We denote by SL2(Z) the group of 2 by 2 matrices with
integer coefficients and determinant 1.
The group PSL2(Z) is the projectivization of SL2(Z).
For any integer N we have groups
Γ0(N)={pmatrix(a & b
c & d)∈SL2(Z) Big|
pmatrix(a & b
c & d) ≡ pmatrix(* & *
0 & * ) mod N}
Γ1(N)={pmatrix(a & b
c & d)∈SL2(Z) Big|
pmatrix(a & b
c & d) ≡ pmatrix(1 & *
0 & 1) mod N}
Γ(N)={pmatrix(a & b
c & d)∈SL2(Z) Big|
pmatrix(a & b
c & d) ≡ pmatrix(1 & 0
0 & 1) mod N}
Γ1(N)={pmatrix(a & b
c & d)∈SL2(Z) Big|
pmatrix(a & b
c & d) ≡ pmatrix(1 & 0
* & 1) mod N}
Γ0(N)={pmatrix(a & b
c & d)∈SL2(Z) Big|
pmatrix(a & b
c & d) ≡ pmatrix(* & 0
* & * ) mod N}
A congruence subgroup
is any discrete subgroup Γ of SL2(R)
which is commensurable with SL2(Z), that is, Γ∩SL2(Z) has
finite index in Γ and in SL2(Z), and such that Γ(N) is
contained in G for some N.
The level N of a congruence subgroup G is the greatest
integer N such that Γ(N) is contained in Γ.
We will abuse notation and also refer to the projectivizations of
these groups by the same names.
Returns PSL2(R), the projective linear group over the ring R.
The group Γ0(N) for any positive integer N.
The group Γ1(N) for any positive integer N.
The group Γ0(N) for any positive integer N.
The group Γ1(N) for any positive integer N.
The group Γ(N) for any positive integer N.
For a positive integer N and i=0, 1, 2, 3, or 4, this is the group Γ0(N), Γ1(N), Γ(N), Γ1(N) or Γ0(N) respectively.
This is the congruence subgroup consisting of 2 by 2 matrices with integer coefficients [a, b, c, d] with b = 0 mod P, c = 0 mod N, and a = d = 1 mod M. It is required that M divides NP.
The intersection of congruence subgroups G and H.
Examples of defining different congruence subgroups:
> G := PSL2(Integers()); > H := CongruenceSubgroup([2,3,6]); > H; Gamma_0(2) intersection Gamma^1(3) intersection Gamma^0(2) > K := CongruenceSubgroup(0,5); > K meet H; Gamma_0(10) intersection Gamma^1(3) intersection Gamma^0(2)
Returns true if and only if the congruence subgroups G and H are equal.
For congruence subgroups G and H contained in PSL2(Z), returns true if and only if H is a subgroup of G.
For congruence subgroups G and H, returns the index of G in H provided G is a subgroup of H.
For G a congruence subgroup in PSL2(Z), returns the index in PSL2(Z).
The level of a congruence subgroup G.
Returns true if and only if G is a congruence subgroup.
Returns true if and only if G is equal to Γ0(N) for some integer N.
Returns true if and only if G is equal to Γ1(N) for some integer N.
Returns the base ring over which matrices of the congruence subgroup G are defined.
Returns the identity matrix in the congruence subgroup G.