Supersingularity and singularity

Supersingularity is not to be confused with singularity.

When an elliptic curve is defined over real numbers, singularity of the curve is related to its smoothness. More specifically, a curve is singular if its slope at a point is not defined.

Supersingularity, on the other hand, is related to the order of E2^n and how this order relates to the number of points in the underlying finite field.