Given some natural number $n$ and the elliptic curve $E:y^2=x^3-x-n$, how can I find the minimal model of $E$?

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 $n\leq20,000$.

Given some natural number $n$ and the elliptic curve $E:y^2=x^3-x-n$, how can I find the minimal model of $E$? Obviously we can reduce all sixth powers of $p$, which reduces to another curve of the form $y^2=x^3-x-n$ with now

$$n'=\prod_{p^{\alpha}|n}p^{\alpha\,\,\mathrm{mod}(6)}$$

Is the curve $E':y^2=x^3-x-n!$ obtained in this fashion always going to be minimal?

Given some natural number $n$ and the elliptic curve $E:y^2=x^3-x-n$, how can I find the minimal model of $E$?