cp's OEIS Frontend

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

%I A362570 #29 Oct 23 2023 08:30:29
%S A362570 5,7,9,11,13,15,17,17,19,21,23,25,25,27,27,29,31,31,33,33,35,35,37,37,
%T A362570 39,41,41,41,41,43,45,45,47,47,49,49,51,51,51,53,53,53,55,55,57,57,59,
%U A362570 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
%N A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).
%C A362570 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.
%C A362570 Thus, by the Hasse bound, a(n) is the number of integers with absolute value bounded by 2*sqrt(prime(n)).
%H A362570 Robin Visser, <a href="/A362570/b362570.txt">Table of n, a(n) for n = 1..10000</a>
%H A362570 Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Univ. Hamburg 14 (1941), 197-272.
%H A362570 J. H. Silverman, <a href="https://doi.org/10.1007/978-0-387-09494-6">The Arithmetic of Elliptic Curves</a>, Second edition. Graduate Texts in Mathematics, 106. Springer, Dordrecht, 2009.
%F A362570 a(n) = 2*floor(2*sqrt(prime(n))) + 1.
%F A362570 a(n) = 2*A247485(n) - 1.
%e A362570 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.
%e A362570 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.
%t A362570 2Floor[2Sqrt[Prime[Range[100]]]]+1 (* _Paolo Xausa_, Oct 23 2023 *)
%o A362570 (Magma) [2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];
%o A362570 (PARI) a(n) = 2*sqrtint(4*prime(n)) + 1;
%Y A362570 Cf. A247485, A362198, A362201, A362243, A364681.
%K A362570 nonn
%O A362570 1,1
%A A362570 _Robin Visser_, Apr 25 2023