Elliptic curves with trivial Mordell–Weil group over certain fields. I am looking for elliptic curves $E,E'$ defined over $\mathbb{F}_{3}$ and $\mathbb{F}_{4}$ respectively and given by a Weierstrass equation such that their Mordell-Weil group is trivial, i.e. such that 
$$
E(\mathbb{F}_{3})=\{\mathcal{O}\}\quad\text{and}\quad E'(\mathbb{F}_{4})=\{\mathcal{O}\},
$$
where $\mathcal{O}$ is the point at infinity. Is there any procedure to find them? Or is the only way just to change the coefficients until you find one satisfying the requirements?

Let $K=\Bbb{F}_4=\{0,1,\alpha,\alpha+1\}$ with $\alpha^2=\alpha+1$.
For all $x\in K*$ we have $x^3=1$. Therefore the cubic $x^3+\alpha$ only takes values in $K\setminus\Bbb{F}_2$. On the other hand, $y^2+y\in\Bbb{F}_2$ for all $y\in K$. 
Therefore the equation
$$
y^2+y=x^3+\alpha
$$
has no solutions $(x,y)\in K^2$. The point at infinity is thus the only $K$-rational point of this elliptic curve.
