Given $2P$ find $P$ on elliptic curve
If $C:y^2=x^3-4$ over $\mathbb Q$ then using division polynomials find $P$ s.t. $2P=(5,11)$, With the formulas Here my problem is reduced to finding one solution $(x,y)$ with,
$4xy^2-3x^4+48x-20y^2=0$ and $4y(x^6-8x^3-128)-352y^4=0$ and wolframalpha yields this and since the values are approximated I cannot test which of them satisfy the desired equation, can you help ?
I have not checked the accuracy of the formulas you are using for the multiplication-by-2 map, but here is the formula for the $x$-coordinate of $2P$ that appears in Silverman's "The Arithmetic of Elliptic Curves" (Ch III, Group Law Algorithm 2.3), when simplified for your curve: $$x(2P) = \frac{x^4+32x}{4x^3-16}.$$ Since $x(2P)=5$ in our case, we need $$x^4+32x=5(4x^3-16)$$ or, equivalently, we need $x^4-20x^3+32x+80=0$. This polynomial has a unique linear factor over $\mathbb{Q}[x]$, and this factor gives you a unique rational root which corresponds to the $x$-coordinate of the point $P$ you are looking for.