If $E$ is a non-CM elliptic curve over $Q$, then it is a famous theorem of Serre that there is some integer $M(E)$ such that for any prime $\ell > M(E)$, the image of the Galois representations $\rho_{E,l}: G_{\mathbb Q} \rightarrow Gl_2(\mathbb F_\ell)$ on the $\ell$-torsion points of $E$ is surjectibe. If we define $M(E)$ as the smallest integer having this property, then Serre's bounded uniformity conjecture is that $M(E)$ is bounded above by an absolute constant (41 perhaps) when $E$ varies over all non-CM-elliptic curve over $\mathbb Q$.

Let $N(E)$ be the product of all primes where $E$ has bad reduction. My question is:

What are the current best bounds (if any) of $M(E)$ in terms of $N(E)$, both unconditionally and under GRH (the second being the case that interests me most)?

I kind of get lost in the immense literature on the subject. As is well-known, there are four types of proper maximal subgroups of $Gl_2(\mathbb F_\ell)$, which are (1) Borel subgroups, (2) (resp. (3)) Normalizer of split (resp. non-split) Cartan subgroups, (4) exceptional ones (icsoahedral, dodecahedral, etc), so if $\rho_{E,\ell}$ fails to be surjective, it lands in a subgroup of one of those type, and we can define four integers $M^i(E)$ for $i=1,..,4$ ,as the smallest integer such that for $\ell$ larger that $M^i(E)$, $\rho_{E,\ell}$ does not fall into a subgroup of type (i). Please tell me if I am not correct (I am troubled by Lemma 17 page 197 of this paper of Serre) , but it is known that $M^4(E)$,$M^1(E)$,$M^2(E)$ are bounded by an absolute constant (independent of $E$) due to results of Serre, Mazur, and Bilu-Parent (respectively and in chronological order, the last one being very recent). So the only problem that remains for Serre's uniformity conjecture would be to bound $M^3(E)$ uniformly, and for my question to bound it at least in terms of $N(E)$.