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A362570
a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).
4
5, 7, 9, 11, 13, 15, 17, 17, 19, 21, 23, 25, 25, 27, 27, 29, 31, 31, 33, 33, 35, 35, 37, 37, 39, 41, 41, 41, 41, 43, 45, 45, 47, 47, 49, 49, 51, 51, 51, 53, 53, 53, 55, 55, 57, 57, 59, 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
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
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