magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -5, 25, 5, -50, -1, 35, 0, -10, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^10 - 10*x^8 + 35*x^6 - x^5 - 50*x^4 + 5*x^3 + 25*x^2 - 5*x - 1)
gp: K = bnfinit(x^10 - 10*x^8 + 35*x^6 - x^5 - 50*x^4 + 5*x^3 + 25*x^2 - 5*x - 1, 1)
Normalized defining polynomial
\( x^{10} - 10 x^{8} + 35 x^{6} - x^{5} - 50 x^{4} + 5 x^{3} + 25 x^{2} - 5 x - 1 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $10$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[10, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(762939453125=5^{17}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $15.43$ | 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: | $5$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(25=5^{2}\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{25}(1,·)$, $\chi_{25}(4,·)$, $\chi_{25}(6,·)$, $\chi_{25}(9,·)$, $\chi_{25}(11,·)$, $\chi_{25}(14,·)$, $\chi_{25}(16,·)$, $\chi_{25}(19,·)$, $\chi_{25}(21,·)$, $\chi_{25}(24,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$
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: | $9$ | 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: | \( a^{5} - 5 a^{3} + 5 a \), \( a^{8} - a^{7} - 8 a^{6} + 7 a^{5} + 20 a^{4} - 15 a^{3} - 15 a^{2} + 10 a \), \( a^{3} - 3 a \), \( a^{6} - 6 a^{4} + 9 a^{2} - 2 \), \( a^{9} - 9 a^{7} - a^{6} + 27 a^{5} + 5 a^{4} - 30 a^{3} - 5 a^{2} + 10 a \), \( a^{7} - 7 a^{5} + 14 a^{3} - 7 a \), \( a^{2} - 2 \), \( a^{9} - 9 a^{7} + 27 a^{5} - 30 a^{3} + 9 a \), \( a \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 315.247729768 \) | 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
| \(\Q(\sqrt{5}) \), 5.5.390625.1 |
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 | ${\href{/LocalNumberField/2.10.0.1}{10} }$ | ${\href{/LocalNumberField/3.10.0.1}{10} }$ | R | ${\href{/LocalNumberField/7.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/11.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/13.10.0.1}{10} }$ | ${\href{/LocalNumberField/17.10.0.1}{10} }$ | ${\href{/LocalNumberField/19.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/23.10.0.1}{10} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/31.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/37.10.0.1}{10} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }^{2}$ | ${\href{/LocalNumberField/43.2.0.1}{2} }^{5}$ | ${\href{/LocalNumberField/47.10.0.1}{10} }$ | ${\href{/LocalNumberField/53.10.0.1}{10} }$ | ${\href{/LocalNumberField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $5$ | 5.10.17.1 | $x^{10} - 5 x^{8} + 5$ | $10$ | $1$ | $17$ | $C_{10}$ | $[2]_{2}$ |