The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 47

The first one, with B = 432, was obtained in Section 2.5.2 from the Fermat

equation x 3 + y 3 + z 3 = 0. The second curve, once with A =

−

− 25 and once

with A = 4, appeared in Chapter 1.

The curves with j =0andwith j = 1728 have automorphisms (bijective
group homomorphisms from the curve to itself) other than the one defined by
( x, y ) → ( x, −y ), which is an automorphism for any elliptic curve in Weier-
strass form.
1. y 2

−

= x 3 + B has the automorphism ( x, y )

→ ( ζx,−y ), where ζ is a

nontrivial cube root of 1.

2. y 2 = x 3 + Ax has the automorphism ( x, y ) x, iy ), where i 2 = → ( − − 1.

(See Exercise 2.17.)
Note that the j -invariant tells us when two curves are isomorphic over an
algebraically closed field. However, if we are working with a nonalgebraically
closed field K , then it is possible to have two curves with the same j -invariant
that cannot be transformed into each other using rational functions with co-
ecients in K . For example, both y 2 = x 3 − 25 x and y 2 = x 3 − 4 x have

j = 1728. The first curve has infinitely points with coordinates in Q ,for
example, all integer multiples of ( − 4 , 6) (see Section 8.4). The only rational
points on the second curve are ∞ ,(2 , 0), ( − 2 , 0), and (0 , 0) (see Section 8.4).
Therefore, we cannot change one curve into the other using onl y rational func-

tions defined over Q . Of course, we can use the field Q ( √ 10) to change one
curve to the other via ( x, y ) ( μ 2 x, μ 3 y ), where μ = √ 10 / 2.

If two different elliptic curves defined over a field K have the same j -
invariant, then we say that the two curves are twists of each other.
Finally, we note that j is the j -invariant of

→

3 j
1728 − j x +

2 j
1728 − j

y 2 = x 3 +

(2.9)

when j =0 , 1728. Since y 2 = x 3 +1 and y 2 = x 3 + x have j -invariants 0
an d 1728, we find the j -invariant gives a bijection between elements of K and
K -isomorphism classes of elliptic curves defined over K (that is, each j K

corresponds to an elliptic curve defined over K , and any two elliptic curves
defined over K andwiththesame j -invariant c an be transformed into each
other by a change of variables (2.8) defined over K ).
If the characteristic of K is 2 or 3, the j -invariant can also be defined, and
results similar to the above one hold. See Section 2.8 and Exercise 2.18.

∈

2.8 Elliptic Curves in Characteristic 2

Since we have been using the Weierstrass equation rather than the gener-
alized Weierstrass equation in most of the preceding sections, the formulas