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Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?

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    Uh, I don't really understand your question. An elliptic curve over a field $k$ is a smooth projective curve $E$ of genus $1$ together with a distinguished $k$-rational point $0\in E(k)$ which serves as the identity for the group law on $E$.2012-11-21
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    @tm1rbrt, the question is: does that help *you*? The point at infinity in projective space can hardly be recognized as an actual point on the elliptic curve (or on any projective curve, for that matter). We use it basically to provide Poincare's group with a neutral element (so that it is *actually* a group), and of course you can think of it as "the point at infinity on the curve", as it's usually done...but you won't spot it on the curve!2012-11-21

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