# Properties

 Label 4.0.8110512.3 Degree $4$ Signature $[0, 2]$ Discriminant $2^{4}\cdot 3^{2}\cdot 151\cdot 373$ Root discriminant $53.37$ Ramified primes $2, 3, 151, 373$ Class number $2$ Class group $[2]$ Galois Group $D_{4}$ (as 4T3)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![56323, 0, 415, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 415*x^2 + 56323)
gp: K = bnfinit(x^4 + 415*x^2 + 56323, 1)

## Normalizeddefining polynomial

$$x^{4}$$ $$\mathstrut +\mathstrut 415 x^{2}$$ $$\mathstrut +\mathstrut 56323$$

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

## Invariants

 Degree: $4$ magma: Degree(K); sage: K.degree() gp: poldegree(K.pol) Signature: $[0, 2]$ magma: Signature(K); sage: K.signature() gp: K.sign Discriminant: $$8110512=2^{4}\cdot 3^{2}\cdot 151\cdot 373$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $53.37$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 3, 151, 373$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is not Galois over $\Q$. This is not a CM field.

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

$1$, $a$, $\frac{1}{133} a^{2} + \frac{8}{133}$, $\frac{1}{133} a^{3} + \frac{8}{133} a$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

## Class group and class number

Multiplicative Abelian group isomorphic to C2, order $2$

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

## Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
 Rank: $1$ magma: UnitRank(K); sage: UK.rank() gp: K.fu Torsion generator: $$\frac{1}{133} a^{2} + \frac{274}{133}$$ (order $6$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental unit: $$\frac{3684927973862518896149239114376}{133} a^{3} - \frac{45305224396369944231562678436904}{133} a^{2} + \frac{304815001027523047204060595221828}{133} a - \frac{17506260175775636125019075808113317}{133}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$148.108045608$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$D_4$ (as 4T3):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 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: data not computed Degree 4 sibling: data not computed

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R R ${\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/23.4.0.1}{4} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }$

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

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];
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]])

## Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2} 33.2.1.2x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2} 151$$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
$\Q_{151}$$x + 5$$1$$1$$0$Trivial$[\ ]$
151.2.1.2$x^{2} + 755$$2$$1$$1$$C_2$$[\ ]_{2}$
373Data not computed