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Let $V(G)$ be a complex algebraic curve given by $$G=X_0^2X_1^2-X_0^2X_2^2+X_1^2X_2^2$$ find a parametrization for $V(G)$.

The curve defined by $G$ has three singular points $(1:0:0)$, $(0:1:0)$, $(0:0:1)$ of multiplicity $2$, and I think I know the procedure to find a rational parametrization, that is, constructing a linear system of conics, $\Lambda$, that goes through the three singular points and another point $a\in V(G)$.

Theoretically this should work. However, I don't know how to find the point $a\in C$ to construct $\Lambda$, and even if I knew what point to choose, I don't think I know how to construct $\Lambda$.

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    Over what field are you working? To find $a$, you can set $X_0 = X_1 = 1$ and then solve the resulting quadratic, but if your base field field isn't algebraically closed, this may require a field extension. If you're working over $\mathbb{C}$, then this works fine, though. – 2017-02-06

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