What means a birational isomorphism from an elliptic curve to $P^1$?

Question

What means a birational isomorphism from an elliptic curve to $P^1$? An elliptic curve is given by equation $y^2=x^3+p\cdot x+ q$.

Answer 1

Let's work over an algebraically closed field $K$ of characteristic $\neq 2, 3$, so that the equation of the elliptic curve $C$ can be assumed of the form $$y^2 = x(x - 1)(x - c).$$ after some change of coordinates, as you mentioned in the comments.

Next, assume for a contradiction that there exists a birational map from $\mathbf P^1_K$ to the elliptic curve $C$. This is the same as saying that $K(t)$, the function field of $\mathbf P^1_K$, is isomorphic to the function field of $C$, which is $K(x,y)$ - the fraction field of $$K[x, y]/(y^2 - x(x-1)(x-c)).$$ Let $f, g \in K(t)$ corresponds to $x$ and $y$ under this isomorphism. Writing $f = p/q$ and $g = r/s$, for polynomials $p, q, r, s \in K[t]$, we have $$\left(\frac r s\right)^2 = \frac p q \left(\frac p q - 1\right)\left(\frac p q - c\right)$$ or, equivalently, $$q^3r^2 = s^2p(p -q)(p -cq).$$

Would you now know how to proceed?