A362198 -id:A362198 - cp's OEIS frontend

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

Robin Visser, Apr 25 2023

nonn

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)).

			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.

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.

Cf. A247485, A362198, A362201, A362243, A364681.

[2*Floor(2*Sqrt(p)) + 1 : p in PrimesUpTo(500)];

2Floor[2Sqrt[Prime[Range[100]]]]+1 (* Paolo Xausa, Oct 23 2023 *)

PARI
```
a(n) = 2*sqrtint(4*prime(n)) + 1;
```

a(n) = 2*floor(2*sqrt(prime(n))) + 1.

a(n) = 2*A247485(n) - 1.

A362201 a(n) = number of isogeny classes of dimension 3 abelian varieties over the finite field of order prime(n).

Original entry on oeis.org

215, 677, 2953, 7979, 30543, 50371, 112283, 156589, 277517, 555843, 678957, 1153875, 1569637, 1810805, 2364089, 3389675, 4675707, 5167277, 6846631, 8147047, 8855295, 11222313, 13014767, 16045439, 20772343, 23449327, 24870063, 27880975, 29473619, 32839031, 46617799, 51162221

Offset: 1

Robin Visser, Apr 10 2023

nonn

Two abelian varieties over a finite field are isogenous if and only if their Hasse-Weil zeta functions coincide.

Thus a(n) is the number of degree 6 integer polynomials with leading coefficient prime(n)^3 and whose (complex) roots all have absolute value 1/sqrt(prime(n)).

Robin Visser, Table of n, a(n) for n = 1..50
S. A. DiPippo and E. W. Howe, Real polynomials with all roots on the unit circle and abelian varieties over finite fields, arXiv:math/9803097 [math.NT], 1998-2000.
T. Dupuy, K. Kedlaya, D. Roe, and C. Vincent, Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB, arXiv:2003.05380 [math.NT].
S. Haloui, The characteristic polynomials of abelian varieties of dimensions 3 over finite fields, arXiv:1003.0374 [math.AG], 2020.
LMFDB, Abelian variety count results.
W. C. Waterhouse, Abelian varieties over finite fields, Ann. Sci. École Norm. Sup. (4) 2 (1969), 521-560.

Cf. A362198, A362570.

from sage.rings.polynomial.weil.weil_polynomials import WeilPolynomials
def a(n):
    p = Primes()[n-1]
    return len(list(WeilPolynomials(6, p)))

def a(n):
    R. = PolynomialRing(CC)
    num_solutions = 0
    p = Primes()[n-1]
    for a1 in range(ceil(-6*sqrt(p)), floor(6*sqrt(p))+1):
        for a2 in range(ceil(-15*p), floor(15*p)+1):
            for a3 in range(ceil(-20*p*sqrt(p)), floor(20*p*sqrt(p))+1):
                L_poly = 1+a1*x+a2*x^2+a3*x^3+p*a2*x^4+p^2*a1*x^5+p^3*x^6
                for r in L_poly.roots():
                    if (abs(abs(r[0]) - 1/sqrt(p)) > 1e-12):
                        break
                else:
                    num_solutions += 1
    return num_solutions

a(n) ~ (1024/45) * prime(n)^3.

cp's OEIS Frontend

A362570 a(n) is the number of isogeny classes of elliptic curves over the finite field of order prime(n).

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