# Properties

 Label 4.0.140608.2 Degree $4$ Signature $[0, 2]$ Discriminant $2^{6}\cdot 13^{3}$ Root discriminant $19.36$ Ramified primes $2, 13$ Class number $2$ Class group $[2]$ Galois Group $C_4$ (as 4T1)

# Related objects

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![52, 0, 26, 0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^4 + 26*x^2 + 52)
gp: K = bnfinit(x^4 + 26*x^2 + 52, 1)

## Normalizeddefining polynomial

$$x^{4}$$ $$\mathstrut +\mathstrut 26 x^{2}$$ $$\mathstrut +\mathstrut 52$$

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: $$140608=2^{6}\cdot 13^{3}$$ magma: Discriminant(K); sage: K.disc() gp: K.disc Root discriminant: $19.36$ magma: Abs(Discriminant(K))^(1/Degree(K)); sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) Ramified primes: $2, 13$ magma: PrimeDivisors(Discriminant(K)); sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ This field is Galois and abelian over $\Q$. Conductor: $$104=2^{3}\cdot 13$$ Dirichlet character group: $\lbrace$$\chi_{104}(1,·), \chi_{104}(21,·), \chi_{104}(5,·), \chi_{104}(25,·)$$\rbrace$ This is a CM field.

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

$1$, $a$, $\frac{1}{6} a^{2} - \frac{1}{3}$, $\frac{1}{6} a^{3} - \frac{1}{3} a$

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

## Class group and class number

$C_{2}$, which has 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: $$-1$$ (order $2$) magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); sage: UK.torsion_generator() gp: K.tu[2] Fundamental unit: $$\frac{1}{6} a^{2} + \frac{2}{3}$$ magma: [K!f(g): g in Generators(UK)]; sage: UK.fundamental_units() gp: K.fu Regulator: $$2.38952643457$$ magma: Regulator(K); sage: K.regulator() gp: K.reg

## Galois group

$C_4$ (as 4T1):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
 A cyclic group of order 4 The 4 conjugacy class representatives for $C_4$ Character table for $C_4$

## Intermediate fields

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

## Frobenius cycle types

 $p$ Cycle type 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 R ${\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/31.4.0.1}{4} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.4.0.1}{4} }$

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$2.4.6.3$x^{4} + 2 x^{2} + 20$$2$$2$$6$$C_4$$[3]^{2} 1313.4.3.1x^{4} - 13$$4$$1$$3$$C_4$$[\ ]_{4}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.2e3_13.4t1.2c1$1$ $2^{3} \cdot 13$ $x^{4} + 26 x^{2} + 52$ $C_4$ (as 4T1) $0$ $-1$
* 1.13.2t1.1c1$1$ $13$ $x^{2} - x - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.2e3_13.4t1.2c2$1$ $2^{3} \cdot 13$ $x^{4} + 26 x^{2} + 52$ $C_4$ (as 4T1) $0$ $-1$

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 *.