The elliptic discriminant determines whether a cubic curve in Weierstrass form is nonsingular. Such a model has an equation of the form
 |
(1)
|
When the discriminant defined below is nonzero, the affine solutions together with the point at infinity
form an elliptic curve. By changing variables,
,
assuming the field characteristic is not
2, the equation becomes
 |
(2)
|
where
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
Define also the quantity
 |
(6)
|
then the discriminant is given by
 |
(7)
|
The discriminant depends on the choice of equations, and can change after a change of variables, unlike the j-invariant.
If the field characteristic is neither 2 nor 3, then its equation can be written as
 |
(8)
|
in which case, the discriminant is given by
 |
(9)
|
For the short equation 
, the discriminant is nonzero
when the right-hand side has three distinct polynomial
roots. In the classical case of an elliptic curve
over the complex numbers, the discriminant has
a geometric interpretation. If 
, then the projective elliptic
curve, including its point at infinity, is
nonsingular and has curve genus 1. Topologically,
it is a torus (Milne 2021). The affine solution set omits
that point at infinity and is a punctured torus. If 
, the cubic model is singular and is not an elliptic
curve. If also 
, it has a cusp, with one tangent
direction. If instead 
, it has an ordinary
double point (or node), with two distinct tangent directions.
Note that the discriminant of an elliptic curve is not the same as the polynomial discriminant
of the corresponding polynomial, but the two kinds
of discriminants vanish for the same values of 
and 
.
