Coordinate ring and singularity of the semicubical parabola

The homomorphismhas kernel $(y^2-x^3)$ and image $k[t^2,t^3]$. The quotient is therefore an integral domain, so the curve is irreducible. In characteristic zero its gradient $(-3x^2,2y)$ vanishes at the origin, making the origin its unique singular point of an algebraic variety.