Properties

Label 4.2.976.1
Degree $4$
Signature $[2, 1]$
Discriminant $-\,2^{4}\cdot 61$
Root discriminant $5.59$
Ramified primes $2, 61$
Class number $1$
Class group Trivial
Galois Group $S_4$ (as 4T5)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 3, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 + 3*x^2 - 1)
gp: K = bnfinit(x^4 - 2*x^3 + 3*x^2 - 1, 1)

Normalized defining polynomial

\(x^{4} \) \(\mathstrut -\mathstrut 2 x^{3} \) \(\mathstrut +\mathstrut 3 x^{2} \) \(\mathstrut -\mathstrut 1 \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $4$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[2, 1]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(-976=-\,2^{4}\cdot 61\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $5.59$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 61$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$

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:  \( a \),  \( a^{3} - a^{2} + 2 a + 2 \)
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 0.990699215006 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$S_4$ (as 4T5):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 24
The 5 conjugacy class representatives for $S_4$
Character table for $S_4$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Galois closure: data not computed
Degree 6 siblings: 6.2.238144.1, 6.0.58107136.1
Degree 8 sibling: 8.0.56712564736.2
Degree 12 siblings: 12.2.55351463182336.2, 12.0.3376439254122496.4

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R ${\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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];
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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
$61$61.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
61.2.1.2$x^{2} + 122$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_61.2t1.1c1$1$ $ 2^{2} \cdot 61 $ $x^{2} + 61$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_61.3t2.1c1$2$ $ 2^{2} \cdot 61 $ $x^{3} + x - 6$ $S_3$ (as 3T2) $1$ $0$
3.2e6_61e2.6t8.1c1$3$ $ 2^{6} \cdot 61^{2}$ $x^{4} - 2 x^{3} + 3 x^{2} - 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e4_61.4t5.1c1$3$ $ 2^{4} \cdot 61 $ $x^{4} - 2 x^{3} + 3 x^{2} - 1$ $S_4$ (as 4T5) $1$ $1$

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 *.