Szpiro ratio (awaiting review)

The (modified) Szpiro ratio of an elliptic curve $E$ is defined as \[ \frac{\log \max(|c_4|^3, |c_6|^2)}{\log N}, \] where $N$ is the conductor of $E$ and $c_4$ and $c_6$ are defined as for the $j$-invariant. The (modified) Szpiro conjecture is that, for any $\epsilon > 0$, there are only finitely many elliptic curves with Szpiro ratio larger than $6+\epsilon$. In [MR:992208], Oesterlé proves that this conjecture is equivalent to the $abc$ conjecture.

In Oesterlé's paper cited above, there is another conjecture, that the ratio $$ \frac{\log \Delta}{\log N}, $$ also has the property of only taking values larger than $6+\epsilon$ finitely many times (here $\Delta$ is the minimal discriminant of $E$). This conjecture is implied by the modified Szpiro conjecture (and thus the $abc$ conjecture), but it is not currently known to be equivalent. All of the Szpiro ratios in the LMFDB are computed in terms of $c_4$ and $c_6$ rather than $\Delta$ for this reason.