Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 125 / 1 / a / Solution

The Weierstrass equation of an elliptic curve over a field $K$ has the general formwith nonzero elliptic-curve discriminant. Admissible change of Weierstrass coordinates relates two such equations defining isomorphic pointed curves over $K$:where $u\in K^{\times }$ and $r,s,t\in K$.

If $\mathrm{char}K\ne 2,3$, completing the square and translating $x$ simplify every equation. Short Weierstrass form isTwo short equations are $K$-isomorphic precisely when, for some $u\in K^{\times }$,the isomorphism from the first curve to the second is $(x,y)\mapsto (u^{2}x,u^{3}y)$.