OFFSET
1,2
COMMENTS
a(n) is the number of isomorphism classes of elliptic curves E over the finite field F_p such that E has exactly p+1 points over F_p.
LINKS
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
R. Schoof, Nonsingular plane cubic curves over finite fields, J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211.
W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
FORMULA
a(n) = A259825(4*prime(n))/12 if n > 2.
EXAMPLE
For n = 1, the unique a(1) = 1 elliptic curve over F_2 whose trace of Frobenius is zero is y^2 + y = x^3.
For n = 2, the a(2) = 2 elliptic curves over F_3 whose trace of Frobenius is zero are y^2 = x^3 + x and y^2 = x^3 + 2*x.
For n = 3, the a(3) = 2 elliptic curves over F_5 whose trace of Frobenius is zero are y^2 = x^3 + 1 and y^2 = x^3 + 2.
PROG
(PARI) a(n) = if (n<=2, n, qfbhclassno(4*prime(n)));
(Sage) # A brute force computation of a(n)
def a(n):
if n==1: return 1
p, ECs = Primes()[n-1], []
for A, B in ((x, y) for x in range(p) for y in range(p)):
if ((4*A^3 + 27*B^2)%p != 0):
E = EllipticCurve(GF(p), [A, B])
if (E.trace_of_frobenius()==0):
if not any([E.is_isomorphic(Ei) for Ei in ECs]): ECs.append(E)
return len(ECs)
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Feb 05 2024
STATUS
approved