The modified Szpiro conjecture is described in
[Wikipedia](https://en.wikipedia.org/wiki/Szpiro_conjecture)
and [here](http://modular.math.washington.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01/node2.html) and [here](http://www.encyclopediaofmath.org/index.php/Szpiro%27s_conjecture).



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> The modified Szpiro conjecture states that: given $\varepsilon > 0$, there exists a constant $C(\varepsilon)$ such that for any elliptic curve $E$ defined over $\mathbb{Q}$ with invariants $c4, c6$ and conductor $f$, we have
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$$\max\{ \vert c\_4 \vert^3, \vert c\_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon } $$


The original Szpiro conjecture requires minimal model, while
the modified one appears to doesn't require minimal model.



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> Does the modified Szpiro conjecture require minimal model?
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> Does the modified Szpiro conjecture allow $a\_1 \ne 0$ and/or $a\_3 \ne 0$?
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> Reference for it and other names?
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Does the modified Szpiro conjecture require minimal model?
The conductor **does not** depend on the model. The $c\_4$ and $c\_6$ do. In fact as $E$ varies among the different integral models for the same elliptic curve, the left-hand side of the inequality differs by an arbitrary $12$-th power of an integer. The conjecture certainly allows for $a\_1 \ne 0$ and $a\_3 \ne 0$. If we impose $a\_1=a\_3=0$ we can get a model that is close to being minimal, and then the left-hand side is multiplied by a bounded factor that can be absorbed into the $C(\epsilon)$.


In fact, the conjecture still makes sense if you restrict to models for the form
$$
Y^2=X^3+AX+B, \qquad A,B \in \mathbb{Z}, \qquad 4A^3+27B^2 \ne 0
$$ 
assuming that the model satisfies the following minimality condition: there is no integer $u>1$ such that $u^4 \mid A$ and $u^6 \mid B$. In this case you can write the conjecture in the form
$$
\max(A^3,B^2) \le C^\prime(\epsilon) \cdot f^{6+\epsilon}.
$$
The $C^\prime(\epsilon)$ has to differ slightly from $C(\epsilon)$ to take account of the fact the model is not entirely minimal at $2$ and $3$. It is an easy exercise to translate between the version the OP gives for minimal models and the version I give for short Weierstrass models (that satisfy the stated minimality condition).


