Are finite field computations slow?
I ran the following in Sage 4.5.3 on a MacBook Pro (running OSX):
p = 5; pow = 2; k = GF(5 ** pow, 'z'); f(x, y) = y^3 + y + 4*x; def count_pts(): retval = 0; for a in k: for b in k: if (f(a, b) == 0): retval = retval + 1; return retval + 1; count_pts()
It took around 10-20 seconds to return the correct answer of 26. With pow = 1 it terminated much more quickly. The loop is only over 625 pairs of elements of GF(25); is there some reason this program runs so slowly? (And can the performance be improved?)