I am trying to follow case 2 of the procedure given in Cohen:
for the cubic $f(x,y,z) = x^3 + 3 y^3 - 11 z^3$ using the rational point $P_0 = (2 : 1 : 1)$. The tangent at this point is $y = - \tfrac{12}{9}(x-2) + 1$ and it intersects the curve again at the point $P = (28:-19:5)$.
So I built the linear change of variables $(x:y:z) = (\bar x + 297825\bar y + 28\bar z:\bar x + 215600\bar y - 19\bar z :\bar x + 1698144\bar y + 5\bar z)$ which produces
$\begin{align} \bar{f}(\bar x,\bar y,\bar z) =& -7 \bar{x}^3 - 53204877 \bar{x}^2 \bar{y} - 252 \bar{x}^2 \bar{z} \\\\ & - 94477421044413 \bar{x} \bar{y}^2 - 584088120 \bar{x} \bar{y} \bar{z} + 4776 \bar{x} \bar{z}^2 \\\\ & - 53809704024840479199 \bar{y}^3 - 476307229568940 \bar{z} \bar{y}^2 \end{align}$.
As expected $(0:0:1)$ is a root and the tangent to the curve at that point is $\bar{x}=0$. The problem is there are no linear terms in $\bar{y}$ so $y L(0,1) + y^2 Q(0,1) + y^3 C(0,1)$ has $y=0$ as a double root and another point ($y=-8.8516976\ldots \times 10^{-6}$) contradicting the assertion, further $Q(0,1)^2-4 L(0,1) C(0,1)$ is not zero (since $L(0,1)$ is zero, it would have to be $-950232922990035300/901521983$ for that expression to be zero).
Am I making a mistake somewhere or following the procedure wrong or can I find some simpler instructions elsewhere?
I tried to take what you apply the "final trick" and I got this:
? l(1,t) + x*q(1,t) + x^2*c(1,t) % = (165/64*t^3 + 81/64*t^2 - 81/64*t + 27/64)*x^2 + (-99/4*t^2 - 189/10*t + 189/20)*x + (297/5*t + 1764/25) ? (2*x*c(1,t)+q(1,t))^2 - q(1,t)^2 + 4*c(1,t)*l(1,t) % = (27225/1024*t^6 + 13365/512*t^5 - 20169/1024*t^4 - 1053/256*t^3 + 10935/1024*t^2 - 2187/512*t + 729/1024)*x^2 + (-16335/64*t^5 - 20493/64*t^4 + 20331/160*t^3 + 16281/160*t^2 - 5103/64*t + 5103/320)*x + (9801/16*t^4 + 82269/80*t^3 + 22599/400*t^2 - 102789/400*t + 11907/100)
neither of which seem to be Weierstrass form.
? w(s,t) % = s^2 + 1485/16*t^3 + 2673/16*t^2 + 8019/80*t + 11907/400 ? s = (90*x^3 - 756*x^2 + (270*y^3 + 1539*y^2 + 495))/(80*x^2 + (120*y - 440)*x + (45*y^2 - 330*y + 605)); ? t = (15*y + 57)/(20*x + (15*y - 55)); ? w(s,t)*(1600*u)/(64*25*4*c(1,t)) % = x^3 + (3*y^3 - 11) ? c(1,'t) % = 165/64*t^3 + 81/64*t^2 - 81/64*t + 27/64 ? c(1,t) % = (675*x^3 - 11340*x^2 + 63504*x + (2025*y^3 + 23085*y^2 + 87723*y - 7425))/(1600*x^3 + (3600*y - 13200)*x^2 + (2700*y^2 - 19800*y + 36300)*x + (675*y^3 - 7425*y^2 + 27225*y - 33275))