Question: For all of the following questions show your sage input/output. Compute the order of the curve defined by y^2 = x^3 7*x 25 over the finite field with 47 elements On the curve defined by

Question:

For all of the following questions show your sage input/output.

1. Compute the order of the curve defined by y^2 = x^3 7*x 25 over the finite field with 47 elements
2. On the curve defined by y^2 x*y = x^3 x over GF(2^8) compute the inverse of the point (1,1)
3. On the curve defined by y^2 y = x^3 x^2 x 1 over the finite field with 701 elements, find a generator and show its order.
4. On the curve defined by y^2 = x^3 4187*x 3814 over finite field of size 6421 compute the sum of the points (3711,373) and (4376,2463).
5. On the elliptic curve defined by y^2 = x^3 3361*x 6370 over finite field of size 8461 compute 1001 times the point (1735, 3464).
6. On the elliptic curve defined by y^2 = x^3 1800*x 1357 over finite field of size 8191, let P1 = (1794, 1318) and P2 = (3514, 409), compute the sum of 13 times P1 plus 28 times P2.