A quick follow-up to the previous post: D. Zywina has found the value of the Chebotarev invariant for the Weyl group of E8, which is
So you would typically expect to need to look at a bit more than four Frobenius conjugacy classe, on average, to check that a Galois extension with group a subgroup of W(E8) is not smaller. As we managed to do it with only two with F. Jouve in our particular case, this means that we were indeed somewhat lucky.
(Thanks to Nguyen Ngoc Dong Quan for also proposing to do the computation with Magma).