# Properties

 Label 4.0.784.1 Degree $4$ Signature $[0, 2]$ Discriminant $784$ Root discriminant $$5.29$$ 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 - 3*x^2 + 4)

gp: K = bnfinit(x^4 - 3*x^2 + 4, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4, 0, -3, 0, 1]);

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

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: $$784$$ 784 $$\medspace = 2^{4}\cdot 7^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $$5.29$$ 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))); $\card{ \Gal(K/\Q) }$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$28=2^{2}\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{28}(1,·), \chi_{28}(27,·), \chi_{28}(13,·), \chi_{28}(15,·)$$\rbrace$ This is a CM field.

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

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

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: $1$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$-\frac{1}{2} a^{3} + \frac{1}{2} a$$ -(1)/(2)*a^(3) + (1)/(2)*a  (order $4$) sage: UK.torsion_generator()  gp: K.tu[2]  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $a^{3}+a^{2}-2a-3$ a^3 + a^2 - 2*a - 3 sage: UK.fundamental_units()  gp: K.fu  magma: [K!f(g): g in Generators(UK)]; Regulator: $$2.76865938331$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{2}\cdot 2.76865938331 \cdot 1}{4\sqrt{784}}\approx 0.975913315523$

## 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 R ${\href{/padicField/3.2.0.1}{2} }^{2}$ ${\href{/padicField/5.2.0.1}{2} }^{2}$ R ${\href{/padicField/11.2.0.1}{2} }^{2}$ ${\href{/padicField/13.2.0.1}{2} }^{2}$ ${\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.2.0.1}{2} }^{2}$ ${\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.2.0.1}{2} }^{2}$ ${\href{/padicField/37.1.0.1}{1} }^{4}$ ${\href{/padicField/41.2.0.1}{2} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{2}$ ${\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.2.0.1}{2} }^{2}$

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
$$2$$ 2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2] 2.2.2.1x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
$$7$$ 7.4.2.1$x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$
* 1.28.2t1.a.a$1$ $2^{2} \cdot 7$ $$\Q(\sqrt{7})$$ $C_2$ (as 2T1) $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $$\Q(\sqrt{-1})$$ $C_2$ (as 2T1) $1$ $-1$
* 1.7.2t1.a.a$1$ $7$ $$\Q(\sqrt{-7})$$ $C_2$ (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 *.