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.