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This is a continuation of the following question: Szpiro ratios of elliptic curves over $\mathbb{Q}$

In that question I asked whether Szpiro ratio

$$\displaystyle \beta_E = \frac{\log |\Delta_{\min}(E)|}{\log N(E)}$$

where $E/\mathbb{Q}$ is an elliptic curve over the rationals, $\Delta_\min(E)$ its minimal discriminant, and $N(E)$ its conductor, is dense over the interval $[1,6]$ as $E$ varies over all elliptic curves over $\mathbb{Q}$. Mike Bennett gave a construction using quadratic twists that answers this question in the affirmative.

This question asks a refined version of the question, which is to ask whether $\beta_E$ is dense over $[1,6]$ as $E$ varies over semi-stable elliptic curves, given in short Weierstrass model, say. Recall that an elliptic curve is semi-stable if and only if it does not have any additive bad reduction.

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    $\begingroup$ You can probably prove density in $[1,6]$ by working with parametrized families of Mordell curves $y^2=x^3+k$ of positive rank, where you can control the size of a point of infinite order. Then multiples of this point will give you tuples $(x,y,z)$ of integers with $y^2=x^3+k z^6$, where the short Weierstrass curve $Y^2=X^3+(-3x) X +2y$ has Szpiro ratio tending upwards to $6$ (as $z$ increases). There are, of course, lots of technical details to handle, as well as extra care required (via local conditions) to avoid additive reduction at $2$ and $3$. $\endgroup$
    – Mike Bennett
    Commented Feb 25, 2022 at 21:53

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