The base field K of the Galois representation A: Gal(bar K/K)toGLm(C).
Degree (=dimension) m of a Galois representation A: Gal(bar K/K)toGLm(C)
> K:=pAdicField(3,20);
> R<x>:=PolynomialRing(K);
> F:=ext<K|x^3-3>;
> list:=GaloisRepresentations(F,K);
> forall{A: A in list | BaseField(A) eq K};
true
> [Degree(A): A in list];
[ 1, 1, 2 ]
Finite Galois group Gal(F/K) that computes the finite part of a Galois representation, where F is Field(A) and K is BaseField(A).
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > list1:=[PrincipalCharacter(K),CyclotomicCharacter(K),SP(K,3)]; > [Degree(A): A in list1]; [ 1, 1, 3 ] > [GroupName(Group(A)): A in list1]; [ C1, C1, C1 ] > list2:=GaloisRepresentations(x^4-2); > [Degree(A): A in list2]; [ 1, 1, 1, 1, 2 ] > [GroupName(Group(A)): A in list2]; [ D4, D4, D4, D4, D4 ] > list1[1] eq list2[1]; true
An arithmetic Frobenius element of Group(A) for a Galois representation A.
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > A:=GaloisRepresentations(x^4+x^3+x^2+x+1)[4]; A; 1-dim unramified Galois representation (1,-1,-zeta(4)_4,zeta(4)_4) with G=C4, I=C1 over Q2[20] > frob:=FrobeniusElement(A); frob; (1, 2, 4, 3) > F<u>:=Field(A); > Valuation(Automorphism(A,frob)(u)-u^2) gt 0; true
Character of the finite part of a Galois representation A. For this to be well-defined, A must have only one component (and not several components with different characters).
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > A1:=PrincipalCharacter(K); > Character(A1); ( 1 ) > A2:=CyclotomicCharacter(K); > Character(A2); ( 1 ) > A3:=PermutationCharacter(ext<K|3>,K); > Character(A3); ( 3, 0, 0 ) > A1+A2+A3; 5-dim unramified Galois representation Unr(1-3/2*x+1/2*x^2) + (3,0,0) with G=C3, I=C1 over Q2[20]
Given a Galois representation A, return the p-adic field F such that the finite part of A factors through Gal(F/K), K = BaseField(A).
For a Galois representation A over K, returns a polynomial over K whose splitting field is F=Field(A). The Galois group Gal(F/K) is represented as a permutation group Group(A) on the roots of this polynomial.
> K:=pAdicField(5,20); > E:=EllipticCurve([K|0,5]); > A:=GaloisRepresentation(E); A; 2-dim Galois representation Unr(sqrt(5)*i)*(2,-2,0,0,-1,1) with G=D6, I=C6, conductor 5^2 over Q5[20] > Field(A); Totally ramified extension defined by the polynomial x^6 - 5 over Unramified extension defined by the polynomial x^2 + 4*x + 2 over 5-adic field mod 5^20 > DefiningPolynomial(A); x^6 + O(5^20)*x^5 + O(5^20)*x^4 + O(5^20)*x^3 + O(5^20)*x^2 + O(5^20)*x - 5 + O(5^20)
The automorphism of Field(A)/BaseField(A) given by g.
> R<x>:=PolynomialRing(Rationals()); > K:=NumberField(x^6-5); > a:=ArtinRepresentations(K)[6]; > A:=GaloisRepresentation(a,5); A; 2-dim Galois representation (2,-2,0,0,-1,1) with G=D6, I=C6, conductor 5^2 over Q5[40] > F:=Field(A); F; Totally ramified extension defined by the polynomial x^6 - 5 over Unramified extension defined by the polynomial x^2 + 4*x + 2 over 5-adic field mod 5^40 > DefiningPolynomial(A); x^6 - 5
The 12 elements σ∈Gal(F/K) isomorphic to D6 act on π=root 6 of 5 by multiplying it by 6th roots of unity, with σ(π)=π for 2 of them, and v(σ(π) - π)=1 for the other 10.
> pi:=UniformizingElement(F); > autF:=[* Automorphism(A,g): g in Group(A) *]; > [Valuation(sigma(pi)-pi): sigma in autF]; [ 241, 1, 1, 1, 1, 1, 240, 1, 1, 1, 1, 1 ]
R: Rng Default:
Euler factor (=local polynomial) of a Galois representation A over a p-adic field K. It is defined byP(T) = det (1 - FrobK - 1T|AIK),
and has degree Dimension(A) if and only if A is unramified. The coefficient ring of P (rational/complex/cyclotomic field) may be specified with the optional parameter R.
> G<chi>:=DirichletGroup(5); > A:=GaloisRepresentation(chi,2); > EulerFactor(A); x + 1 > A:=GaloisRepresentation(chi,5); > EulerFactor(A); 1
Return true if A is the Galois representation 0.
Return true if A is the trivial 1-dimensional Galois representation.
Returns the list of tuples < χi,ni,ρi >, where A is the direct sum over i of twists by (SP)(ni) by unramified representations with Euler factor χi, and a finite Weil representation given by a character ρi of Group(A).
> R<x>:=PolynomialRing(ComplexField()); // prettier print for complex polys > K:=pAdicField(2,20); > S:=SP(K,2); > S; Factorization(S); 2-dim Galois representation SP(2) over Q2[20] [* <-x + 1, 2, ( 1 )> *] > A:=Semisimplification(S); > A; Factorization(A); 2-dim unramified Galois representation Unr(1-3/2*x+1/2*x^2) over Q2[20] [* <1/2*x^2 - 3/2*x + 1, 1, ( 1 )> *] > [Factorization(I)[1]: I in Decomposition(A)]; [ <-x + 1, 1, ( 1 )>, <-1/2*x + 1, 1, ( 1 )> ]
For a Galois representation A over a p-adic field K this is the image of inertia IK⊂Gal(bar K/K) under the semisimplification of A. Equivalently, ifA = ψ tensor SP(n) tensor R,
as in ParaNotation and Printing, with ψ unramified and R a representation of a finite Galois group Gal(F/K), this is the image of the inertia subgroup of Gal(F/K) under R. If F is chosen to be minimal possible (so that R is faithful), then InertiaGroup(A) simply is the inertia subgroup of Gal(F/K).
> K:=pAdicField(3,20); > R<x>:=PolynomialRing(K); > list:=GaloisRepresentations(x^6-3); > [GroupName(InertiaGroup(A)): A in list]; [ C1, C2, C1, C2, S3, S3 ]
The nth (lower) ramification subgroup of InertiaGroup(A).
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > list:=GaloisRepresentations(x^8-2); > a:=list[#list]; a; 2-dim Galois representation (2,-2,0,0,0,zeta(8)_8^3+zeta(8)_8, -zeta(8)_8^3-zeta(8)_8) with G=SD16, I=SD16, conductor 2^10 over Q2[20] > [GroupName(InertiaGroup(a,n)): n in [1..17]]; [ SD16, SD16, C8, C8, C4, C4, C4, C4, C2, C2, C2, C2, C2, C2, C2, C2, C1 ]
Return true if a Galois representation is unramified.
> K:=pAdicField(2,20); > IsUnramified(CyclotomicCharacter(K)); true > IsUnramified(SP(K,2)); false > IsUnramified(Semisimplification(SP(K,2))); true
Return true if a Galois representation is ramified.
> K:=pAdicField(2,20); > IsRamified(CyclotomicCharacter(K)); false > IsRamified(SP(K,2)); true > IsRamified(Semisimplification(SP(K,2))); false
Return true if a Galois representation A over a p-adic field K is tamely ramified. Equivalently, InertiaGroup(A) has order prime to p (and is then automatically cyclic).
Return true if a Galois representation A over a p-adic field K is wildly ramified, i.e. not tamely ramified. Equivalently, InertiaGroup(A) has non-trivial p-Sylow.
> E:=EllipticCurve("75a1");
> A5:=GaloisRepresentation(E,5); A5;
2-dim Galois representation Unr(sqrt(5)*i)*(2,0,-1) with G=S3, I=C3, conductor
5^2 over Q5[40]
> IsWildlyRamified(A5);
false
> E:=EllipticCurve("256a1");
> A2:=GaloisRepresentation(E,2); A2;
2-dim Galois representation Unr(sqrt(2))*(2,-2,0,0,0) with G=D4, I=C4,
conductor 2^8 over Q2[40]
> IsWildlyRamified(A2);
true
Inertia invariants of a Galois representation A. This is an unramified Galois representation.
> K:=pAdicField(5,20);
> E:=BaseChange(EllipticCurve("15a1"),K);
> A:=GaloisRepresentation(E); A;
2-dim Galois representation Unr(5)*SP(2) over Q5[20]
> I:=InertiaInvariants(A); I;
1-dim trivial Galois representation 1 over Q5[20]
> Dimension(A),Dimension(I);
2 1
Conductor exponent of a Galois representation.
Conductor of a Galois representation.
> K:=pAdicField(2,40);
> E:=BaseChange(EllipticCurve("256a1"),K);
> A:=GaloisRepresentation(E); A;
2-dim Galois representation Unr(sqrt(2))*(2,-2,0,0,0) with G=D4, I=C4,
conductor 2^8 over Q2[40]
> ConductorExponent(A);
8
> Conductor(A);
2^8 + O(2^48)
> Conductor(E); // same, by definition
2^8 + O(2^48)
Epsilon-factor ε(A) of a Galois representation over a p-adic field. Currently only implemented in a few basic cases, and returns 0 otherwise. See also Example H57E52 in ParaExample: Local and Global Epsilon Factors for Dirichlet Characters.
Root number ε(A)/|ε(A)| of a Galois representation. Currently only implemented in a few basic cases, and returns 0 otherwise. See also Example H57E52 in ParaExample: Local and Global Epsilon Factors for Dirichlet Characters.
> E:=EllipticCurve("98a1");
> A:=GaloisRepresentation(E,7); A;
2-dim Galois representation Unr(7)*SP(2)*(1,-1) with G=C2, I=C2, conductor 7^2
over Q7[40]
> RootNumber(A);
-1
> RootNumber(E,7); // same
-1
Return true if a Galois representation A is irreducible.
> R<x>:=PolynomialRing(Rationals());
> K:=NumberField(x^8-6);
> GroupName(GaloisGroup(K));
C8:C2^2
> assert exists(A){A: A in ArtinRepresentations(K) | Degree(A) eq 4};
> A;
Artin representation C8:C2^2: (4,-4,0,0,0,0,0,0,0,0,0) of ext<Q|x^8-6>
> [IsIrreducible(GaloisRepresentation(A,p)): p in PrimesUpTo(10)];
[ true, false, false, false ]
Return true if a Galois representation A is indecomposable (for semisimple representations, i.e. Weil representations, same as irreducible).
Return true if a Galois representation A is semisimple, i.e. a Weil representation.
Semisimplification of a Galois representation A.
Decompose A into indecomposable (for semisimple representations same as irreducible) consituents and return them as a sequence, possibly with repetitions.
> K:=pAdicField(2,20); > S:=SP(K,2); > IsIndecomposable(S); true > IsIrreducible(S); false > IsSemisimple(S); false > Decomposition(S); [ 2-dim Galois representation SP(2) over Q2[20] ] > Decomposition(Semisimplification(S)); [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim unramified Galois representation Unr(1/2) over Q2[20] ]