Constructing subgroups by intersection
I'd like to construct a subgroup of Sp(4,Z) of the form:
G0(N)=M(N)∩Sp(4,Z)
where M(N) is a 4×4 matrix over the integer ring with elements that are multiples of the integer N. I think I know how to construct such an M(N) for a given N, but how does one then construct such a subgroup G0(N)? Thanks!
With your definition M(N) is not a group as it does not contain the identity... Do you meant the principal congruence subgroup?
Sorry, I should have put curly brackets around M(N), so it's just a matrix. Specifically, I want to construct:
G0(N)=(ZZZNZ NZZNZN2Z ZZZNZ ZZZZ)∩Sp(4,Z)
We can define congruence subgroups of the modular group in this way, but I want to do the same thing for subgroups of Sp(4,Z). Thanks for your help!
The answer really depends on what kind of computations you want to achieve. Could you precise it in your question? Building such a group in Sage will require some non-trivial amount of work and the only non-trivial operations you might get will come from the software GAP (which is shipped with Sage and used a lot for everything related to group theory). You should have a look at it.
Okay, thanks for your response! What I really want to look at are the generators for such subgroups..