The discriminant of a singular elliptic curve
An elliptic curve defined by
y2 = x3 + ax + b
D = 4 a3 + 27 b2
and is singular when D = 0.
When this happens we have that
b2 = (-4/27) a3
If we think of that as a polynomial in a and b, that’s another singular elliptic curve, isn’t it? Now it's in a and b instead of in x and y.
I'll have to think about that for a while. It may or may not actually be interesting.