# Properties

 Label 2.2.1001.1 Degree $2$ Signature $[2, 0]$ Discriminant $1001$ Root discriminant $31.64$ Ramified primes $7, 11, 13$ Class number $2$ Class group $[2]$ 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 - x - 250)

gp: K = bnfinit(x^2 - x - 250, 1)

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-250, -1, 1]);

$$x^{2} - x - 250$$

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: $$1001$$$$\medspace = 7\cdot 11\cdot 13$$ sage: K.disc()  gp: K.disc  magma: Discriminant(Integers(K)); Root discriminant: $31.64$ 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: $7, 11, 13$ 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: $$1001=7\cdot 11\cdot 13$$ Dirichlet character group: $\lbrace$$\chi_{1001}(1,·)$$\chi_{1001}(1000,·)$$\rbrace This is not a CM field. ## Integral basis (with respect to field generator $$a$$) 1, a sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); ## Class group and class number C_{2}, which has order 2 sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); ## Unit group sage: UK = K.unit_group() magma: UK, f := UnitGroup(K);  Rank: 1 sage: UK.rank() gp: K.fu magma: UnitRank(K); Torsion generator: $$-1$$ (order 2) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); Fundamental unit: $$67064 a + 1027373$$ sage: UK.fundamental_units() gp: K.fu magma: [K!f(g): g in Generators(UK)]; Regulator: $$14.567780056$$ 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 14.567780056 \cdot 2}{2\sqrt{1001}}\approx 1.8417739603 ## 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 {\href{/LocalNumberField/2.1.0.1}{1} }^{2} {\href{/LocalNumberField/3.2.0.1}{2} } {\href{/LocalNumberField/5.1.0.1}{1} }^{2} R R R {\href{/LocalNumberField/17.1.0.1}{1} }^{2} {\href{/LocalNumberField/19.2.0.1}{2} } {\href{/LocalNumberField/23.1.0.1}{1} }^{2} {\href{/LocalNumberField/29.2.0.1}{2} } {\href{/LocalNumberField/31.1.0.1}{1} }^{2} {\href{/LocalNumberField/37.2.0.1}{2} } {\href{/LocalNumberField/41.2.0.1}{2} } {\href{/LocalNumberField/43.2.0.1}{2} } {\href{/LocalNumberField/47.1.0.1}{1} }^{2} {\href{/LocalNumberField/53.1.0.1}{1} }^{2} {\href{/LocalNumberField/59.1.0.1}{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 77.2.1.2x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
$11$11.2.1.1$x^{2} - 11$$2$$1$$1$$C_2$$[\ ]_{2} 1313.2.1.1x^{2} - 13$$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$1$ $1$ $x$ $C_1$ $1$ $1$
* 1.1001.2t1.a.a$1$ $7 \cdot 11 \cdot 13$ $x^{2} - x - 250$ $C_2$ (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 *.