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How to decide whether a given curve is elliptic

I have the equation $y^2+2y=x^3+x^2-x-2$ (over $\mathbb C$) then which condition is true;

If the RHS (polynomial in $x$) has no repeated roots, then the curve is elliptic, (I think this holds if the LHS has a single term, so $2y$ destroys everything)

If there is no singular point then it is elliptic, well for example the point $(-1,1)$ is a singular point, but it does not necesssarily lie on the curve

user1161
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1 Answers1

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You can complete the square to get a single term on the lefthand side: $$ \tilde{y}^2 = (y+1)^2 = x^3 + x^2 - x - 1 = (x-1)(x+1)^2 \, , $$ where we have let $\tilde{y} = y+1$. The new righthand side has a repeated root, so the curve is singular.

Viktor Vaughn
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