# Properties

 Label 4.0.441.1 Degree $4$ Signature $[0, 2]$ Discriminant $441$ Root discriminant $$4.58$$ Ramified primes see page Class number $1$ Class group trivial Galois group $C_2^2$ (as 4T2)

# Related objects

Show commands: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

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

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

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

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

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $4$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 2]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$441$$ 441 $$\medspace = 3^{2}\cdot 7^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$4.58$$ 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: $$3$$, $$7$$ 3, 7 sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $\card{ \Gal(K/\Q) }$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$21=3\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{21}(8,·), \chi_{21}(1,·), \chi_{21}(20,·), \chi_{21}(13,·)$$\rbrace$ This is a CM field.

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

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

sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

 Monogenic: No Index: $2$ Inessential primes: $2$

## 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: $1$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{2} a + 2$$ -(1)/(2)*a^(3) - (1)/(2)*a^(2) + (1)/(2)*a + 2  (order $6$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $\frac{1}{2}a^{3}-\frac{1}{2}a^{2}+\frac{1}{2}a-1$ 1/2*a^3 - 1/2*a^2 + 1/2*a - 1 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$1.56679923697$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{2}\cdot 1.56679923697 \cdot 1}{6\sqrt{441}}\approx 0.490910750629$

## Galois group

$C_2^2$ (as 4T2):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 An abelian group of order 4 The 4 conjugacy class representatives for $C_2^2$ Character table for $C_2^2$

## Intermediate fields

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

## Multiplicative Galois module structure

 $U_{K^{gal}}/\textrm{Tors}(U_{K^{gal}}) \cong$ $A_1$

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type ${\href{/padicField/2.2.0.1}{2} }^{2}$ R ${\href{/padicField/5.2.0.1}{2} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$$3$$ 3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} $$7$$ 7.2.1.2x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $3$ $$\Q(\sqrt{-3})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.21.2t1.a.a$1$ $3 \cdot 7$ $$\Q(\sqrt{21})$$ $C_2$ (as 2T1) $1$ $1$

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 *.