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I am trying to compute the number of 3 x 3 matrices A over the field of p elements (p prime) such that A^2 = I. I wrote the following code which gives me the answers for p=2 and p=3, but it is taking for ever to compute when for p=5. Is there a better way to do this job which can speed up the computations? Many thanks for help.

def num_invol(n): G = GL(n,GF(3)); sum = 0 for A in G: if A^2 == G.one(): sum = sum +1 return sum

for p in (2..11): if p.is_prime(): print (p, num_invol(p))

I am trying to compute the number of 3 x 3 matrices A over the field of p elements (p prime) such that A^2 = I. I wrote the following code which gives me the answers for p=2 and p=3, but it is taking for ever to compute when for p=5. Is there a better way to do this job which can speed up the computations? Many thanks for help.

def num_invol(n):
    G = GL(n,GF(3));
    sum = 0
    for A in G:
        if A^2 == G.one():
                sum = sum +1
    return sum
sum
  
for p in (2..11):
    if p.is_prime():
        print (p, num_invol(p))num_invol(p))