A362243 a(n) = number of isomorphism classes of elliptic curves over the finite field of order prime(n).

5, 8, 12, 18, 22, 32, 36, 42, 46, 60, 66, 80, 84, 90, 94, 108, 118, 128, 138, 142, 152, 162, 166, 180, 200, 204, 210, 214, 224, 228, 258, 262, 276, 282, 300, 306, 320, 330, 334, 348, 358, 368, 382, 392, 396, 402, 426, 450, 454, 464, 468, 478, 488, 502, 516, 526, 540, 546, 560, 564, 570

Offset: 1

Author

Robin Visser, Apr 12 2023

Keywords

• nonn

Examples

			For n = 1, the a(1) = 5 elliptic curves over F_2 can be given by their Weierstrass models as: y^2 + y = x^3, y^2 + y = x^3 + x, y^2 + y = x^3 + x + 1, y^2 + x*y = x^3 + 1, y^2 + x*y + y = x^3 + 1.
For n = 2, the a(2) = 8 elliptic curves over F_3 can be given by their Weierstrass models as: y^2 = x^3 + x, y^2 = x^3 + 2*x, y^2 = x^3 + 2*x + 1,  y^2 = x^3 + 2*x + 2,  y^2 = x^3 + x^2 + 1, y^2 = x^3 + x^2 + 2, y^2 = x^3 + 2*x^2 + 1, y^2 = x^3 + 2*x^2 + 2.
For n = 3, the a(3) = 12 elliptic curves over F_5 can be given by their Weierstrass models as: y^2 = x^3 + 1, y^2 = x^3 + 2, y^2 = x^3 + x, y^2 = x^3 + x + 1, y^2 = x^3 + x + 2, y^2 = x^3 + 2*x, y^2 = x^3 + 2*x + 1, y^2 = x^3 + 3*x, y^2 = x^3 + 3*x + 2, y^2 = x^3 + 4*x, y^2 = x^3 + 4*x + 1, y^2 = x^3 + 4*x + 2.
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..10000
• H. W. Lenstra, Factoring integers with elliptic curves, Ann. of Math. (2) 126 (1987), no. 3, 649-673.
• S. Marseglia, Computing abelian varieties over finite fields isogenous to a power, arXiv:1808.03673 [math.AG], 2018.

Programs

Mathematica

A362243list[nmax_]:=Map[2#+{6,1,2,0,2,0,4,0,0,0,0}[[Mod[#,12]]]&,Prime[Range[nmax]]];A362243list[100] (* Paolo Xausa, Aug 28 2023 *)

Sage

def a(n):
    if n == 1:
        return 5
    if n == 2:
        return 8
    p = Primes()[n-1]
    r = [1, 5, 7, 11]
    C = [6, 2, 4, 0]
    return 2*p + C[r.index(p%12)]

Formula

a(1) = 5, a(2) = 8, and for n > 2, a(n) = 2*prime(n) + C, where C is 6, 2, 4, 0 if prime(n) is 1, 5, 7, 11 mod 12 respectively.