Weierstrass equation or model (reviewed)

A Weierstrass equation or Weierstrass model over a field $k$ is a plane curve $E$ of the form $$y^2 + a_1xy + a_3y = x^3 + a_2 x^2 + a_4 x + a_6,$$ with $a_1, a_2, a_3, a_4, a_6 \in k$.

The Weierstrass coefficients of this model $E$ are the five coefficients $a_i$. These are often displayed as a list $[a_1, a_2, a_3, a_4, a_6]$.

It is common not to distinguish between the affine curve defined by a Weierstrass equation and its projective closure, which contains exactly one additional point at infinity, $[0:1:0]$.

A Weierstrass model is smooth if and only if its discriminant $\Delta$ is nonzero. In this case, the plane curve $E$ together with the point at infinity as base point, define an elliptic curve defined over $k$.

Two smooth Weierstrass models define isomorphic elliptic curves if and only if they are isomorphic as Weierstrass models.