# Properties

 Label 4.2.848.1 Degree $4$ Signature $[2, 1]$ Discriminant $-\,2^{4}\cdot 53$ Root discriminant $5.40$ Ramified primes $2, 53$ Class number $1$ Class group Trivial Galois Group $S_4$ (as 4T5)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

## Normalizeddefining polynomial

$$x^{4}$$ $$\mathstrut -\mathstrut x^{2}$$ $$\mathstrut -\mathstrut 2 x$$ $$\mathstrut +\mathstrut 1$$

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: $$-848=-\,2^{4}\cdot 53$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $5.40$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 53$ 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: $$a$$,  $$a^{3} + a^{2} - 2$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$0.992994938315$$ 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: 6.2.179776.1, 6.0.38112512.1 Degree 8 sibling: 8.0.32319410176.2 Degree 12 siblings: 12.2.27406859829248.2, 12.0.1452563570950144.5

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2} 5353.2.0.1x^{2} - x + 5$$1$$2$$0$$C_2$$[\ ]^{2}$
53.2.1.2$x^{2} + 106$$2$$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_53.2t1.1c1$1$ $2^{2} \cdot 53$ $x^{2} + 53$ $C_2$ (as 2T1) $1$ $-1$
2.2e2_53.3t2.1c1$2$ $2^{2} \cdot 53$ $x^{3} - x^{2} + 4 x - 2$ $S_3$ (as 3T2) $1$ $0$
3.2e6_53e2.6t8.3c1$3$ $2^{6} \cdot 53^{2}$ $x^{4} - x^{2} - 2 x + 1$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e4_53.4t5.1c1$3$ $2^{4} \cdot 53$ $x^{4} - x^{2} - 2 x + 1$ $S_4$ (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 *.