I was studying M. Bhargava Et al's seminal paper titled "[Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves][1]"

And came across a very fascinating observation that they made in the paper. Theorem 1.2 states: 

The number of integral points on an elliptic curve, E, denoted by $N(E)$ is at most $O_\epsilon(\left| \text{Disc}(E) \right|^{0.1117... + \epsilon})
$

However, following the proof is a little above my pay-grade right now, and I'm not entirely sure as to how they derive the proof, which is given in section 5, Theorems 5.1 and 5.2.

I have a couple of questions: 

Is there a fixed value of $\epsilon$ that can act as an upper bound? Am I correct in asserting that $\epsilon$ must be quite small in nature? By quite small, I mean $0.1117... + \epsilon < 1$. The authors state that the earlier upper bound which was given by Venkatesh, Helgott etc. was $O_\epsilon(\left| \text{Disc}(E) \right|^{0.2007... + \epsilon})$. So, can I reasonably assume that $\epsilon + 0.1117... < 0.2007....$

As you can see, I am desperately trying to simply get rid of the $\epsilon$ factor, and would really appreciate a hard upper bound which is not expressed in terms of $\epsilon$ or a constant.

Edit: I know that $N(E) = C(\left| \text{Disc}(E) \right|^{0.1118})$
And would like help in computing $C$. Of course, a weak lower bound is achievable by letting $N(E) = D$, but I would like to know the least number such that $C$ satisfies the condition $\forall E$.

Any help will be much appreciated. 
Thank you!




[1]: https://arxiv.org/abs/1701.02458