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

1,1

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.

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.

Example

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.

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 *)

Prog

(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)]

Crossrefs

Sequence in context: A314411 A359353 A360404 * A069102 A001043 A118775

Adjacent sequences: A362240 A362241 A362242 * A362244 A362245 A362246

Keyword

nonn

Author

Robin Visser, Apr 12 2023

Status

approved