I am not entirely sure this question is research-level question, but I have tried stack exchange and have received no response there.
During a conversation with a professor, I was informed of the following statement: $x^4+kx^2y^2+y^4=z^2$ parametrizes all elliptic curves and fixes all the 4-torsion points. Here by fixing all the 4-torsion points I believe he meant that for any $t$, the projection $E\rightarrow \mathbb{P}^1$ sends any 4-torsion point on $E$ to one of $\{0,1,-1,i,-i,\infty\}$. I wonder if this is true, and is there any reference or proof of this.
I have thought about the following verification: by $X=\frac{x^2}{y^2}$ and $Y=\frac{zx}{y^3}$ one gets $X^3+kX^2+X=Y^2$, and solve the 4-torsion condition via explicit computations. But firstly, as professor Elkies mentioned here, this is not an isomorphism but a 2-isogeny, I am not sure if this will affect the verification. And secondly, I think there might be some insights in this other than just computations.
