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

In order to derive the equations f n (b, c) = 0 in special cases, we make use of the
following formulas.
(4.5) Calculations. On the curve E(b, c) we have the following:

2
P = (0, 0), 2P = (b, bc), 3P = (c, b − c), 4P = d(−1), d (c − d + 1) ,
2
2

−P = (0, b), −2P = (b, 0), −3P = (c, c ), −4P = d(d − 1), d(d − 1) ,
where d = b/c in the formulas for 4P and −4P. Finally, introducing e = c/(d −1),
wehave

2
2
−5P = de(e − 1), d e(e − 1) 2 and 5P = de(e − 1), de (d − e) .
(4.6) Examples. We have the following formulas for f n (b, c) arising from the con-
dition that nP = 0.
(a) 4P = 0 is equivalent to 2P =−2P which by (4.5) reduces to the relation
c = 0. Thus f 4 (b, c) = c is the equation of a projective line. Moreover, the equation
for the family becomes
2
3
E(b, 0) : y + xy − by = x − bx 2
4
with discriminant = b (1 + 16b) = 0. For a given x,the y-coordinate of a point
2
2
3
(x, y) on E(b, 0) satisfies the quadratic equation y + (x − b)y + (bx − x ) = 0.
The point (x, y) has order 2 if and only if this equation in y has a double root, or, in
other words, the discriminant of the quadratic equation in zero.
2
2
(x − b) − 4x (b − x) = 0.
One solution is 2P = (b, bc) = (b, 0) =−2P for c = 0. The other solutions are
points whose x-coordinates satisfy

1 1
2 2
0 = 4x + x − b = 4x + x + − b + .
16 16
There are two other 2-division points on E(b, 0) over a field other than coming from
2
2P =−2P if and only if b + 1/16 is a square v . Moreover, v can take on any value
except 0, 1/4, and −1/4since b must be unequal to 0 and −1/16. The x values for
these two points are

1 v
x =− ± .
8 2
(b) 5P = 0 if equivalent to 3P =−2P which by (4.5) reduces to the relation
b = c. Thus f 5 (b, c) = b − c is also the equation of a projective line. Moreover, the
equation for the family becomes

2
3
E(b, b) : y + (1 − b)xy − by = x − bx 2