# Properties

 Label 4.0.144.1 Degree $4$ Signature $[0, 2]$ Discriminant $144$ Root discriminant $3.46$ Ramified primes $2, 3$ Class number $1$ Class group trivial Galois group $C_2^2$ (as 4T2)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

This is the quartic field with Galois group $C_2^2$ with the smallest absolute discriminant.

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^4 - x^2 + 1)

gp: K = bnfinit(x^4 - x^2 + 1, 1)

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

$$x^{4} - x^{2} + 1$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 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: $$144$$$$\medspace = 2^{4}\cdot 3^{2}$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $3.46$ 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: $2, 3$ 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: $$12=2^{2}\cdot 3$$ Dirichlet character group: $\lbrace$$\chi_{12}(1,·), \chi_{12}(11,·), \chi_{12}(5,·)$$\chi_{12}(7,·)$$\rbrace This is a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a, a^{2}, a^{3} sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number Trivial group, which has order 1 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 1 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$a$$ (order 12) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $$a - 1$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$1.31695789692$$ 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 1.31695789692 \cdot 1}{12\sqrt{144}}\approx 0.361051484875 ## 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 R {\href{/LocalNumberField/5.2.0.1}{2} }^{2} {\href{/LocalNumberField/7.2.0.1}{2} }^{2} {\href{/LocalNumberField/11.2.0.1}{2} }^{2} {\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/37.1.0.1}{1} }^{4} {\href{/LocalNumberField/41.2.0.1}{2} }^{2} {\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/59.2.0.1}{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 22.4.4.1x^{4} + 8 x^{2} + 4$$2$$2$$4$$C_2^2$$[2]^{2}$
$3$3.4.2.1$x^{4} + 9 x^{2} + 36$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.12.2t1.a.a$1$ $2^{2} \cdot 3$ $x^{2} - 3$ $C_2$ (as 2T1) $1$ $1$
* 1.4.2t1.a.a$1$ $2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 1.3.2t1.a.a$1$ $3$ $x^{2} - x + 1$ $C_2$ (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 *.