Compute points of finite order on an elliptic curve

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Examples of curves with various torsion groups (taken from Silverman, p. 238):

(1) y2 = x3 - 2. Cyclic of order 1.

(2) y2 = x3 + 8. Cyclic of order 2.

(3) y2 = x3 + 4. Cyclic of order 3.

(4) y2 = x3 + 4x. Cyclic of order 4.

(5) y2-y = x3 - x2. Cyclic of order 5.
(In order to check with the above applet, substitute x/4 for x and (y+4)/8 for y to turn the equation into y2 = x3 - 4x2 + 16.)

(6) y2 = x3 + 1. Cyclic of order 6.

(7) y2 = x3 - 43x + 166. Cyclic of order 7.

(8) y2+7xy = x3 + 16x. Cyclic of order 8.
(Turn the equation into y2 = x3 + 49x2 + 256x.)

(9) y2+xy+y = x3 - x2 - 14x + 29. Cyclic of order 9.
(Turn the equation into y2 = x3 - 3x2 - 216x + 1872.)

(10) y2+xy = x3 - 45x + 81. Cyclic of order 10.
(Turn the equation into y2 = x3 + x2 - 720x + 5184.)

(11) y2+43xy-210y = x3 - 210x2.

(12) y2 = x3 - 4x. Product C2 x C2.

(13) y2 = x3 + 2x2 - 3x. Product C2 x C4.

(14) y2+5xy-6y = x3 - 3x2. Product C2 x C6.
(Turn the equation into y2 = x3 + 13x2 - 240x + 576.)

(15) y2+17xy-120y = x3 - 60x2.