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A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.

%I A364681 #14 Aug 04 2023 15:47:55
%S A364681 5,7,9,9,11,9,13,13,15,13,17,17,19,20,17,21,23,15,25,25,27,27,27,29,
%T A364681 31,31,21,33,33,35,35,29,37,37,39,41,41,41,41,43,45,37,45,25,45,47,47,
%U A364681 49,49,51,51,51,50,53,53,53,55,55,57,57,59,59,61,61,61,61,63,45,63,37,65,65
%N A364681 a(n) is the number of isogeny classes of elliptic curves over GF(q), where q = A246655(n) is the n-th prime power > 1.
%C A364681 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 A364681 Thus a(n) is the number of integers k such that there exists an elliptic curve over GF(q) with trace k, where q = A246655(n).
%H A364681 Robin Visser, <a href="/A364681/b364681.txt">Table of n, a(n) for n = 1..10000</a>
%H A364681 Max Deuring, <a href="https://doi.org/10.1007/BF02940746">Die Typen der Multiplikatorenringe elliptischer Funktionenkörper</a>, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
%H A364681 W. C. Waterhouse, <a href="https://doi.org/10.24033/asens.1183">Abelian varieties over finite fields</a>, Ann Sci. E.N.S., (4) 2 (1969), 521-560.
%F A364681 a(n) = 2*floor(2*sqrt(q)) + 1 if q is prime, where q = A246655(n).
%e A364681 For n = 1, the a(1) = 5 isogeny classes of elliptic curves over GF(2) are parametrized by the 5 possible values for the trace of Frobenius: -2, -1, 0, 1, 2.
%e A364681 For n = 2, the a(2) = 7 isogeny classes of elliptic curves over GF(3) are parametrized by the 7 possible values for the trace of Frobenius: -3, -2, -1, 0, 1, 2, 3.
%o A364681 (Sage)
%o A364681 for q in range(1, 1000):
%o A364681     if Integer(q).is_prime_power():
%o A364681         p, ans = Integer(q).prime_factors()[0], 0
%o A364681         for a in range(-floor(2*sqrt(q)), floor(2*sqrt(q))+1):
%o A364681             if (a%p != 0) or (Integer(q).is_square() and ((abs(a) == 2*sqrt(q))
%o A364681                   or ((p%3 != 1) and (abs(a) == sqrt(q))) or ((p%4 != 1) and
%o A364681                   (a==0)))) or ((not Integer(q).is_square()) and
%o A364681                   (((p in [2,3]) and (abs(a) == sqrt(p*q))) or (a==0))):
%o A364681                 ans += 1
%o A364681         print(ans)
%Y A364681 Cf. A005523, A362570.
%K A364681 nonn
%O A364681 1,1
%A A364681 _Robin Visser_, Aug 02 2023