= j-invariant of an elliptic curve
{c}
{title2=$j(E)=c_4^3/\Delta$}
The j-invariant is unchanged by admissible changes of <Weierstrass equation of an elliptic curve>. With $b_2=a_1^2+4a_2$ and $b_4=a_1a_3+2a_4$, 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.
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