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.

5, 7, 9, 9, 11, 9, 13, 13, 15, 13, 17, 17, 19, 20, 17, 21, 23, 15, 25, 25, 27, 27, 27, 29, 31, 31, 21, 33, 33, 35, 35, 29, 37, 37, 39, 41, 41, 41, 41, 43, 45, 37, 45, 25, 45, 47, 47, 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

Offset: 1

Author

Robin Visser, Aug 02 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 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).

Examples

			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.
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.
		

Links

• Robin Visser, Table of n, a(n) for n = 1..10000
• Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197-272.
• W. C. Waterhouse, Abelian varieties over finite fields, Ann Sci. E.N.S., (4) 2 (1969), 521-560.

Crossrefs

Cf. A005523, A362570.

Programs

Sage

for q in range(1, 1000):
    if Integer(q).is_prime_power():
        p, ans = Integer(q).prime_factors()[0], 0
        for a in range(-floor(2*sqrt(q)), floor(2*sqrt(q))+1):
            if (a%p != 0) or (Integer(q).is_square() and ((abs(a) == 2*sqrt(q))
                  or ((p%3 != 1) and (abs(a) == sqrt(q))) or ((p%4 != 1) and
                  (a==0)))) or ((not Integer(q).is_square()) and
                  (((p in [2,3]) and (abs(a) == sqrt(p*q))) or (a==0))):
                ans += 1
        print(ans)

Formula

a(n) = 2*floor(2*sqrt(q)) + 1 if q is prime, where q = A246655(n).