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 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).
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.
FORMULA
a(n) = 2*floor(2*sqrt(q)) + 1 if q is prime, where q = A246655(n).
EXAMPLE
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.
PROG
(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)
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Aug 02 2023
STATUS
approved