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
KEYWORD
nonn
AUTHOR
Robin Visser, Apr 25 2023
STATUS
approved