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

1,1

Links

Table of n, a(n) for n=1..27.

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

Cf. A362243, A363840.

Sequence in context: A234786 A234779 A264475 * A262790 A184680 A129626

Adjacent sequences: A363840 A363841 A363842 * A363844 A363845 A363846

Keyword

nonn

Author

Robin Visser, Jun 23 2023

Status

approved