# Elliptic curves, inflection points and divisors

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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Let the base field be $F_2$ (hopefully you're fine with a finite base field). Let the curve be $y^2+y=x^3$ and the line $y=0$. The function $y$ has a pole of order 3 at the point of infinity ${\mathcal O}$ and a triple zero at origin ${\mathcal P}=(0,0)$, so the divisor of $y$ is $3{\mathcal P}-3{\mathcal O}$ as prescribed.

Edit: D'oh. The OP asked for examples in other characteristics. I'm apparently at a my dullest. Doesn't the same example work in any characteristic? (Except at char 3, because then the curve has a singular point). See a figure of the real points below.

The origin looks like an inflection point to me :-)

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Jyrki: Curious - it appears this answer is not associated with any user, so I cannot even merge it to your account. I recommend reposting, and creating a bug report on meta. –  Zev Chonoles Sep 12 '11 at 7:37
@Zev: Thanks for tending to the matter. I will try to follow your advice. –  Jyrki Lahtonen Sep 12 '11 at 7:40
@J.M. It was deleted by Jeff Atwood. I am not sure what the reasoning is for this, but hopefully we will find out soon. –  Zev Chonoles Sep 12 '11 at 7:57
Everything looks like it's better now. –  Zev Chonoles Sep 12 '11 at 7:58
@Jasiu: This kind of examples abound in all characteristics. The condition is equivalent to finding a point ${\mathcal P}$ of order 3. I would just have to think a bit harder to find one, as my experience is mostly with char 2 (for reasons similar to yours:-) –  Jyrki Lahtonen Sep 12 '11 at 8:25