A369902 Number of isomorphism classes of elliptic curves over the finite field of order prime(n) whose trace of Frobenius is zero.

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

Offset: 1

Author

Robin Visser, Feb 05 2024

Keywords

• nonn

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.

Examples

			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.
		

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.

Crossrefs

Cf. A259825, A362243.

Programs

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)

Formula

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