# Properties

 Label 4.0.1016.1 Degree $4$ Signature $[0, 2]$ Discriminant $2^{3}\cdot 127$ Root discriminant $5.65$ Ramified primes $2, 127$ 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![2, -2, 1, -1, 1]);

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^3 + x^2 - 2*x + 2)

gp: K = bnfinit(x^4 - x^3 + x^2 - 2*x + 2, 1)

## Normalizeddefining polynomial

$$x^{4} - x^{3} + x^{2} - 2 x + 2$$

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: $$1016=2^{3}\cdot 127$$ magma: Discriminant(Integers(K));  sage: K.disc()  gp: K.disc Root discriminant: $5.65$ 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: $2, 127$ magma: PrimeDivisors(Discriminant(Integers(K)));  sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~ $|\Aut(K/\Q)|$: $1$ 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: $1$ 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: $$a^{3} + a - 1$$ magma: [K!f(g): g in Generators(UK)];  sage: UK.fundamental_units()  gp: K.fu Regulator: $$1.27428442991$$ 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: 6.2.1048772096.1, 6.2.1032256.1 Degree 8 sibling: 8.0.1065552449536.3 Degree 12 siblings: Deg 12, Deg 12

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.4.0.1}{4} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.3.0.1}{3} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

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$2.2.3.4$x^{2} + 10$$2$$1$$3$$C_2$$[3] 2.2.0.1x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
$127$$\Q_{127}$$x + 9$$1$$1$$0Trivial[\ ] \Q_{127}$$x + 9$$1$$1$$0Trivial[\ ] 127.2.1.1x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3_127.2t1.1c1$1$ $2^{3} \cdot 127$ $x^{2} - 254$ $C_2$ (as 2T1) $1$ $1$
2.2e3_127.3t2.1c1$2$ $2^{3} \cdot 127$ $x^{3} - x^{2} - 6 x + 2$ $S_3$ (as 3T2) $1$ $2$
3.2e6_127e2.6t8.1c1$3$ $2^{6} \cdot 127^{2}$ $x^{4} - x^{3} + x^{2} - 2 x + 2$ $S_4$ (as 4T5) $1$ $-1$
* 3.2e3_127.4t5.1c1$3$ $2^{3} \cdot 127$ $x^{4} - x^{3} + x^{2} - 2 x + 2$ $S_4$ (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 *.