An elliptic curve given by $E: y^2=x^3+ax+b$ with $a,b \in K$ and $Δ(E)=-16(4a^3+27b^2) \neq 0$ is adequate for elliptic curves with $char\neq2,3$ Because of the factor -16 in the definition of $Δ(E)$, according to this definition there are no elliptic curves in characteristic 2.
Ok I understand the short Weierstrass equation, and I understand the discriminant equation. So my questions are:
- A long Weierstrass equation is needed only for curves with char=2 or 3?
- Why does the factor of -16 imply that curves of characteristic 2 do not exist?
- I read in another paper that "$E: y^2=x^3+ax+b$ does not apply in char 2 because $E: y^2=x^3+ax+b$ is always singular in char 2" but I'm not sure what this means. Singular means the discriminant =0 right? I guess this is related to #2, how does is the value of the characteristic used in the discriminant equation?