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

2. For j = 0 take λ =−ρ. Then the curve has equation
2
y = x(x − 1)(x + ρ),
and make a change of variable x + (1 − ρ)/3 for x. This gives the equation

2 1 − ρ −2 − ρ 1 + 2ρ
y = x + x + x + .
3 3 3
Observe that this is just
i
2 3
y = x − √
3 3
2
3
which is one of the family y = x + a studied in (3.3). It is closely related to the
3
3
3
Fermat curve u + v = w .
4 2
2
The discriminant for the family E λ is given by λ = 2 λ (λ − 1) .Thisisthe
special case of the discriminant of an elliptic curve considered in (3.1).
Exercises
4 2 2
1. Verify the expression 2 λ (λ − 1) for the discriminant of E λ .
2. Verify that j(λ) is the j-invariant of E λ , see (1.4).

§2. Families of Curves with Points of Order 3:
The Hessian Family

Returning to the general normal form of the cubic defining an elliptic curve, we
observe that (0, 0) is a point on the curve if and only if a 6 = 0. The family of these
cubics reduces to
2 3 2
E 0 : y + a 1 xy + a 3 y = x + a 2 x + a 4 x.

From the calculation of the derivative y in the relation
2
(2y + a 1 x + a 3 ) y = 3x a 2 x + a 4 − a 1 y

we see that the slope of the tangent line at (0, 0) is a 4 /a 3 on E 0 .