magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-67, 2081, -378, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 - 378*x^2 + 2081*x - 67)
gp: K = bnfinit(x^4 - x^3 - 378*x^2 + 2081*x - 67, 1)
Normalized defining polynomial
\( x^{4} - x^{3} - 378 x^{2} + 2081 x - 67 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $4$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[4, 0]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1027243729=1009^{3}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $179.03$ | 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: | $1009$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(1009\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{1009}(1008,·)$, $\chi_{1009}(1,·)$, $\chi_{1009}(540,·)$, $\chi_{1009}(469,·)$$\rbrace$ | ||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $\frac{1}{7} a^{2} - \frac{2}{7}$, $\frac{1}{98} a^{3} - \frac{2}{49} a^{2} + \frac{13}{49} a + \frac{43}{98}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{7}$, which has order $7$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $3$ | 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: | \( \frac{1}{7} a^{2} + 2 a - \frac{156}{7} \), \( \frac{1}{98} a^{3} + \frac{5}{49} a^{2} - \frac{232}{49} a + \frac{15}{98} \), \( \frac{3}{98} a^{3} + \frac{8}{49} a^{2} - \frac{549}{49} a + \frac{563}{98} \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 376.315575617 \) | 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 4 |
| The 4 conjugacy class representatives for $C_4$ |
| Character table for $C_4$ |
Intermediate fields
| \(\Q(\sqrt{1009}) \) |
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 | ${\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/5.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/7.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/11.4.0.1}{4} }$ | ${\href{/LocalNumberField/13.4.0.1}{4} }$ | ${\href{/LocalNumberField/17.4.0.1}{4} }$ | ${\href{/LocalNumberField/19.4.0.1}{4} }$ | ${\href{/LocalNumberField/23.4.0.1}{4} }$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/31.4.0.1}{4} }$ | ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }$ | ${\href{/LocalNumberField/47.4.0.1}{4} }$ | ${\href{/LocalNumberField/53.4.0.1}{4} }$ | ${\href{/LocalNumberField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
| 1009 | Data not computed | ||||||
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| * | 1.1009.4t1.1c1 | $1$ | $ 1009 $ | $x^{4} - x^{3} - 378 x^{2} + 2081 x - 67$ | $C_4$ (as 4T1) | $0$ | $1$ |
| * | 1.1009.2t1.1c1 | $1$ | $ 1009 $ | $x^{2} - x - 252$ | $C_2$ (as 2T1) | $1$ | $1$ |
| * | 1.1009.4t1.1c2 | $1$ | $ 1009 $ | $x^{4} - x^{3} - 378 x^{2} + 2081 x - 67$ | $C_4$ (as 4T1) | $0$ | $1$ |
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 *.