Properties

Label 1.1.1.1
Degree 1
Signature $[1, 0]$
Discriminant $1$
Ramified primes $\textrm{None}$
Class number 1
Class group Trivial
Galois Group Trivial

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![0, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x)
gp: K = bnfinit(x, 1)

Normalized defining polynomial

\(x \)

magma: DefiningPolynomial(K);
sage: K.defining_polynomial()
gp: K.pol

Invariants

Degree:  $1$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 0]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(1\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Ramified primes:  $\textrm{None}$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is Galois and abelian over $\Q$.
Conductor:  \(1\)
Dirichlet group:    $\lbrace$$\chi_{1}(1,·)$$\rbrace$

Integral basis

$1$

magma: IntegralBasis(K);
sage: K.integral_basis()
gp: K.zk

Class group and class number

Trivial Abelian group, order 1

magma: ClassGroup(K);
sage: K.class_group().invariants()
gp: K.clgp

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $0$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
Regulator:  \( 1 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

Trivial:

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A cyclic group of order 1
Conjugacy class representatives for Trivial
Character table for Trivial

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 ${\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.1.0.1}{1} }$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
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]])

Local algebras for ramified primes

There are no ramified primes.

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$

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 *.