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Math 346/646 - Homework 9 hints 1. Consider the case that P1 or P2 is equal to T separately. Otherwise, it may be helpful to use the formula (on page 80 in edition 1) for (x, y) + (0, 0). 2. (a) This should be similar to the example in class where the rank is zero. (b) The rank is positive. (c) This curve has a lot of points. 3. Just use the method we have for computing |im α| and |im α|. It may be helpful to remember that the points (p, 0), (0, 0), (−p, 0) are on C. It will turn out that im α = {±1, ±p}. On the other hand im α will be {1}. Showing the others aren’t possible basically boils down to the equations N 2 = 2M 4 + 2p2 e4 and N 2 = 4M 4 + p2 e4 and N 2 = 2pM 4 + 2pe4 . In the first case, look at the equation modulo 16 and the in the second and third cases, you can get a contradiction using the fact that x2 ≡ −1 (mod p) is not solvable. 1