A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

5, 7, 9, 11, 13, 15, 17, 17, 19, 21, 23, 25, 25, 27, 27, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 41, 41, 41, 41, 43, 45, 45, 47, 47, 49, 49, 51, 51, 51, 53, 53, 53, 55, 55, 57, 57, 59, 59, 61, 61, 61, 61, 63, 63, 65, 65, 65, 65, 67, 67, 67, 69, 71, 71, 71, 71, 73, 73, 75, 75, 75, 75, 77

Offset: 1

Author

Robin Visser, Apr 25 2023

Keywords

• nonn

Comments

Two elliptic curves over a finite field F_q are isogenous if and only if they have the same trace of Frobenius, or equivalently, have the same number of points over F_q.

Thus, by the Hasse bound, a(n) is the number of integers with absolute value bounded by 2*sqrt(prime(n)).

Examples

			For n = 1, the a(1) = 5 isogeny classes of elliptic curves are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
For n = 2, the a(2) = 7 isogeny classes of elliptic curves are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.
		

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
• J. H. Silverman, The Arithmetic of Elliptic Curves, Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.

Crossrefs

Cf. A247485, A362198, A362201, A362243, A364681.

Programs

Magma

[2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];

Mathematica

2Floor[2Sqrt[Prime[Range[100]]]]+1 (* Paolo Xausa, Oct 23 2023 *)

PARI

a(n) = 2*sqrtint(4*prime(n)) + 1;

Formula

a(n) = 2*floor(2*sqrt(prime(n))) + 1.

a(n) = 2*A247485(n) - 1.