I want to calculate the cardinality of $E_p(\mathbb F_p)$ where $E_p$ is the elliptic curve defined by the reduction of $Y^2=X^3-17$ modulo $p$.
Somehow I don't get the right results/the results the computer tells me to be true.
If $p=2$, then I need $Y^2 \equiv X^3-1 \mod 2$. To me this seems fulfilled only for $(0,1), (1,0) \in \mathbb F_2$.
Including the point at infinity we get $\text{card } E_p(\mathbb F_p)=3$, but the computer tells me $\text{card } E_p(\mathbb F_p)=2$.
If $p=3$, then I need $Y^2 \equiv X^3-2 \mod 3$. To me this seems fulfilled only for $(0,1), (0,2), (2,0) \in \mathbb F_2$.
Including the point at infinity we get $\text{card } E_p(\mathbb F_p)=4$, but the computer tells me $\text{card } E_p(\mathbb F_p)=3$.
If $p=5$, then I need $Y^2 \equiv X^3-2 \mod 5$. To me this seems fulfilled only for $(1,2), (1,3), (2,1), (2,4), (3,0) \in \mathbb F_2$.
Including the point at infinity we get $\text{card } E_p(\mathbb F_p)=6$, this time this is the right result.
I am quite convinced that my calculations are right. Did I miss something conceptually? Would be nice, if anybody could check my calculations.