Weierstrass equation of an elliptic curve

A Weierstrass equation presents an elliptic curve aswith nonzero discriminant, together with the point at infinity $O$.

Over a discretely valued field, a minimal Weierstrass equation has integral coefficients and has least possible discriminant valuation among integral Weierstrass equations related by admissible changes of variables.

Reducing a minimal integral Weierstrass equation modulo the maximal ideal gives a projective cubic over the residue field. Reducing primitive projective coordinates defines the reduction map on rational points.