Discriminant

The discriminant of an elliptic curve in Weierstraß normal form

$y^{2}=x^{3}+ax+b$

is defined to be

$\Delta =-64a^{3}-432b^{2}$

or

$\Delta ={\frac {-110592a^{3}}{j}}$

in terms of the j-invariant when the field characteristic is not 2 or 3 .

The discriminant of the cubic polynomial in the same form is defined somewhat differently from that of the “elliptic curve” as such, and in fact $\Delta =-1728D$, where

$D={\frac {4a^{3}+27b^{2}}{108}}$

or

$D={\frac {64a^{3}}{j}}$

in terms of the j-invariant.