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
Keywords
Examples
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.
Links
- 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.
Programs
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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
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)].
Comments