I was studying M. Bhargava's seminal paper titled "Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves"
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 would like to know the way to caculate the effectively computational constant $C$ for which this $N(E) < C(\left| \text{Disc}(E) \right|^{0.1118})$.
Of course, a weak lower bound for $C$ 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!