 20, -7, 16, -2, 5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - x^9 + 5*x^8 - 2*x^7 + 16*x^6 - 7*x^5 + 20*x^4 + x^3 + 12*x^2 - 3*x + 1)
gp: K = bnfinit(x^10 - x^9 + 5*x^8 - 2*x^7 + 16*x^6 - 7*x^5 + 20*x^4 + x^3 + 12*x^2 - 3*x + 1, 1)
Normalized defining polynomial
\( x^{10} - x^{9} + 5 x^{8} - 2 x^{7} + 16 x^{6} - 7 x^{5} + 20 x^{4} + x^{3} + 12 x^{2} - 3 x + 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 5]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(-52089208083=-\,3^{5}\cdot 11^{8}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $11.79$ | 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: | $3, 11$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(33=3\cdot 11\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{33}(1,·)$, $\chi_{33}(4,·)$, $\chi_{33}(5,·)$, $\chi_{33}(14,·)$, $\chi_{33}(16,·)$, $\chi_{33}(20,·)$, $\chi_{33}(23,·)$, $\chi_{33}(25,·)$, $\chi_{33}(26,·)$, $\chi_{33}(31,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{13333} a^{9} + \frac{5318}{13333} a^{8} - \frac{6179}{13333} a^{7} - \frac{258}{13333} a^{6} + \frac{1013}{13333} a^{5} + \frac{24}{199} a^{4} + \frac{6519}{13333} a^{3} - \frac{4571}{13333} a^{2} + \frac{6255}{13333} a + \frac{4507}{13333}$
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: | $4$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{4101}{13333} a^{9} + \frac{3670}{13333} a^{8} - \frac{19287}{13333} a^{7} + \frac{4751}{13333} a^{6} - \frac{61082}{13333} a^{5} + \frac{280}{199} a^{4} - \frac{68419}{13333} a^{3} - \frac{27193}{13333} a^{2} - \frac{39062}{13333} a + \frac{9664}{13333} \) (order $6$) | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 26.1711060094 \) | 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 10 |
| The 10 conjugacy class representatives for $C_{10}$ |
| Character table for $C_{10}$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{11})^+\) |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }$ | R | ${\href{/LocalNumberField/5.10.0.1}{10} }$ | ${\href{/LocalNumberField/7.5.0.1}{5} }^{2}$ | R | ${\href{/LocalNumberField/13.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/17.10.0.1}{10} }$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/29.10.0.1}{10} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/41.10.0.1}{10} }$ | ${\href{/LocalNumberField/43.1.0.1}{1} }^{10}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }$ | ${\href{/LocalNumberField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $3$ | 3.10.5.2 | $x^{10} - 81 x^{2} + 243$ | $2$ | $5$ | $5$ | $C_{10}$ | $[\ ]_{2}^{5}$ |
| $11$ | 11.10.8.5 | $x^{10} - 2321 x^{5} + 2033647$ | $5$ | $2$ | $8$ | $C_{10}$ | $[\ ]_{5}^{2}$ |
Artin representations
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 *.