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102 4. Families of Elliptic Curves and Geometric Properties of Torsion Points

2
2
2
3
2
(6.4) Example. The point (t , st ) is the order 3 on E = E[(s−t) −3t , t (t−2s)]
and the point
2 2 3 2 2
3t + 2st − 2s , 3t + 2st − 2s t
2
2
is of order 6. We do the calculations (s , s t) + (0, 0) = (x, tx) where

3
2
2
3
2 2
t x = x + (s − t) − 3t 2 x + t (t − 2s)
and

3
2
2
0 = x − 4t − (s − t) 2 x + ··· .
2
2
2
2
The three roots which add up to 4t −(s−t) are 0, s ,and x. Thus x = 3t +2st−2s 2
holds as asserted.
Some of the above examples were brought to our attention by Hans Peter Kraft
and were worked out in a seminar in Basel, October 1984 by him, Friedrich Knopp,
Gisela Menzel, and Erhar Senn. The reader can find further examples by combining
the above results with the statements in 1(3, Ex. 2).