Recall that an elliptic curve over a field $K$ is a projective nonsingular curve $E$ defined over $K$ of genus $1$ together with a point $O\in E$ defined over $K$ .

All elliptic curves have a Weierstrass model in $\mathbb{P}^2(K)$ , the projective plane over $K$ . This is a simple application of the Riemann Roch theorem for curves:

Moreover, the following proposition specifies any possible change of variables.

Once we have one Weierstrass model for a given elliptic curve $E/K$ , and as long as the characteristic of $K$ is not $2$ or $3$ , there exists a change of variables (of the form given in the previous proposition) which simplifies the model considerably.

Finally, remember that the $j$ -invariant of an elliptic curve is invariant under isomorphism, but the discriminant depends on the model chosen.

Proposition 2   Let $E/K$ be an elliptic curve and let $$ E_1: y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6, \quad E_2: y'^2+a_1x'y'+a_3y'=x'^3+a_2x'^2+a_4x'+a_6$$ be two distinct Weierstrass models for $E/K$ . Then (by Prop. ) there exists a change of variables $(x,y)\mapsto (x',y')$ of the form: $$x=u^2x'+r,\quad y=u^3y'+su^2x'+t$$ with $u,r,s,t\in K$ and $u\neq 0$ . Moreover, $j(E_1)=j(E_2)$ , i.e. the $j$ invariants are equal ($j(E)$ is defined in this entry) and $\Delta(E_1)=u^{12}\Delta(E_2)$ , where $\Delta(E_i)$ is the discriminant (as defined in here).