$abc$ quality (awaiting review)

Given a triple $a,b,c$ of nonzero coprime integers, the quality of the triple is defined as $$ Q = \frac{\log \max(|a|, |b|, |c|)}{\log \operatorname{rad}(abc)}, $$ where $\operatorname{rad}(abc)$ is the product of the primes dividing $abc$. The $abc$ conjecture stipulates that for any $\epsilon > 0$ there are only finitely many relatively prime triples $a,b,c$ with quality larger than $1+\epsilon$.

The $abc$ quality of an elliptic curve $E$ is the quality of an $a,b,c$ triple determined by its $j$-invariant, namely the one defined by writing $\frac{j}{1728} = \frac{a}{c}$ in lowest terms and setting $b = c - a$. Note that the $abc$ quality is undefined for $j=0$ and $j=1728$.

The reason for defining the quality of $E$ in this way comes from the equivalence of the $abc$ conjecture with the modified Szpiro conjecture. For elliptic curves with small conductor, $j$-invariants often have unusually large quality.