Page 112 -

§2. Families of Curves with Points of Order 3: The Hessian Family 89

(2.1) Remark. For the cubic E 0 in normal form with (0, 0) on the curve, the point
(0, 0) is singular if and only if a 3 = a 4 = 0. The point (0, 0) is nonsingular and of
order 2 in the group E if and only if a 3 = 0and a 4 = 0, the case of a vertical tangent
at (0, 0). The family of these cubics reduces to
2
2
3
E 00 : y + a 1 xy = x + a 2 x + a 4 x.
Now we assume that (0, 0) is a nonsingular point which is not of order 2. By a
change of variable of the form

a 2

y = y + x and x = x .
a 3
the equation for E 0 takes the form
3
2
2

E : y + a 1 xy + a 3 y = x + a 2 x ,
where the tangent line to the curve at (0, 0) has slope equal to 0.

(2.2) Remark. The point (0, 0) on E has order 3 if and only if a 2 = 0and a 3 = 0.

This is the condition for the curve E to have a third-order intersection with the
tangent line y = 0at (0, 0). The family reduces to
2 3
E (a 1 , a 3 ) : y + a 1 xy + a 3 y = x .
For these curves some of the basic invariants defined in 3(3.1) and 3(3.3) are the
2 2
following: b 2 = a , b 4 = a 1 a 3 , b 6 = a ,and b 8 = 0. Further we have =
1 3
4
3 3
3
3
a a − 27a and c 4 = a 1 (a − 24a 3 ) with j = c / .
1 3 3 1 4
Since a 3 = 0 in the curve E(a 1 , a 3 ), we can normalize a 3 = 1, and we obtain an
important family of elliptic curves with given point of order 3.
3
2
(2.3) Definition. The Hessian family of elliptic curves E α : y + αxy + y = x is
defined for any field of characteristic different from 3 with j-invariant j(α) = j(E α )
of E α given by
3
α 3 α − 24 3
j(α) = 3 .
α − 27
3
The curve E α is nonsingular for α = 27, that is, if α is not in 3µ 3 , where µ 3
is the group of third roots of unity. Over the line k minus 3 points, k − 3µ 3 ,the
family E α consists of elliptic curves with a constant section (0, 0) of order 3 where
2(0, 0) = (0, −1). The Hessian family E α has three singular fibres which are nodal
2
2
cubics at the points of 3µ 3 ={3, 3ρ, 3ρ },where ρ + ρ + 1 = 0.
2
Now we consider conditions on a 1 and a 3 in the cubic equation y +a 1 xy+a 3 y =
3
x such that both y = 0and y = x + u intersect the cubic with points generating
distinct subgroups of order 3. The line y = x + u has a triple intersection point
(v, v + u) with the cubic if and only if