I'm running into some trouble trying to get some practice working with elliptic curves. Say $F$ is some field and $\lambda\in F$ is an element with $\lambda^3 \neq 27$. Let $E$ be the elliptic curve over $F$ with affine Weierestrass equation $$ y^2 + \lambda x y + y = x^3. $$ I'd like to understand the $2$-torsion of $E$ explicitly. See the question in bold below for what I mean by this.
If $P\in E$, then $2P=0$ if and only if $-P = P$. Using $$ -(x,y) = (x,-y-\lambda x - 1), $$ a little computation gives the following. First, if $-(x,y) = (x,y)$ then $y\neq 0$. So let's switch to $(z,w)$ coordinates, where $$ z = -\frac{x}{y},\qquad w = -\frac{1}{y}. $$ Now $E$ has equation $$ 0 = -w + z^3+\lambda zw+w^2, $$ and the $2$-torsion points are of the form $(z,-z^3)$ where $z$ satisfies $z^4-\lambda z^2 + 2z = 0$.
Question. Let $P_1 = (z_1,-z_1^3)$ and $P_2 = (z_2,-z_2^3)$ be two $2$-torsion points on $E$. What is a general formula for the $z$-coordinate of $P_1+P_2$ in terms of $z_1$ and $z_2$?
(To phrase the question in fancier terms, view the elliptic curve as defined over $R = \mathbb{Z}[\lambda,(\lambda^3-27)^{-1}]$. Then the $2$-torsion $E[2]$ should be a group scheme over $R$. The above description shows it's actually affine, with coordinate ring $A = R[z]/(z^4-\lambda z^2 + 2z)$. My question is equivalent to asking for the coproduct on the Hopf algebra $A$ corresponding to the group structure on $E[2]$.)
This should be easy: just follow the recipe for addition in Silverman. Because $P_1$ and $P_2$ are $2$-torsion, if we let $L$ be the line through $P_1$ and $P_2$, then $P_1+P_2=P_3$ where $P_3 = (z_3,-z_3^3)$ is the third point in the intersection $L\cap E$. The line $L$ has equation $w = mz + b$ with $$ m = \frac{-z_2^3 - (-z_1^3)}{z_2-z_1} = -(z_1^2+z_1z_2+z_2^2). $$ Plugging into the equation for $E$ gives $$ 0 = -(mz+b) + z^3 + \lambda z(mz+b) + (mz+b)^2. $$ This should have the roots $z_1$, $z_2$, and $z_3$, and solving for $z_3$ in $$ -(mz+b) + z^3 + \lambda z(mz+b) + (mz+b)^2 = c(z-z_1)(z-z_2)(z-z_3) $$ reveals $$ z_3 = -z_1-z_2-\lambda m - m^2 = z_1+z_2+\lambda z_1z_2-2(z_1^3z_2+z_1z_2^3)-3z_1^2z_2^2. $$
But this can't be right. Since $P_1$ and $P_2$ are $2$-torsion, setting $z_1 = z_2 = z$ should give $0$, but instead it gives $$ 2z+\lambda z^2 - 7 z^4 = 16 z - 6 \lambda z^2. $$
So I've made a mistake or have some misunderstanding somewhere, but I can't figure out where.
$$E:Y^2 + (1+\lambda X)Y - X^3=0$$ $$(x,y)\mapsto (x,-1-\lambda x-y)$$ is an automorphism of $E$, leaving $O=(\infty,\infty)$ fixed.
This must be our $(x,y)\mapsto -(x,y)$ map.
You can check that $\{ (x,y), y = -1-\lambda x-y\} $ gives a cubic equation in $x$ so it contains 3 elements $P_1,P_2,P_3$.
Then $E[2] = \{ O,P_1,P_2,P_3\}$ and the group law is $2P_i =O, P_i+P_j = P_l,i\ne j\ne l$.