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

The idea is that v behaves like the derivative of u,and y like the derivative of x.
Substituting into the quartic equation we obtain
(ad − bc) 2 ax + b
2 2
v = y = f 4
(cx + d) 4 cx + d
or

4
ax + b 4 4−i i

2 2
4
(ad − bc) y = f (cx + d) = a i (ax + b) (cx + d)
cx + d
i=0
a 4

4 4
= c f x + f 3 (x),
c
3
where f 3 (x) is a cubic polynomial whose coefficient of x is c f (a/c).For a/c a
3
4
2
simple root of f 4 and ad − bc = 1, we reduce to the equation y = f 3 (x) a cubic in
x.
(3.2) Definition. The Jacobi family of quartic curves is given by

u 2
2 2 2 2 4
J σ : v = 1 − σ u 1 − = 1 − 2ρu + u
σ 2
2
2
over any field of characteristic different from 2. Here ρ = (1/2)(σ + 1/σ ) so that
2
ρ + 1 = (1/2)(σ + 1/σ) .
2
We map J σ → E λ where E λ : y = x(x − 1)(x − λ) is a Legendre family with
2
λ = (1/4)(σ + 1/σ) by the following change of variable
2
4
σ + 1 u − σ σ − 1 v
x = and y = .
2σ 2 u − 1/σ 4σ 3 (u − 1/σ) 2
The points on J σ with u-coordinates
1
0, ∞, ±σ, ± , ±1, ±i
σ
2
map to the 16 = 4 points of order 4 on the elliptic curve E λ .
The detailed calculations relating the Jacobi family to the Legendre family are
left as an exercise to the reader.
§4. Tate’s Normal Form for a Cubic with a Torsion Point

Now we return to the normal form of the cubic E . Assume that (0, 0) is on the curve

with tangent line of slope 0.
2
3
2
E : y + a 1 xy + a 3 y = x + a 2 x .