A369902 - OEIS

%I #7 Feb 14 2024 01:56:22

%S 1,2,2,2,4,2,4,4,6,6,6,2,8,4,10,6,12,6,4,14,4,10,12,12,4,14,10,12,6,8,

%T 10,20,8,12,14,14,6,4,22,14,20,10,26,4,10,18,12,14,20,10,12,30,12,28,

%U 16,26,22,22,6,20,12,18,12,38,8,10,12,8,20,14,16,38,18,10,12,34,22,6,20,16

%N Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

%C 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.

%H Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.

%H R. Schoof, <a href="https://doi.org/10.1016/0097-3165(87)90003-3">Nonsingular plane cubic curves over finite fields</a>, J. Combin. Theory Ser. A 46 (1987), no. 2, 183-211.

%H W. C. Waterhouse, <a href="https://doi.org/10.24033/asens.1183">Abelian varieties over finite fields</a>, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

%F a(n) = A259825(4*prime(n))/12 if n > 2.

%e 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.

%e 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.

%e 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.

%o (PARI) a(n) = if (n<=2, n, qfbhclassno(4*prime(n)));

%o (Sage) # A brute force computation of a(n)

%o def a(n):

%o     if n==1: return 1

%o     p, ECs = Primes()[n-1], []

%o     for A,B in ((x, y) for x in range(p) for y in range(p)):

%o         if ((4*A^3 + 27*B^2)%p != 0):

%o             E = EllipticCurve(GF(p), [A,B])

%o             if (E.trace_of_frobenius()==0):

%o                 if not any([E.is_isomorphic(Ei) for Ei in ECs]): ECs.append(E)

%o     return len(ECs)

%Y Cf. A259825, A362243.

%K nonn

%O 1,2

%A _Robin Visser_, Feb 05 2024