magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, -10, 2, 5, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^5 - x^4 + 5*x^3 + 2*x^2 - 10*x - 6)
gp: K = bnfinit(x^5 - x^4 + 5*x^3 + 2*x^2 - 10*x - 6, 1)
Normalized defining polynomial
\( x^{5} - x^{4} + 5 x^{3} + 2 x^{2} - 10 x - 6 \)
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: | \(465124=2^{2}\cdot 11^{2}\cdot 31^{2}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $13.60$ | 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: | $2, 11, 31$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$
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: | $2$ | 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: | \( a + 1 \), \( a^{4} + 10 a^{2} - 8 a - 13 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 23.4142361411 \) | 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 non-solvable group of order 60 |
| The 5 conjugacy class representatives for $A_5$ |
| Character table for $A_5$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 6 sibling: | 6.2.225120016.2 |
| Degree 10 sibling: | 10.2.104708722321984.1 |
| Degree 12 sibling: | Deg 12 |
| Degree 15 sibling: | Deg 15 |
| Degree 20 sibling: | Deg 20 |
| Degree 30 sibling: | 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 | ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/5.5.0.1}{5} }$ | ${\href{/LocalNumberField/7.5.0.1}{5} }$ | R | ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/17.5.0.1}{5} }$ | ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ | ${\href{/LocalNumberField/23.5.0.1}{5} }$ | ${\href{/LocalNumberField/29.5.0.1}{5} }$ | R | ${\href{/LocalNumberField/37.5.0.1}{5} }$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.5.0.1}{5} }$ | ${\href{/LocalNumberField/47.5.0.1}{5} }$ | ${\href{/LocalNumberField/53.5.0.1}{5} }$ | ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
| $2$ | 2.2.0.1 | $x^{2} - x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 2.3.2.1 | $x^{3} - 2$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $11$ | 11.2.0.1 | $x^{2} - x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 11.3.2.1 | $x^{3} - 11$ | $3$ | $1$ | $2$ | $S_3$ | $[\ ]_{3}^{2}$ | |
| $31$ | $\Q_{31}$ | $x + 7$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | |
| 31.2.1.1 | $x^{2} - 31$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 3.2e2_11e2_31e2.12t33.1c1 | $3$ | $ 2^{2} \cdot 11^{2} \cdot 31^{2}$ | $x^{5} - x^{4} + 5 x^{3} + 2 x^{2} - 10 x - 6$ | $A_5$ (as 5T4) | $1$ | $-1$ | |
| 3.2e2_11e2_31e2.12t33.1c2 | $3$ | $ 2^{2} \cdot 11^{2} \cdot 31^{2}$ | $x^{5} - x^{4} + 5 x^{3} + 2 x^{2} - 10 x - 6$ | $A_5$ (as 5T4) | $1$ | $-1$ | |
| * | 4.2e2_11e2_31e2.5t4.1c1 | $4$ | $ 2^{2} \cdot 11^{2} \cdot 31^{2}$ | $x^{5} - x^{4} + 5 x^{3} + 2 x^{2} - 10 x - 6$ | $A_5$ (as 5T4) | $1$ | $0$ |
| 5.2e4_11e4_31e2.6t12.1c1 | $5$ | $ 2^{4} \cdot 11^{4} \cdot 31^{2}$ | $x^{5} - x^{4} + 5 x^{3} + 2 x^{2} - 10 x - 6$ | $A_5$ (as 5T4) | $1$ | $1$ |
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 *.