# Properties

 Label 4.2.21952.1 Degree $4$ Signature $[2, 1]$ Discriminant $-21952$ Root discriminant $$12.17$$ Ramified primes see page Class number $1$ Class group trivial Galois group $D_{4}$ (as 4T3)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 2*x^3 + 5*x^2 - 4*x - 10)

gp: K = bnfinit(x^4 - 2*x^3 + 5*x^2 - 4*x - 10, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-10, -4, 5, -2, 1]);

$$x^{4} - 2x^{3} + 5x^{2} - 4x - 10$$

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: $$-21952$$ -21952 $$\medspace = -\,2^{6}\cdot 7^{3}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$12.17$$ 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: $$2$$, $$7$$ 2, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Aut(K/\Q) }$: $2$ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{2}{5}a$

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: $1$ Inessential primes: None

## Class group and class number

Trivial group, which has order $1$

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: $$-1$$ -1  (order $2$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $\frac{2}{5}a^{3}-\frac{3}{5}a^{2}+\frac{1}{5}a+1$, $\frac{2}{5}a^{3}-\frac{3}{5}a^{2}+\frac{1}{5}a-1$ 2/5*a^3 - 3/5*a^2 + 1/5*a + 1, 2/5*a^3 - 3/5*a^2 + 1/5*a - 1 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$16.2166373091$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{2}\cdot(2\pi)^{1}\cdot 16.2166373091 \cdot 1}{2\sqrt{21952}}\approx 1.37541457028$

## 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: 8.0.481890304.3 Degree 4 sibling: 4.0.2744.1

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.4.0.1}{4} }$ ${\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.4.0.1}{4} }$ ${\href{/padicField/19.4.0.1}{4} }$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ ${\href{/padicField/37.2.0.1}{2} }^{2}$ ${\href{/padicField/41.4.0.1}{4} }$ ${\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }$

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
$$2$$ 2.2.3.3$x^{2} + 2$$2$$1$$3$$C_2$$[3] 2.2.3.3x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$$7$$ 7.4.3.2$x^{4} - 7$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.56.2t1.a.a$1$ $2^{3} \cdot 7$ $$\Q(\sqrt{14})$$ $C_2$ (as 2T1) $1$ $1$
1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
1.8.2t1.b.a$1$ $2^{3}$ $$\Q(\sqrt{-2})$$ $C_2$ (as 2T1) $1$ $-1$
* 2.392.4t3.c.a$2$ $2^{3} \cdot 7^{2}$ 4.2.21952.1 $D_{4}$ (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 *.