Posters: the poster plan (title, abstract, references, group expectations) is due soon. Check the Posters tab for the dates and for what goes in the plan. Not in a poster group yet? Please come talk to me (after class, office hours, or email) so we can sort that out before the plan is due.

Lecture notes: my notes from Friday’s lecture on elliptic curves and the group law are posted (also on the Notes/Videos page and the Archive).

Solutions: solutions to the daily posts due September 28th and 30th and October 2nd, 5th and 7th are now posted (Sep 28, Sep 30, Oct 2, Oct 5, Oct 7). All daily post solutions are collected on the Archive page.

Tools: the Elliptic Curve Tools page (Sage) will add points for you, and the Elliptic Curve Playground draws the whole group as a picture. Do the problems below by hand first, then use these to check.

Relevant Textbook Sections:

5.1 (What is an elliptic curve?), 5.2 (Double points).

To Do:

1. Compute the sum of the following two points on this elliptic curve. $E: y^2 = x^3 + x + 1$ modulo $7$. $P = (0,1)$ and $Q = (2,2)$. Compute $P+Q$.
2. What is $P + \infty$ on this curve?
3. What is $-P$?
4. Suppose, on the same curve, we want to compute the sum of $Q$ with itself. That means we need to find a tangent line! Compute $Q+Q$.
5. Show that if $P=(x,0)$ is a point on an elliptic curve, then $2P = \infty$.
6. On the curve $y^2 = x^3 +2x + 1$ modulo $5$, compute the sum of the two points $P = (3,3)$ and $Q = (1,2)$ by hand. Check that your answer is on the curve.
7. On the curve $y^2 = x^3 +2x + 1$ modulo $7$, try to compute the sum of the two points $P = (0,6)$ and $Q = (0,1)$ by hand. The result should be the point at infinity (because these are inverses of each other), so the computation will “fail” — explain what fails that lets you know the result is the point at infinity.
8. (A preview of things to come) On the curve $y^2 = x^3 +2x + 1$ modulo $35$, try to compute the sum of the two points $P = (28,13)$ and $Q = (21,22)$ by hand. The result will fail. Explain why it fails, and how the failure factors $35$. Explain the relationship to the previous two problems.