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264 13. Elliptic Curves over Finite Fields

and dividing by 6, we obtain the relation

 
p − 1   α β
 
1 + + ,
12 2 3
= 
supersingular E 

up to isomorphism
with j(E) =0.1728
3
where α = 0or depending on E with j(E) = 12 , namely α = 1 if and only if
E is supersingular, and where β = 0or 1 depending on E with j(E) = 0, namely
β = 1 if and only if E is supersingular. Dividing further by 2 and interpretating the
numbers in the denominators as orders of Aut(E), we obtain the following theorem.
(4.1) Theorem. For a prime p the following sum taken over supersingular curves
defined over F p up to isomorphism
¯
p − 1 1
=
24 #Aut(E)
E supersingular mod p
up to isomorphism
holds.
The above discussion proves the theorem for p > 3. The theorem predicts one
supersingular curve for p = 2 with an automorphism group having 24 elements and
one supersingular curve for p = 3 with an automorphism group having 12 elements.
This is the case by (2.2) and 3(6.2) for p = 2 and by (2.3) and 3(5.2) for p = 3.
To make the above theorem more explicit, we decompose (p − 1)/24 = m/2 +
α/4 + β/6, where α and β are 0or 1 and m = [p/12]. Let n(p) denote the number
of supersingular curves in characteristic p.

Table 1. For supersingular curves in characteristic p > 3.

p (mod 12) p = 1(12) p = 5(12) p = 7(12) p = 11(12)
n(p) = number of p − 1 p − 5 p − 7 p − 11
+ 1 + 1 + 2
supersingular curves 12 12 12 12
p (mod 4) p = 1(4) p = 1(4) p =−1(4) p =−1(4)
Hasse invariant of
3
2
3
y = x − x ( j = 12 ) 1 1 0supersingular 0supersingular
p (mod 3) p = 1(3) p =−1(3) p = 1(3) p =−1(3)
Hasse invariant of
2
3
y = x − 1 ( j = 0) 1 0supersingular 1 0supersingular