Let $E: y^2z = x^3 - Axz^2 - Bz^3$ be a plane elliptic curve. I want to calculate the intersection of $E$ with the coordinate hyperplanes $H_i = \{x_i = 0\}$, $i=1,2,3$. I write $H_x = \{x=0\}, H_y = \{y=0\}, H_z = \{z=0\}$.
Let $0 = [0:1:0]$ be the point at infinity and $E[2]$ be the $2$-torsion.
We have $E \cap H_y = E[2] \setminus \{0\}$ and $E \cap H_z = \{0\}$ (with multiplicity 3). For $E \cap H_x$, we get $\{0\} \cup \{[0:\pm\sqrt{B}:1]\}$.
Now my question is: What is the order of the points $[0:\pm\sqrt{B}:1]$ in $E(k)$?