# Properties

 Label 1.1.1.1 Degree $1$ Signature $[1, 0]$ Discriminant $1$ Root discriminant $$1.00$$ Ramified primes $$Class number 1 Class group trivial Galois group Trivial (as 1T1) # Related objects # Downloads # Learn more Show commands: Magma / Oscar / PariGP / SageMath ## Normalizeddefining polynomial sage: x = polygen(QQ); K.<a> = NumberField(x) gp: K = bnfinit(y, 1) magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x); oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x) $$x$$ sage: K.defining_polynomial() gp: K.pol magma: DefiningPolynomial(K); oscar: defining_polynomial(K) ## Invariants  Degree: 1 sage: K.degree() gp: poldegree(K.pol) magma: Degree(K); oscar: degree(K) Signature: [1, 0] sage: K.signature() gp: K.sign magma: Signature(K); oscar: signature(K) Discriminant: $$1$$ 1 sage: K.disc() gp: K.disc magma: OK := Integers(K); Discriminant(OK); oscar: OK = ring_of_integers(K); discriminant(OK) Root discriminant: $$1.00$$ sage: (K.disc().abs())^(1./K.degree()) gp: abs(K.disc)^(1/poldegree(K.pol)) magma: Abs(Discriminant(OK))^(1/Degree(K)); oscar: (1.0 * dK)^(1/degree(K)) Galois root discriminant: 1 Ramified primes: None sage: K.disc().support() gp: factor(abs(K.disc))[,1]~ magma: PrimeDivisors(Discriminant(OK)); oscar: prime_divisors(discriminant((OK))) Discriminant root field: $$\Q$$ \card{ \Gal(K/\Q) }: 1 sage: K.automorphisms() magma: Automorphisms(K); oscar: automorphisms(K) This field is Galois and abelian over \Q. Conductor: $$1$$ Dirichlet character group: \lbrace$$\chi_{1}(1,·)\rbrace$This is not a CM field. ## Integral basis$1$sage: K.integral_basis() gp: K.zk magma: IntegralBasis(K); oscar: basis(OK)  Monogenic: Yes Index:$1$Inessential primes: None ## Class group and class number Trivial group, which has order$1$sage: K.class_group().invariants() gp: K.clgp magma: ClassGroup(K); oscar: class_group(K) ## Unit group sage: UK = K.unit_group() magma: UK, fUK := UnitGroup(K); oscar: UK, fUK = unit_group(OK)  Rank:$0$sage: UK.rank() gp: K.fu magma: UnitRank(K); oscar: rank(UK) Torsion generator: $$-1$$ -1 (order$2) sage: UK.torsion_generator() gp: K.tu[2] magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K); oscar: torsion_units_generator(OK) Regulator: $$1$$ sage: K.regulator() gp: K.reg magma: Regulator(K); oscar: regulator(K) ## Class number formula \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr =\mathstrut & \frac{2^1 (2\pi)^0 \cdot 1\cdot 1}{2\cdot\sqrt 1}\cr= \mathstrut & 1 \end{aligned} # self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK)))) # self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))] /* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK))); # self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK)))) ## Galois groupC_1$(as 1T1): sage: K.galois_group(type='pari') gp: polgalois(K.pol) magma: G = GaloisGroup(K); oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)  A cyclic group of order 1 The conjugacy class representative for Trivial Character table for Trivial ## Intermediate fields  The extension is primitive: there are no intermediate fields between this field and$\Q$. sage: K.subfields()[1:-1] gp: L = nfsubfields(K); L[2..length(b)] magma: L := Subfields(K); L[2..#L]; oscar: subfields(K)[2:end-1] ## Frobenius cycle types $p235711131719232931374143475359$Cycle type${\href{/padicField/2.1.0.1}{1} }{\href{/padicField/3.1.0.1}{1} }{\href{/padicField/5.1.0.1}{1} }{\href{/padicField/7.1.0.1}{1} }{\href{/padicField/11.1.0.1}{1} }{\href{/padicField/13.1.0.1}{1} }{\href{/padicField/17.1.0.1}{1} }{\href{/padicField/19.1.0.1}{1} }{\href{/padicField/23.1.0.1}{1} }{\href{/padicField/29.1.0.1}{1} }{\href{/padicField/31.1.0.1}{1} }{\href{/padicField/37.1.0.1}{1} }{\href{/padicField/41.1.0.1}{1} }{\href{/padicField/43.1.0.1}{1} }{\href{/padicField/47.1.0.1}{1} }{\href{/padicField/53.1.0.1}{1} }{\href{/padicField/59.1.0.1}{1} }$Cycle lengths which are repeated in a cycle type are indicated by exponents. # to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_K$for$p=7$in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)] \\ to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_K$for$p=7$in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]]) // to obtain a list of$[e_i,f_i]$for the factorization of the ideal$p\mathcal{O}_K$for$p=7 in Magma:

p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:

p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]

## Local algebras for ramified primes

There are no ramified primes.

## Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $$\Q$$ $C_1$ $1$ $1$

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 *.