A363840 a(n) is the number of isomorphism classes of genus 2 hyperelliptic curves over the finite field of order prime(n).

20, 69, 285, 749, 2813, 4589, 10149, 14119, 24907, 49675, 60613, 102749, 139613, 160951, 209947, 300667, 414357, 457813, 606151, 721011, 783511, 992477, 1150629, 1418037, 1834951, 2071011, 2196269, 2461749, 2602157, 2898789, 4113149, 4513613, 5161749, 5390839, 6638395, 6909013, 7764749

Offset

1,1

Comments

Any (smooth, projective, geometrically irreducible) curve of genus 2 can be given by a Weierstrass equation of the form: y^2 + h(x)y = f(x), where h(x) and f(x) are polynomials satisfying deg(h) <= 3 and deg(f) <= 6.

Links

Table of n, a(n) for n=1..37.

G. Cardona, On the number of curves of genus 2 over a finite field, Finite Fields Appl. 9 (2003), no.4, 505-526.

G. Cardona, E. Nart, and J. Pujolàs, Curves of genus two over fields of even characteristic, Math. Z. 250 (2005), no.1, 177-201.

E. Nart, Counting hyperelliptic curves, Adv. Math. 221 (2009), no.3, 774-787.

R. Steinberg, Enumerating Curves of Genus 2 Over Finite Fields, (2018). UVM Honors College Senior Theses. 259.

Formula

a(1) = 20, and for n > 1, a(n) = 2*prime(n)^3 + prime(n)^2 + 2*prime(n) - 2 + 2*[prime(n) == 1 (mod 3)] + 8*[prime(n) == 1 (mod 5)] + 2*[prime(n) == 5] + 2*[prime(n) == 1 or 3 (mod 8)].

Example

For n = 1, the a(1) = 20 genus 2 curves over F_2 can be given by their Weierstrass models as: y^2 + y = x^5, y^2 + (x^2 + 1)y = x^5, y^2 + (x^3 + x^2 + 1)y = x^5, y^2 + (x^3 + x + 1)y = x^5, y^2 + (x^3 + x^2 + x + 1)y = x^5, y^2 + y = x^5 + x^4, y^2 + (x+1)y = x^5 + x^4, y^2 + (x^2 + x + 1)y = x^5 + x^4, y^2 + (x^3 + x^2 + 1)y = x^5 + x^4, y^2 + y = x^5 + x^4 + x^3, y^2 + (x^2 + x + 1)y = x^5 + x^4 + x^3, y^2 + xy = x^5 + x^4 + x, y^2 + (x^2)y = x^5 + x^4 + x, y^2 + (x^2 + x)y = x^5 + x^4 + x, y^2 + y = x^5 + x^4 + 1, y^2 + (x^2 + x + 1)y = x^5 + x^4 + 1, y^2 + (x^3 + x^2 + 1)y = x^5 + x^4 + 1, y^2 + y = x^5 + x^3 + 1, y^2 + (x^3 + x^2 + 1)y = x^5 + x^3 + 1, and y^2 + (x^3 + x^2 + 1)y = x^6 + x^5 + 1.

Prog

(Sage)
def a(n):
    if n == 1: return 20
    p = Primes()[n-1]
    ans = 2*p^3 + p^2 + 2*p - 2
    if p%3 == 1: ans += 2
    if p%5 == 1: ans += 8
    if p == 5: ans += 2
    if p%8 in [1, 3]: ans += 2
    return ans

Crossrefs

Cf. A362243, A363843.

Sequence in context: A044539 A297597 A186735 * A238101 A153728 A234367

Adjacent sequences: A363837 A363838 A363839 * A363841 A363842 A363843

Keyword

nonn

Author

Robin Visser, Jun 23 2023

Status

approved