The OEIS is supported by the many generous donors to the OEIS Foundation.

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A363843 a(n) is the number of isomorphism classes of genus 3 hyperelliptic curves over the finite field of order prime(n). 1
 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 (list; graph; refs; listen; history; text; internal format)
 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified March 4 05:23 EST 2024. Contains 370522 sequences. (Running on oeis4.)