# Properties

 Label 2.0.4.1 Degree $2$ Signature $[0, 1]$ Discriminant $-4$ Root discriminant $2.00$ Ramified prime $2$ Class number $1$ Class group trivial Galois group $C_2$ (as 2T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

This is the field of Gaussian rational numbers.

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^2 + 1)

gp: K = bnfinit(x^2 + 1, 1)

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

$$x^{2} + 1$$ sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

## Invariants

 Degree: $2$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[0, 1]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$-4$$$$\medspace = -\,2^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $2.00$ 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: $2$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $2$ This field is Galois and abelian over $\Q$. Conductor: $$4=2^{2}$$ Dirichlet character group: $\lbrace$$\chi_{4}(1,·), \chi_{4}(3,·)$$\rbrace$ This is a CM field.

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

$1$, $a$ sage: K.integral_basis()

gp: K.zk

magma: IntegralBasis(K);

## 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: $0$ sage: UK.rank()  gp: K.fu  magma: UnitRank(K); Torsion generator: $$a$$ (order $4$) sage: UK.torsion_generator()  gp: K.tu  magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Regulator: $$1$$ sage: K.regulator()  gp: K.reg  magma: Regulator(K);

## Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) =\frac{2^{0}\cdot(2\pi)^{1}\cdot 1 \cdot 1}{4\sqrt{4}}\approx 0.785398163397448$

## Galois group

$C_2$ (as 2T1):

sage: K.galois_group(type='pari')

gp: polgalois(K.pol)

magma: GaloisGroup(K);

 A cyclic group of order 2 The 2 conjugacy class representatives for $C_2$ Character table for $C_2$

## Intermediate fields

 The extension is primitive: there are no intermediate fields between this field and $\Q$.

## Frobenius cycle types

 $p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$ Cycle type R ${\href{/padicField/3.2.0.1}{2} }$ ${\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.2.0.1}{2} }$ ${\href{/padicField/11.2.0.1}{2} }$ ${\href{/padicField/13.1.0.1}{1} }^{2}$ ${\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.2.0.1}{2} }$ ${\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.2.0.1}{2} }$ ${\href{/padicField/37.1.0.1}{1} }^{2}$ ${\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.2.0.1}{2} }$ ${\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }$

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], idealfactors[j]])

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, Valuation(Norm(primefactor), p)> : primefactor in idealfactors];

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content