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 A363840 a(n) is the number of isomorphism classes of genus 2 hyperelliptic curves over the finite field of order prime(n). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

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