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

1,2

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

Table of n, a(n) for n=1..80.

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

Cf. A259825, A362243.

Sequence in context: A368548 A375783 A240039 * A130127 A366771 A217982

Adjacent sequences: A369899 A369900 A369901 * A369903 A369904 A369905

Keyword

nonn

Author

Robin Visser, Feb 05 2024

Status

approved