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

3
2
3
x − (x + u) − (a 1 x + a 3 )(x + u) = (x − v) .
1
2
0
Comparing coefficients of x , x ,and x yields the relations
3v = 1 + a 1 ,
2
−3v = 2u + a 1 u + a 3 ,
3
2
v = u + a 3 u.
Multiply the second relation by u, we obtain
2 2 2
−3uv = 2u + a 1 u + a 3 u,
and subtracting it from the third relation yields

3 2 2 2 3 3
v + 3v u =− (1 + a 1 ) u =−3vu or just (v + u) = u .
3
This means that the second point of order 3 has the form (v, v +u) where (v +u) =
3
2
u . Since v = 0, we must have v+u = ρu where ρ +ρ+1 = 0. Thus v = (ρ = 1)u,
2
2
2
and u = (ρ−1) −1 v.Since T +T +1 = (T −ρ)(T −ρ ),sothat3 = (1−ρ)(1−ρ ),
we have
1 2 1
−1
u = (ρ − 1) v = ρ − 1 v =− (ρ + 2)v
3 3
and
1
u + v = (1 − ρ)v.
3
In terms of the parameter v with v = 0wehaveabasis
1

(0, 0), v, (1 − ρ)v
3
2
3
of the group of points of order 3 on the curve with equation y + a 1 xy + a 3 y = x .
Now solving for a 1 and a 3 in terms of the parameter which is the x-coordinate of the
second point of order 3, we obtain a one-parameter family of curves with a basis for
the 3-division points, i.e., the points of order 3 on the curve.

(2.4) Assertion. The family of cubic curves
2
3
E γ : y + a 1 (γ )xy + a 3 (γ )y = x ,
where a 1 (γ ) = 3γ − 1and a 3 (γ ) = γ(ρ − 1)(γ − 1/3(ρ + 1)) defines for (γ )
3
= (a 1 (γ ) − 27a 3 (γ ))a 3 (γ ) = 0 a family of elliptic curves with a basis (0, 0),
(γ, 1/3(1 − ρ)γ ) for the subgroup of points of order 3 on E γ .