In algebraic geometry, enumerating or finding superspecial curves in positive characteristic $p$ is important both in theory and in computation. In this paper, we propose feasible algorithms to enumerate or find superspecial hyperelliptic curves of genus $4$ with automorphism group properly containing the Klein $4$-group. Executing the algorithms on Magma, we succeeded in enumerating such superspecial curves for every $p$ with $19 \leq p < 500$, and in finding a single one for every $p$ with $19 \leq p < 7000$.

翻译：在代数几何中，对正特征$p$下超特殊曲线进行枚举或寻找无论在理论还是计算上都具有重要意义。本文提出了可行的算法，用于枚举或寻找自同构群真包含Klein四元群的亏格$4$超特殊超椭圆曲线。通过Magma执行这些算法，我们成功枚举了所有满足$19 \leq p < 500$的此类超特殊曲线，并为每个满足$19 \leq p < 7000$的$p$找到了一条单曲线。