I am trying to compute the number of prime divisors of the order of $E_8(q)$. I am interested in the general solution, but in particular, my problem calls for $q=p^{15}$ (for prime $p$) and $q\equiv 0,1,$ or $ 4 \mod 5$, if this helps at all.
So, the order is $|E_8(q)|=q^{120}(q^{30}-1)(q^{24}-1)(q^{20}-1)(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{8}-1)(q^{2}-1)$ (ref: Wilson, The Finite Simple Groups). Is there any more efficient algorithm than the standard to factorize integers of this form? I am primarily interested in knowing the number of prime divisors, but the divisors themselves would also be very useful.
So, among other things, you want to know the (number of) prime factors of $p^{450}-1$ for various primes $p$. Of course the polynomial $x^{450}-1$ factors into lots of smaller pieces, but there's still an irreducible part of degree 120. I can't imagine there would be a general formula for the number of prime factors of $f(p)$ for such a polynomial $p$, nor even much chance of computing them for $p$ with more than 2 digits.
You may find some useful information in the book, Brillhart, Lehmer, Selfridge, Tuckerman, Wagstaff, Factorizations of $b^n\pm1$.
Also, I'm not certain what algorithm you refer to when you write, "the standard." The standard algorithm for factoring that kind of number is probably the Special Number Field Sieve; is that what you had in mind?