Let $K$ be a field of characteristic zero, $V/K$ a smooth projective variety and $F=\overline{K}(V)$ be the function field of $V$ over $\overline{K}$. For any elliptic curve $E/F$, it has a Weiertrass equation of the form $$E: y^2=x^3+Ax+B, \qquad A,B\in F. $$ Is it possible to find a Weierstrass equation such that for all prime divisors $D \in Div_{\overline{K}}(V)$, we have i) $ord_{D}(A)<0$ or ii) $ord_D(B)<0$ or iii) $0\leq ord_D(A)<4$ or iv) $0\leq ord_D(B)<6$ ? In other words, is it always possible to find a global Weierstrass equation that is minimal w.r.t. $D$ whenever the Weierstrass equation is integral w.r.t. $D$ ?
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