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68 3. Elliptic Curves and Their Isomorphisms

(2.2) Invariant Differential. For an elliptic curve E given by the equation 0 =
f (1, x, y) in normal form the invariant differential is given by the following two
expressions:
dx dx dy dy
ω = = =− = 2 .
2y + a 1 x + a 3 f y f x 3x + 2a 2 x + a 4 − a 1 y
Henceforth when we speak of an elliptic curve we will always mean a cubic curve
E with O at 0 : 0 : 1 and the equation in normal form. We still need to study to what
extend these forms are unique. This is related to transformations of an elliptic curve,
or from another point of view, changes of variable which preserves the normal form.
We make this precise with the next definition.
(2.3) Definition. An admissible change of variables in the equation of an elliptic
curve is one of the form
3
2
2
x = u ¯x + r and y = u ¯y + su ¯x + t,
where u,r, s,and t in k with u invertible, i.e., nonzero.
(2.4) Remark. Substitution by an admissible change of variables into the equation
0 = f (w, x, y) in normal form yields a new form of the equation in terms of the
variables ¯x and ¯y:
2 3 2
¯ y +¯a 1 xy +¯a 3 ¯y =¯x +¯a 2 ¯x +¯a 4 ¯x +¯a 6 .
The invariant differential ω is changed to ¯ω and ¯ω = uω.Also,
u ¯a 1 = a 1 + 2s,
2
2
u ¯a 2 = a 2 − sa 1 + 3r − s ,
3
u ¯a 3 = a 3 + ra 1 + 2t = f y (r, t),
4 2
u ¯a 4 = a 4 − sa 3 + 2ra 2 − (t + rs)a 1 + 3r − 2st =− f x (r, t) − sf y (r, t),
6 2 3 2
u ¯a 6 = a 6 + ra 4 + r a 2 + r − ta 3 − rta 1 − t =− f (r, t).
These relations are left to the reader who should keep in mind that a 6 =− f (0, 0),
a 4 =− f x (0, 0),and a 3 = f y (0, 0).

¯
(2.5) Remark. Consider two elliptic curves E and E defined by the equations in ¯a i
and a i , respectively, in normal form. Then φ : E → E is an isomorphism such that
¯
the functions x, y on E composed with φ are related to the functions ¯x, ¯y on E by
¯
3
2
2
xφ = u ¯x + r and yφ = u ¯y + su ¯x + t
as in an admissible change of variable. An easy calculation shows that the compo-
sition of two admissible changes of variable is again one and that the inverse of an
admissible change of variables is again one.
For an admissible change of variables with r = s = t = 0wehave a i =
i
3
2
u ¯a i , x = u ¯x, y = u ¯y,and ω = u −1 ¯ ω. Thus there is a homogeneity where x
has weight 2, y weight 3, a i weight i,and ω weight −1. The polynomial f (x, y) has
weight 6 where f (x, y) = 0 is the equation in normal form.