magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, 11, 6, -1, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 + 6*x^2 + 11*x + 10)
gp: K = bnfinit(x^4 - x^3 + 6*x^2 + 11*x + 10, 1)
Normalized defining polynomial
\(x^{4} \) \(\mathstrut -\mathstrut x^{3} \) \(\mathstrut +\mathstrut 6 x^{2} \) \(\mathstrut +\mathstrut 11 x \) \(\mathstrut +\mathstrut 10 \)
magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol
Invariants
| Degree: | $4$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 2]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(110889=3^{4}\cdot 37^{2}\) | magma: Discriminant(K);
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $18.25$ | magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $3, 37$ | magma: PrimeDivisors(Discriminant(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 \(a\))
$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} - \frac{1}{3}$
magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk
Class group and class number
$C_{2}$, which has order $2$
magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp
Unit group
magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
| Rank: | $1$ | 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 unit: | \( a^{3} - 129 a^{2} - 203 a - 147 \) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 14.5871081372 \) | 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 12 |
| The 4 conjugacy class representatives for $A_4$ |
| Character table for $A_4$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Galois closure: | Deg 12 |
| Degree 6 sibling: | 6.2.8982009.1 |
Frobenius cycle types
| $p$ | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | 53 | 59 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/LocalNumberField/2.3.0.1}{3} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ | ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ | ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ | ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ | ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ | ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ | ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ | ${\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ | R | ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ | ${\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ | ${\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ | ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ | ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$ |
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];
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)]
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]])
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 |
|---|---|---|---|---|---|---|---|
| $3$ | $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
| 3.3.4.2 | $x^{3} - 3 x^{2} + 3$ | $3$ | $1$ | $4$ | $C_3$ | $[2]$ | |
| $37$ | 37.4.2.1 | $x^{4} + 333 x^{2} + 34225$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |
Artin representations
| Label | Dimension | Conductor | Defining polynomial of Artin field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.1c1 | $1$ | $1$ | $x$ | $C_1$ | $1$ | $1$ |
| 1.3e2.3t1.1c1 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| 1.3e2.3t1.1c2 | $1$ | $ 3^{2}$ | $x^{3} - 3 x - 1$ | $C_3$ (as 3T1) | $0$ | $1$ | |
| * | 3.3e4_37e2.4t4.1c1 | $3$ | $ 3^{4} \cdot 37^{2}$ | $x^{4} - x^{3} + 6 x^{2} + 11 x + 10$ | $A_4$ (as 4T4) | $1$ | $-1$ |
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 *.