Computational enumeration of  superspecial hyperelliptic curves

Problem Setting
Let g be an integer > 1 and p a rational prime.
Given (g,p), enumerating superspecial (s.sp. for short) curves of genus g over a field K of characteristic p is a very important problem in arithmetic and algebraic geometry. 

The word "enumerate" means to list representatives (defining equations) for all isomorphism classes of those curves, where we consider isomorphisms over K, or its algebraic closure L.
Note that only finite such curves exist, up to isomorphism over L.
For the field of definition, the most important case is K=GF(p^2), since any s.sp. curve over K is L-isomorphic to one over GF(p^2).

In this page, we focus on the enumeration of s.sp. hyperelliptic curves of genus > 3.

Magma codes, log files and a table of results

(Genus 4): The following code enumerates s.sp. hyperelliptic curves of genus 4 defined by H_{a,b} : y^2 = f_{a,b} = x^10 + x^7 + a*x^4 + b*x, where a, b in GF(p^2).

The code works over Magma V.2.26-10 (or later versions).

• Code (NKT_enum4.txt)

• Log file (log_NKT_enum4.txt), where our computational environment is described in Section 4 of the paper arXiv:2210.14822.

• Computational results with timing data (NKT_table.pdf);
  At the fist time of executing NKT_enum4.txt, for p = 631 and 797, strangely it took a long time for Step 2 compared with other cases (due to memory issue?). So we re-executed the algorithm for these p (with the code NKT_enum4_p631and797.txt), and the time shown in Table 2 are those for this re-execution (see also log_NKT_enum4_p631and797.txt).

(Genus 5): The following code enumerates s.sp. hyperelliptic curves of genus 5 with automorphism group containing a subgroup isomorphic to C_2^3, where C_n denotes the cyclic group of order n. Isomorphisms classes are those over the algebraic closure of GF(p).

The code works over Magma V.2.28-4 (or later versions).

• Code (13-1000.m)

• Log file (13-1000.log)