I think $x^4+ux^2y^2+y^4=z^2$ is an elliptic curve, so how should I transform it into Weierstrass form? Either by hand or by software like MAGMA is fine. I am new to MAGMA and I tried something similar to this, but it seems like the "Curve" function requires the function to be homogeneous, which I don't know why and how to get around.
Edit: Here $u$ is a parameter, and I meant to ask that for every $u$, what is the corresponding Weierstrass form, with $u$ as a parameter in it. The base field considered is $\mathbb{Q}$.
The good answer by Elkies is missing a little detail which I now supply and explain.
Suppose that $\,u,x,y,z\,$ are integers with $\,y\ne0.\,$ Given the equation $$ x^4 + ux^2y^2 + y^4 = z^2, $$ Elkies suggests to define $\,t = x/y\,$ but omits that he also assumes $\,v=z/y^2.\,$ Thus $$ 1 + u t^2 + t^4 - v^2 = (x^4 + u x^2 y^2 + y^4-z^2)/y^4. $$
Similarly, use $\,X = x^2/y^2,\; Y = xz/y^3\,$ to get the Weierstrass form $$ X^3 \!+\! uX^2 \!+\! X \!-\! Y^2 \!=\! (x^4 \!+\! ux^2y^2 \!+\! y^4 \!-\! z^2)x^2/y^6. $$
Thus, rational solutions to $\, X^3 + uX^2 + X = Y^2 \,$ (assuming $\,X = t^2$) correspond to integer solutions to $\, x^4 + ux^2y^2 + y^4 = z^2 \,$ by using $\,x = y\,t,\; z = Yy^2/t.$
For $\,u=-1\,$ the elliptic curve with LMFDB label 24.a5 $\,y^2 = x^3-x^2+x\,$ has only three rational points which are $\,(0,0),\;(1,1),\;(1,-1)\,$ and also the point at infinity.
For $\,u=1\,$ the elliptic curve with LMFDB label 48.a5 $\,y^2 = x^3+x^2+x\,$ has only one rational point which is $\,(0,0)\,$ and also the point at infinity.