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70 3. Elliptic Curves and Their Isomorphisms

new functions ¯x and ¯y satisfying ¯x = (¯z) −2 +· · · and ¯y =−(¯z) −3 +· · · .Since
2
2
2
x = u (¯z) −2 +· · · , it follows that x = u ¯x up to a constant, i.e., x = u ¯x + r for r
3
3
in k. Since y =−u (¯z) −3 +· · · , it follows that y = u ¯y up to a linear combination
2
3
of x and a constant, i.e., y = u ¯y + su ¯x + t.
We summarize (2.6) and (2.7) in the next theorem.
(2.8) Theorem. Any isomorphism between two elliptic curves is given by an admis-
sible change of variables relative to two given equations in normal form.
Exercises
i
1. Verify the formulas for u ¯a i , where i = 1, 2, 3, 4, and 6 in (2.4).
2. Show that the inverse and the composite of two admissible changes of variable are again
admissible changes of variable.

§3. The Discriminant and the Invariant j

We associate to a cubic equation in normal form
2
2
3
(N 1 ) y + a 1 xy + a 3 y = x + a 2 x + a 4 x + a 6 ,
two new sets of coefficients b i for i = 2, 4, 6, and 8 and c j for j = 4and 6which
arise first from completing the square and second from completing the cube.
(3.1) Notations. Associated to the coefficients a 1 in the cubic (N 1 ) are the coeffi-
cients
2
b 2 = a + 4a 2 ,
1
b 4 = a 1 a 3 + 2a 4 ,
2
b 6 = a + 4a 6 ,
3
2
2
2
b 8 = a a 6 − a 1 a 3 a 4 + 4a 2 a 6 + a 2 a − a .
1 3 4
2
These quantities are related by 4b 8 = b 2 b 6 − b . We also introduce the discriminant
4
in terms of the b i ’s:
2 3 2
=−b b 8 − 8b − 27b + 9b 2 b 4 b 6 .
2 4 6
With the discriminant we can decide whether or not the cubic is nonsingular. It
is nonsingular if and only if = 0. Firstly we study the b i and introduce the c j
coefficients.
(3.2) Remarks. Under an admissible change of variable as in (2.3) and (2.4) we
¯
have, for the corresponding b i and ¯ , the relations
2
u b 2 = b 2 + 12r,
¯