# Properties

 Label 4.2.7975.2 Degree $4$ Signature $[2, 1]$ Discriminant $-\,5^{2}\cdot 11\cdot 29$ Root discriminant $9.45$ Ramified primes $5, 11, 29$ Class number $2$ Class group $[2]$ Galois group $D_{4}$ (as 4T3)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 + x^2 - 20)

gp: K = bnfinit(x^4 - 2*x^3 + x^2 - 20, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-20, 0, 1, -2, 1]);

## Normalizeddefining polynomial

$$x^{4} - 2 x^{3} + x^{2} - 20$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $4$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 1]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-7975=-\,5^{2}\cdot 11\cdot 29$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $9.45$ sage: (K.disc().abs())^(1./K.degree())  gp: abs(K.disc)^(1/poldegree(K.pol))  magma: Abs(Discriminant(Integers(K)))^(1/Degree(K)); Ramified primes: $5, 11, 29$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Aut(K/\Q)|$: $2$ This field is not Galois over $\Q$. This is not a CM field.

## Integral basis (with respect to field generator $$a$$)

$1$, $a$, $\frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{3} + \frac{1}{4} a - \frac{1}{2}$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## Class group and class number

$C_{2}$, which has order $2$

sage: K.class_group().invariants()

gp: K.clgp

magma: ClassGroup(K);

## Unit group

sage: UK = K.unit_group()

magma: UK, f := UnitGroup(K);

 Rank: $2$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-1$$ (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{2}$$,  $$\frac{1}{4} a^{3} - \frac{3}{4} a^{2} + a - 2$$ sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$2.09650187347$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Galois group

$D_4$ (as 4T3):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A solvable group of order 8 The 5 conjugacy class representatives for $D_{4}$ Character table for $D_{4}$

## Intermediate fields

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

## Sibling fields

 Galois closure: 8.0.6472063200625.2 Degree 4 sibling: 4.0.508805.1

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 ${\href{/LocalNumberField/2.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/3.4.0.1}{4} }$ R ${\href{/LocalNumberField/7.4.0.1}{4} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])

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];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$5$5.2.1.1$x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2} 5.2.1.1x^{2} - 5$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2} 11.2.0.1x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
$29$29.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2} 29.2.1.1x^{2} - 29$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.5.2t1.a.a$1$ $5$ $x^{2} - x - 1$ $C_2$ (as 2T1) $1$ $1$
1.319.2t1.a.a$1$ $11 \cdot 29$ $x^{2} - x + 80$ $C_2$ (as 2T1) $1$ $-1$
1.1595.2t1.a.a$1$ $5 \cdot 11 \cdot 29$ $x^{2} - x + 399$ $C_2$ (as 2T1) $1$ $-1$
* 2.1595.4t3.c.a$2$ $5 \cdot 11 \cdot 29$ $x^{4} - 2 x^{3} + x^{2} - 20$ $D_{4}$ (as 4T3) $1$ $0$

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 *.