Does it hold that $E:y^2=x^3-x-n$ is a minimal model for any choice $n$? Using the Sage programming language we can check that $E$ is indeed minimal for every $n\leq20,000$.
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2$\begingroup$ The sentence starting with "Obviously" is incorrect. If you want to change your equation with $p$, the term in front of $x$ would be divided by $p^4$. This means that your equation is minimal except maybe at $2$ and $3$. For those you have to look at Tate's algorithm. From experiments with small $n$, I guess that the answer will be that the equation is indeed globally minimal. $\endgroup$– Chris WuthrichCommented Feb 4, 2021 at 8:55
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3$\begingroup$ It is minimal (indeed good) at 3, as the discriminant is $\Delta=-2^4(27n^2-4)$ so $v_3(\Delta)=0$. Similarly, $v_2(\Delta)\le 7$ implies minimality at 2. $\endgroup$– user171793Commented Feb 4, 2021 at 10:56
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$\begingroup$ @user171793 Thank you for your comment; could you write it up as an answer so I can upvote and accept it? $\endgroup$– Milo MosesCommented Feb 4, 2021 at 15:51
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$\begingroup$ @ChrisWuthrich I removed the "obviously" portion because I realized that my "obvious" argument was flat out incorrect $\endgroup$– Milo MosesCommented Feb 4, 2021 at 16:02
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$\begingroup$ @ChrisWuthrich I don't quite understand your comment; Could you restate it as an answer? $\endgroup$– Milo MosesCommented Feb 6, 2021 at 23:22
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