# Properties

 Label 4.4.112896.2 Degree $4$ Signature $[4, 0]$ Discriminant $112896$ Root discriminant $18.33$ Ramified primes $2, 3, 7$ Class number $2$ Class group $[2]$ Galois group $C_2^2$ (as 4T2)

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Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - 28*x^2 + 49)

gp: K = bnfinit(x^4 - 28*x^2 + 49, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![49, 0, -28, 0, 1]);

$$x^{4} - 28 x^{2} + 49$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 Degree: $4$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[4, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$112896$$$$\medspace = 2^{8}\cdot 3^{2}\cdot 7^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $18.33$ 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: $2, 3, 7$ sage: K.disc().support()  gp: factor(abs(K.disc))[,1]~  magma: PrimeDivisors(Discriminant(Integers(K))); $|\Gal(K/\Q)|$: $4$ This field is Galois and abelian over $\Q$. Conductor: $$168=2^{3}\cdot 3\cdot 7$$ Dirichlet character group: $\lbrace$$\chi_{168}(1,·), \chi_{168}(139,·), \chi_{168}(125,·)$$\chi_{168}(71,·)$$\rbrace This is not a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, \frac{1}{7} a^{2}, \frac{1}{7} a^{3} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number C_{2}, which has order 2 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 3 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-1$$ (order 2) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental units: $$\frac{1}{7} a^{2}$$, $$\frac{1}{7} a^{3} - \frac{1}{7} a^{2} - 4 a + 5$$, $$\frac{1}{7} a^{3} - \frac{2}{7} a^{2} - 4 a + 6$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$14.5823618183$$ 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^{4}\cdot(2\pi)^{0}\cdot 14.5823618183 \cdot 2}{2\sqrt{112896}}\approx 0.694398181824 ## 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 J Galois action is Type III ## Frobenius cycle types  p 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 Cycle type R R {\href{/LocalNumberField/5.2.0.1}{2} }^{2} R {\href{/LocalNumberField/11.1.0.1}{1} }^{4} {\href{/LocalNumberField/13.1.0.1}{1} }^{4} {\href{/LocalNumberField/17.2.0.1}{2} }^{2} {\href{/LocalNumberField/19.2.0.1}{2} }^{2} {\href{/LocalNumberField/23.2.0.1}{2} }^{2} {\href{/LocalNumberField/29.2.0.1}{2} }^{2} {\href{/LocalNumberField/31.2.0.1}{2} }^{2} {\href{/LocalNumberField/37.2.0.1}{2} }^{2} {\href{/LocalNumberField/41.2.0.1}{2} }^{2} {\href{/LocalNumberField/43.2.0.1}{2} }^{2} {\href{/LocalNumberField/47.1.0.1}{1} }^{4} {\href{/LocalNumberField/53.2.0.1}{2} }^{2} {\href{/LocalNumberField/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 pLabelPolynomial e f c Galois group Slope content 22.4.8.4x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2} 77.4.2.1x^{4} + 35 x^{2} + 441$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$