OFFSET
1,1
LINKS
E. Nart, Counting hyperelliptic curves, Adv. Math. 221 (2009), no. 3, 774-787.
E. Nart and D. Sadornil, Hyperelliptic curves of genus three over finite fields of even characteristic, Finite Fields Appl. 10 (2004), no. 2, 198-220.
FORMULA
a(1) = 76, and for n > 1, a(n) = 2*prime(n)^5 + 2*prime(n)^3 - 2 - 2*(prime(n)^2 - prime(n))*[prime(n) == 3 (mod 4)] + 2*(prime(n)-1)*[prime(n) > 3] + 4*[prime(n) == 1 (mod 8)] + 12*[prime(n) == 1 (mod 7)] + 2*[prime(n) == 7] + 2*[prime(n) == 1 or 5 (mod 12)].
EXAMPLE
For n = 1, E. Nart and D. Sadornil showed that there are 76 genus 3 hyperelliptic curves over F_2, so a(1) = 76.
PROG
(Sage)
def a(n):
if n == 1: return 76
p = Primes()[n-1]
ans = 2*p^5 + 2*p^3 - 2
if p%4 == 3: ans -= 2*(p^2 - p)
if p > 3: ans += 2*(p - 1)
if p%8 == 1: ans += 4
if p%7 == 1: ans += 12
if p == 7: ans += 2
if p%12 in [1, 5]: ans += 2
return ans
CROSSREFS
KEYWORD
nonn
AUTHOR
Robin Visser, Jun 23 2023
STATUS
approved