The discriminant of a singular elliptic curve

An elliptic curve defined by

y2 = x3 + ax + b

has discriminant

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.

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