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

1,1

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)).

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.

Formula

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

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

Example

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.

Mathematica

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

Prog

(Magma) [2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];
(PARI) a(n) = 2*sqrtint(4*prime(n)) + 1;

Crossrefs

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

Sequence in context: A343064 A106505 A097985 * A342378 A099047 A174392

Adjacent sequences: A362567 A362568 A362569 * A362571 A362572 A362573

Keyword

nonn

Author

Robin Visser, Apr 25 2023

Status

approved