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I'm studying basics of elliptic curves. I'm reading An Elementary Introduction to Elliptic Curves by Leonard Charlap and David Robbins. It is stated there that the divisor of a line (i.e. a polynomial of the form $ax + by + c$) can have only few forms, among them is $3\langle P \rangle - 3\langle \mathcal{O}\rangle$. I tried to find an example of a curve and a line on it that has such divisor, but to no avail. Can anyone provide an example? If it helps, they suggest that $P$ is an inflection point.

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    That's actually an "if and only if". More precisely, assuming that $\mathcal O$ is an inflection point (which it is if you're using a standard Weierstrass equation), then the divisor of $ax+by+c$ has the form $3\langle P\rangle-3\langle\mathcal O\rangle$ if and only if $P$ is an inflection point and the line $ax+by+c$ is the tangent line at $P$.2016-11-15

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