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§4. Number of Supersingular Elliptic Curves 263

0 if p − 1 does not divide a
a
x =
−1if p − 1 does divide a,
x∈F p
we obtain another version of the above formula
p−1 m
#E λ (F p ) = 1 −{coefficient of x in [x(x − 1)(x − λ)] }.
By (3.4) we have the following assertion.

(3.9) Proposition. For λ ∈ F p and p > 3 the number of rational points on the
curve E λ is given by

#E λ (F p ) ≡ 1 − G p (λ) mod p.
In particular, E λ is supersingular if and only if #E λ (F p ) ≡ 1mod p in which case,
we have #E λ (F p ) = p + 1 by the Riemann hypothesis.

§4. Number of Supersingular Elliptic Curves

We already know from (3.7) that there are (p − 1)/2 values such that the curve E λ :
2
y = x(x − 1)(x − λ) is supersingular in characteristic p = 2 and in characteristic
2
3
2 there is just one supersingular curve y + y = x up to isomorphism. In order
to describe the number of supersingular curves up to isomorphism, we count the j
values of j(λ) as λ ranges over the points on the λ-line such that E is supersingular.
Recall the map from the λ-line to the j-line is given by
(λ 2 − λ + 1) 3
8
j(λ) = 2 .
2
λ (λ − 1) 2
3
This is a map of degree 6 with ramification precisely over the points {∞, 0, 12 }.

has a j-value j 0 = j(λ 0 ), then all the points λ in
If a supersingular curve E λ 0
j −1 ( j 0 ) correspond to supersingular curves E each isomorphic to E λ 0 . With two
. Starting with the
possible exceptions there are six such curves isomorphic to E λ 0
relation
p − 1
= 1,
2
E λ supersingular