Consider the points which satisfy the equation mod p y² = x³ + ax + b b 15, and p where a = 2₁ = = = 19. This curve contains the point P = (4,7). Enter a comma separated list of points (x, y) consisting of all multiples of P in the elliptic curve group with parameters = 2,6 15, and p = 19. (Do not try to enter O, the a = point at infinity, even though it is a multiple of P.) What is the cardinality of the subgroup generated by P?

2. Find the terminal point P (x, y) and the reference number determined by t Round the coordinates of the terminal point to the nearest hundredth, and express the reference number in terms of (x, y) = ( Reference number 61T 6 t=₁ ) <-write decimal round to hundredth

Find the length of the curve for 12x = 4y³ + 3y-¹ where 1 ≤ y ≤ 3 Enter your answer in exact form. If the answer is a fraction, enter it using / as a fraction. Do not use the equation editor to answer.

5. By considering lines through the point (-1,0), find all rational points on the ellipse x²+2y² = 1.