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

 

Logo
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
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
Sequence in context: A234786 A234779 A264475 * A262790 A184680 A129626
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.

License Agreements, Terms of Use, Privacy Policy. .

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