Page 94 -

§3. The Discriminant and the Invariant j 71

4
2
u b 4 = b 4 + rb 2 + 6r ,
¯
3
6
2
u b 6 = b 6 + 2rb 4 + r b 2 + 4r ,
¯
2
8
u b 8 = b 8 + 3rb 6 + 3r b 4 + 3r 4 ,
¯
and u = . Moreover, if k is a field of characteristic different from 2, then for
12 ¯

y = y + (a 1 x + a 3 )/2and x = x the equation in normal form becomes
3
2
2
(N 2 ) (y ) = (x ) + b 2 (x ) + b 4 b 6 .
x +
4 2 4
(3.3) Notations. Associated to the coefficients a i in (N 1 ) and b j in (3.1) are the
coefficients
2
3
c 4 = b − 24b 4 , c 6 =−b + 36b 2 b 4 − 216b 6 .
2 2
For invertible, we introduce the quantity
c 3
j(E) = j = 4 .

3
2
3
We have the following relation 12 = c − c , and, therefore,
4 6
c 3
j = 12 3 3 4 2 .
c − c
4 6
(3.4) Remarks. Under an admissible change of variable as in (2.3) and (2.4) we
¯
have for the corresponding ¯c j and j the relations
4 6
¯
u ¯c 4 = c 4 and u ¯c 6 = c 6 and finally j = j.
¯
For the j-invariant we have j = j which means that j(E) is an invariant of an
elliptic curve E up to isomorphism. If E and E are isomorphic, as in (2.5) and (2.8),
¯
¯
then we have j(E) = j(E). Moreover, if k is a field of characteristic different from

2 and 3, then for y = y and x = x + b 2 /12 the equation in normal form becomes

3
(N 3 ) (y ) = (x ) − c 4 x − c 6

48 864

and ω = dx /2y .
Now we take up the question of when the normal form defines a nonsingular
3
curve. First consider a cubic polynomial f (x) = x + px + q. The discriminant
2
D( f ) is the resultant R( f, f ) where f (x) = 3x + p is the derivative of f (x).We

compute

q p 0 1 0

0 q p 0 1
2 3
D( f ) = p 0 3 0 0 = 27q + 4p .

0 p 0 3 0

0 0 p 0 3