cp's OEIS Frontend

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

%I A363840 #10 Jun 25 2023 08:13:44
%S A363840 20,69,285,749,2813,4589,10149,14119,24907,49675,60613,102749,139613,
%T A363840 160951,209947,300667,414357,457813,606151,721011,783511,992477,
%U A363840 1150629,1418037,1834951,2071011,2196269,2461749,2602157,2898789,4113149,4513613,5161749,5390839,6638395,6909013,7764749
%N A363840 a(n) is the number of isomorphism classes of genus 2 hyperelliptic curves over the finite field of order prime(n).
%C A363840 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.
%H A363840 G. Cardona, <a href="https://doi.org/10.1016/S1071-5797(03)00039-X">On the number of curves of genus 2 over a finite field</a>, Finite Fields Appl. 9 (2003), no.4, 505-526.
%H A363840 G. Cardona, E. Nart, and J. Pujolàs, <a href="https://doi.org/10.1007/s00209-004-0750-0">Curves of genus two over fields of even characteristic</a>, Math. Z. 250 (2005), no.1, 177-201.
%H A363840 E. Nart, <a href="https://doi.org/10.1016/j.aim.2009.01.001">Counting hyperelliptic curves</a>, Adv. Math. 221 (2009), no.3, 774-787.
%H A363840 R. Steinberg, <a href="https://scholarworks.uvm.edu/hcoltheses/259">Enumerating Curves of Genus 2 Over Finite Fields</a>, (2018). UVM Honors College Senior Theses. 259.
%F A363840 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)].
%e A363840 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.
%o A363840 (Sage)
%o A363840 def a(n):
%o A363840     if n == 1: return 20
%o A363840     p = Primes()[n-1]
%o A363840     ans = 2*p^3 + p^2 + 2*p - 2
%o A363840     if p%3 == 1: ans += 2
%o A363840     if p%5 == 1: ans += 8
%o A363840     if p == 5: ans += 2
%o A363840     if p%8 in [1,3]: ans += 2
%o A363840     return ans
%Y A363840 Cf. A362243, A363843.
%K A363840 nonn
%O A363840 1,1
%A A363840 _Robin Visser_, Jun 23 2023