Related to rationally derived polynomials.

Neither Maple nor Magma online couldn't solve it.

Choose $s,h \in \mathbb{Q}$.

I am looking for rational points (possibly of finite order) on this elliptic curve or a Weierstrass model for it:

```
4*u^8 + 4*u^7*v + 16*u^6*v^2 + 28*u^5*v^3 + 40*u^4*v^4 + 28*u^3*v^5 + 16*u^2*v^6 + 4*u*v^7 + 4*v^8 - 18*u^7*h - 30*u^6*v*h - 90*u^5*v^2*h - 150*u^4*v^3*h - 150*u^3*v^4*h - 90*u^2*v^5*h - 30*u*v^6*h - 18*v^7*h + 35*u^6*h^2 + 78*u^5*v*h^2 + 189*u^4*v^2*h^2 + 260*u^3*v^3*h^2 + 189*u^2*v^4*h^2 + 78*u*v^5*h^2 + 35*v^6*h^2 - 36*u^5*h^3 - 84*u^4*v*h^3 - 168*u^3*v^2*h^3 - 168*u^2*v^3*h^3 - 84*u*v^4*h^3 - 36*v^5*h^3 + 16*u^4*h^4 + 32*u^3*v*h^4 + 48*u^2*v^2*h^4 + 32*u*v^3*h^4 + 16*v^4*h^4 - 18*u^7*s - 30*u^6*v*s - 90*u^5*v^2*s - 150*u^4*v^3*s - 150*u^3*v^4*s - 90*u^2*v^5*s - 30*u*v^6*s - 18*v^7*s + 86*u^6*h*s + 204*u^5*v*h*s + 522*u^4*v^2*h*s + 680*u^3*v^3*h*s + 522*u^2*v^4*h*s + 204*u*v^5*h*s + 86*v^6*h*s - 180*u^5*h^2*s - 516*u^4*v*h^2*s - 1032*u^3*v^2*h^2*s - 1032*u^2*v^3*h^2*s - 516*u*v^4*h^2*s - 180*v^5*h^2*s + 200*u^4*h^3*s + 544*u^3*v*h^3*s + 816*u^2*v^2*h^3*s + 544*u*v^3*h^3*s + 200*v^4*h^3*s - 96*u^3*h^4*s - 192*u^2*v*h^4*s - 192*u*v^2*h^4*s - 96*v^3*h^4*s + 35*u^6*s^2 + 78*u^5*v*s^2 + 189*u^4*v^2*s^2 + 260*u^3*v^3*s^2 + 189*u^2*v^4*s^2 + 78*u*v^5*s^2 + 35*v^6*s^2 - 180*u^5*h*s^2 - 516*u^4*v*h*s^2 - 1032*u^3*v^2*h*s^2 - 1032*u^2*v^3*h*s^2 - 516*u*v^4*h*s^2 - 180*v^5*h*s^2 + 408*u^4*h^2*s^2 + 1248*u^3*v*h^2*s^2 + 1872*u^2*v^2*h^2*s^2 + 1248*u*v^3*h^2*s^2 + 408*v^4*h^2*s^2 - 480*u^3*h^3*s^2 - 1248*u^2*v*h^3*s^2 - 1248*u*v^2*h^3*s^2 - 480*v^3*h^3*s^2 + 240*u^2*h^4*s^2 + 384*u*v*h^4*s^2 + 240*v^2*h^4*s^2 - 36*u^5*s^3 - 84*u^4*v*s^3 - 168*u^3*v^2*s^3 - 168*u^2*v^3*s^3 - 84*u*v^4*s^3 - 36*v^5*s^3 + 200*u^4*h*s^3 + 544*u^3*v*h*s^3 + 816*u^2*v^2*h*s^3 + 544*u*v^3*h*s^3 + 200*v^4*h*s^3 - 480*u^3*h^2*s^3 - 1248*u^2*v*h^2*s^3 - 1248*u*v^2*h^2*s^3 - 480*v^3*h^2*s^3 + 576*u^2*h^3*s^3 + 1152*u*v*h^3*s^3 + 576*v^2*h^3*s^3 - 288*u*h^4*s^3 - 288*v*h^4*s^3 + 16*u^4*s^4 + 32*u^3*v*s^4 + 48*u^2*v^2*s^4 + 32*u*v^3*s^4 + 16*v^4*s^4 - 96*u^3*h*s^4 - 192*u^2*v*h*s^4 - 192*u*v^2*h*s^4 - 96*v^3*h*s^4 + 240*u^2*h^2*s^4 + 384*u*v*h^2*s^4 + 240*v^2*h^2*s^4 - 288*u*h^3*s^4 - 288*v*h^3*s^4 + 144*h^4*s^4
```

subject to the constraint $u,v,h$ are all distinct.

Some points are $(u,v,h,s)=(-t,-t,-t,t)$ or $(t,t,-t,t)$.

Modulo my errors it is conjectured that there are no points subject to the constraint.

Suspect that other points (if any) will be of large height.