A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).
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
Keywords
Examples
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.
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.
Programs
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Magma
[2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];
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Mathematica
2Floor[2Sqrt[Prime[Range[100]]]]+1 (* Paolo Xausa, Oct 23 2023 *)
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PARI
a(n) = 2*sqrtint(4*prime(n)) + 1;
Formula
a(n) = 2*floor(2*sqrt(prime(n))) + 1.
a(n) = 2*A247485(n) - 1.
Comments