= = Let K be a field of characteristic + 2. Let a, b € K be such that b#0 and a2 - 46 +0. (1) Show that E: y = 13+ ax? +
Posted: Mon Apr 11, 2022 5:57 am
- Let K Be A Field Of Characteristic 2 Let A B K Be Such That B 0 And A2 46 0 1 Show That E Y 13 Ax 1 (38.42 KiB) Viewed 18 times
= = Let K be a field of characteristic + 2. Let a, b € K be such that b#0 and a2 - 46 +0. (1) Show that E: y = 13+ ax? + bx, Ex : y2 = 23 – 2ax2 + (a? – 4b)... are elliptic curves. (2) Show that (,y= 22 is an isogeny from E, to Ey. (3) Determine ker(6). 22 olx.x) = Comment: For part (1), you may use the formula for the discriminant of a general cubic polynomial given at class. In part (2), for the sake of brevity, when checking that (P1 + P2) = o(P1) + 6(P2), you may just consider the general case in which P1, P2¢ {(0,0), O}. When one among P1, P2 lies in {(0,0), O}, the general argument fails and an additional argument is needed; feel free to disregard this degenerate case.