magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![568, 2060, -90, -65, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - 65*x^3 - 90*x^2 + 2060*x + 568)
gp: K = bnfinit(x^5 - 65*x^3 - 90*x^2 + 2060*x + 568, 1)
Normalized defining polynomial
\( x^{5} - 65 x^{3} - 90 x^{2} + 2060 x + 568 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $5$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[1, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(172225000000=2^{6}\cdot 5^{8}\cdot 83^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $176.69$ | 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: | $2, 5, 83$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2} a + \frac{1}{3}$, $\frac{1}{552} a^{4} + \frac{19}{276} a^{3} - \frac{31}{184} a^{2} + \frac{7}{69} a - \frac{19}{46}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{20}\times C_{20}$, which has order $400$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $2$ | 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: | \( \frac{7}{276} a^{4} - \frac{14}{69} a^{3} - \frac{191}{276} a^{2} + \frac{1231}{138} a - \frac{905}{69} \), \( \frac{1}{138} a^{4} - \frac{4}{69} a^{3} - \frac{47}{138} a^{2} + \frac{166}{69} a + \frac{47}{69} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 109.116786659 \) | 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 solvable group of order 10 |
| The 4 conjugacy class representatives for $D_{5}$ |
| Character table for $D_{5}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | data not computed |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/LocalNumberField/3.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/7.5.0.1}{5} }$ | ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.5.0.1}{5} }$ | ${\href{/LocalNumberField/17.5.0.1}{5} }$ | ${\href{/LocalNumberField/19.5.0.1}{5} }$ | ${\href{/LocalNumberField/23.1.0.1}{1} }^{5}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.5.0.1}{5} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ | ${\href{/LocalNumberField/41.5.0.1}{5} }$ | ${\href{/LocalNumberField/43.5.0.1}{5} }$ | ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| 2.2.3.2 | $x^{2} + 6$ | $2$ | $1$ | $3$ | $C_2$ | $[3]$ | |
| $5$ | 5.5.8.2 | $x^{5} - 5 x^{4} + 5$ | $5$ | $1$ | $8$ | $C_5$ | $[2]$ |
| $83$ | $\Q_{83}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 83.2.1.1 | $x^{2} - 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 83.2.1.1 | $x^{2} - 83$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |