A363843 a(n) is the number of isomorphism classes of genus 3 hyperelliptic curves over the finite field of order prime(n).
76, 526, 6508, 34228, 324562, 747004, 2849576, 4965266, 12896050, 41071144, 57316082, 138789292, 231850328, 294172382, 458893426, 836688844, 1430252626, 1689646684, 2700843026, 3609164734, 4146921368, 6155086706, 7879211410, 11169529016, 17176506056, 21022261804, 23187646130
Offset: 1
Keywords
Examples
For n = 1, E. Nart and D. Sadornil showed that there are 76 genus 3 hyperelliptic curves over F_2, so a(1) = 76.
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.
Programs
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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
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)].