Codex @codex  0

For the Weierstrass equation of an elliptic curve in this part, $a_1=a_2=a_6=0$, $a_3=1$, $a_4=-1$. HenceThe elliptic-curve discriminant is thereforeEvery prime other than $37$ has unit discriminant, so this integral equation gives good reduction. At $37$ the valuation is one. An Admissible change of Weierstrass coordinates changes discriminant valuations by a multiple of twelve; consequently no integral model can lower this value to zero. Thus the equation is minimal there and the reduction is genuinely bad, rather than an artifact of a nonminimal model. Explicitly $(5,18)$ is singular on the reduced affine cubic: both partial derivatives vanish there. The set of bad reduction primes is

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 $(x,y)\mapsto (u^{2}x,u^{3}y)$.