As the title says: For a given $q \in \mathbb{N}$, in $\mathbb{F}_q$, is it true that different curves on it can have different group orders?
I assume yes. Let $q:=5$. Wikipedia gives $9$ points for $x^3+x+1$. For $x^3-x$, calculating it in the same naive way, I get only $8$:
- $(0,0)$ for $x=0$
- $(1,0)$ for $x=1$
- $(2,1)$ and $(2,4)$ for $x=2$
- $(3,2)$ and $(3,3)$ for $x=3$
- $(4,0)$ for $x=4$
- "infinity"
Bonus question: Is this the reason why factorization using elliptic curves is better? Say, if I don't get a factor with one elliptic curve, I can try another one, and since it might have a different order, I might "somehow" get a factor?