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§3. The Jacobi Family 91

This assertion follows from the above calculations. There are other versions of
the Hessian family which in homogeneous coordinates take the form
3
3
3
H µ : u + v + w = 3µuvw,
or in affine coordinates with w =−1, it has the form
3
3
u + v = 1 − 3µuv.
3
3
Ifweset y =−v and x =−uv, we obtain x /y − y = 1 + 3µx,or
3
2
E 3µ : y + 3µxy + y = x .
This change of variable defines what is called a 3-isogeny of H µ onto E 3µ . There are
nine cross-sections of the family H µ given by
2
(0, −1, 1), (0, −ρ, 1), (0, −ρ , 1),
2
2
(1, 0, −1), (ρ, 0, −ρ ), (ρ , 0, −ρ),
2
(−1, 1, 0), (−1,ρ , 0), (1−,ρ, 0).
2
Again ρ is a primitive third root of unity satisfying ρ +ρ +1 = 0. The family H µ is
nonsingular over the line minus µ 3 , and choosing, for example, 0 = (−1, 1, 0), one
can show that these nine points form the subgroup of 3-division points of the family
H µ .

Exercise
3
2
2
1. Show that E t : y − 2txy + ty = x + (1 − 2t)x + tx is a family of cubics with
s 1 (t) = (0, 0) and s 2 (t) = (0, −t) two cross-sections of order 3 inverse to each other.
2
2
Show that E 0 is the nodal cubic y = x (x + 1).
§3. The Jacobi Family

Finally we consider the Jacobi family, which along with the Legendre family and the
Hessian family, give the three basic classical families of elliptic curves. The Jacobi
family is given by a quartic equation and we begin by explaining how to transform a
quartic equation to a cubic equation.

2
3
2
4
(3.1) Remark. Let v = f 4 (u) = a 0 u + a 1 u + a 2 u + a 3 u + a 4 be a quartic
equation. Let
ax + b ad − bc
u = and v = 2 y.
cx + d (cx + d)