# Properties

 Label 2.2.1016.1 Degree $2$ Signature $[2, 0]$ Discriminant $1016$ Root discriminant $31.87$ Ramified primes $2, 127$ Class number $3$ Class group $[3]$ Galois group $C_2$ (as 2T1)

# Related objects

Show commands for: SageMath / Pari/GP / Magma

## Normalizeddefining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 254)

gp: K = bnfinit(x^2 - 254, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-254, 0, 1]);

$$x^{2} - 254$$

sage: K.defining_polynomial()

gp: K.pol

magma: DefiningPolynomial(K);

 Degree: $2$ sage: K.degree()  gp: poldegree(K.pol)  magma: Degree(K); Signature: $[2, 0]$ sage: K.signature()  gp: K.sign  magma: Signature(K); Discriminant: $$1016$$$$\medspace = 2^{3}\cdot 127$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $31.87$ 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: $2, 127$ 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: $$1016=2^{3}\cdot 127$$ Dirichlet character group: $\lbrace$$\chi_{1016}(1,·)$$\chi_{1016}(507,·)$$\rbrace This is not a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number C_{3}, which has order 3 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 1 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-1$$ (order 2) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $$16 a + 255$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$6.23440688102$$ 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^{2}\cdot(2\pi)^{0}\cdot 6.23440688102 \cdot 3}{2\sqrt{1016}}\approx 1.17354443673 ## 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{/LocalNumberField/3.2.0.1}{2} } {\href{/LocalNumberField/5.1.0.1}{1} }^{2} {\href{/LocalNumberField/7.1.0.1}{1} }^{2} {\href{/LocalNumberField/11.1.0.1}{1} }^{2} {\href{/LocalNumberField/13.2.0.1}{2} } {\href{/LocalNumberField/17.1.0.1}{1} }^{2} {\href{/LocalNumberField/19.1.0.1}{1} }^{2} {\href{/LocalNumberField/23.1.0.1}{1} }^{2} {\href{/LocalNumberField/29.1.0.1}{1} }^{2} {\href{/LocalNumberField/31.2.0.1}{2} } {\href{/LocalNumberField/37.2.0.1}{2} } {\href{/LocalNumberField/41.1.0.1}{1} }^{2} {\href{/LocalNumberField/43.2.0.1}{2} } {\href{/LocalNumberField/47.2.0.1}{2} } {\href{/LocalNumberField/53.1.0.1}{1} }^{2} {\href{/LocalNumberField/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][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 22.2.3.3x^{2} + 2$$2$$1$$3$$C_2$$[3]$
$127$127.2.1.1$x^{2} - 127$$2$$1$$1$$C_2$$[\ ]_{2}$

## Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.1016.2t1.a.a$1$ $2^{3} \cdot 127$ $x^{2} - 254$ $C_2$ (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 *.