An elliptic curve is the set of solutions to an equation of the form
 |
(1)
|
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 or 3, then its equation can be written as
 |
(8)
|
in which case, the discriminant is given by
 |
(9)
|
Algebraically, the discriminant is nonzero when the right-hand side has three distinct roots. In the classical case of an elliptic curve
over the complex numbers, the discriminant has
a geometric interpretation. If 
, then the elliptic curve is nonsingular and has curve genus 1, i.e., it is a torus.
If 
and 
,
then it has a cusp singularity, in which case there is one
tangent direction at the singularity. If 
and 
, then its singularity is called an ordinary
double point (or node), in which case the singularity has 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 
.
