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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 $E$. –  Keenan Kidwell Nov 21 '12 at 16:31
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I think i mean the point at infinity. Is that any help? –  tm1rbrt Nov 21 '12 at 16:36
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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! –  DonAntonio Nov 21 '12 at 17:04
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Every elliptic curve $E$ over a field $k$ is isomorphic to the locus in $\mathbf{P}_k^2$ cut out by a nonsingular Weierstrass equation in such a way that the given rational point on $E$ is mapped the point at infinity. So I guess the answer to your question is yes.

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