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A362243 a(n) = number of isomorphism classes of elliptic curves over the finite field of order prime(n). 5
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
LINKS
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
KEYWORD
nonn
AUTHOR
Robin Visser, Apr 12 2023
STATUS
approved

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Last modified February 24 17:27 EST 2024. Contains 370307 sequences. (Running on oeis4.)