Does $0$ lie on an elliptic curve, where $0$ is the identity (e.g. $p + 0 = p$)?
Does 0 lie on elliptic curve?
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elliptic-curves
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1@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
1 Answers
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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.