magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5653, 25691, 22774, -6450, -4247, -450, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^7 - x^6 - 450*x^5 - 4247*x^4 - 6450*x^3 + 22774*x^2 + 25691*x + 5653)
gp: K = bnfinit(x^7 - x^6 - 450*x^5 - 4247*x^4 - 6450*x^3 + 22774*x^2 + 25691*x + 5653, 1)
Normalized defining polynomial
\( x^{7} - x^{6} - 450 x^{5} - 4247 x^{4} - 6450 x^{3} + 22774 x^{2} + 25691 x + 5653 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $7$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[7, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1347771585762793801=1051^{6}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $389.00$ | 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: | $1051$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1051\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1051}(1,·)$, $\chi_{1051}(402,·)$, $\chi_{1051}(801,·)$, $\chi_{1051}(217,·)$, $\chi_{1051}(491,·)$, $\chi_{1051}(396,·)$, $\chi_{1051}(845,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{10854523483} a^{6} + \frac{3381971397}{10854523483} a^{5} - \frac{5228444610}{10854523483} a^{4} - \frac{3622531124}{10854523483} a^{3} - \frac{5239560787}{10854523483} a^{2} + \frac{2373667019}{10854523483} a + \frac{4299553903}{10854523483}$
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: | $6$ | 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 | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 2327124.38908 \) | 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 7 |
| The 7 conjugacy class representatives for $C_7$ |
| Character table for $C_7$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.7.0.1}{7} }$ | ${\href{/LocalNumberField/3.7.0.1}{7} }$ | ${\href{/LocalNumberField/5.7.0.1}{7} }$ | ${\href{/LocalNumberField/7.7.0.1}{7} }$ | ${\href{/LocalNumberField/11.7.0.1}{7} }$ | ${\href{/LocalNumberField/13.7.0.1}{7} }$ | ${\href{/LocalNumberField/17.7.0.1}{7} }$ | ${\href{/LocalNumberField/19.7.0.1}{7} }$ | ${\href{/LocalNumberField/23.7.0.1}{7} }$ | ${\href{/LocalNumberField/29.7.0.1}{7} }$ | ${\href{/LocalNumberField/31.7.0.1}{7} }$ | ${\href{/LocalNumberField/37.7.0.1}{7} }$ | ${\href{/LocalNumberField/41.7.0.1}{7} }$ | ${\href{/LocalNumberField/43.7.0.1}{7} }$ | ${\href{/LocalNumberField/47.7.0.1}{7} }$ | ${\href{/LocalNumberField/53.7.0.1}{7} }$ | ${\href{/LocalNumberField/59.7.0.1}{7} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 1051 | Data not computed | ||||||