j-invariant of an elliptic curve ($j(E)=c_4^3/\Delta$)

The j-invariant is unchanged by admissible changes of Weierstrass equation of an elliptic curve. With $b_2=a_1^2+4a_2$ and $b_4=a_1{a}_3+2a_4$, set $c_4=b_2^2-24b_4$ and $j=c_4^3/\Delta$, where $\Delta$ is the elliptic-curve discriminant. For good reduction over a local field, an integral equation with unit discriminant shows that $j$ lies in the valuation ring. Thus a negative valuation of $j$ proves bad reduction, even if the displayed equation has not been proved minimal.