发表机构
Prince of Songkla University; Khon Kaen University; Mahidol University International College（宋卡王子大学; 孔敬大学; 玛希隆大学国际学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
对与D(n)-三元组相关的分裂三次椭圆曲线在有限域上进行确定性普查，记录了1640条约化记录并审计点计数，为后续研究提供数据基础。
AI 中文摘要
我们记录了与正整数D(n)-三元组相关的分裂三次椭圆曲线$E_{a,b,c}^{(n)}:y^2=(ax+n)(bx+n)(cx+n)$的有限域普查。对于$n\in\{-3,3,5,8,12,20\}$，存在164个这样的三元组，满足$1\le a<b<c\le100$，允许零平方。它们在十个素数$\operatorname{NextPrime}(10^k)$（$3\le k\le12$）处的约化产生了1640条记录，其中45条相对于群阶的最大素因子具有余因子4。我们在显式归一化$\lambda=b(c-a)/(c(b-a))$和$\delta=cn(b-a)$下收集了标准分裂三次和二次扭曲恒等式，并利用它们来审计点计数。该档案记录了群阶、因子分解、余因子、嵌入度和转移场位长。一个实例比较了主曲线与其非平方扭曲。该普查是确定性和描述性的；它既不提供渐近密度估计，也不提供密码学参数建议。
英文摘要
We record a finite-field census of the split-cubic elliptic curves $E_{a,b,c}^{(n)}:y^2=(ax+n)(bx+n)(cx+n)$ attached to positive integral $D(n)$-triples. For $n\in\{-3,3,5,8,12,20\}$ there are $164$ such triples with $1\le a<b<c\le100$, allowing zero squares. Their reductions at the ten primes $\operatorname{NextPrime}(10^k)$, $3\le k\le12$, give $1640$ records, of which $45$ have cofactor four relative to the largest prime divisor of the group order. We collect the standard split-cubic and quadratic-twist identities in the explicit normalization $λ=b(c-a)/(c(b-a))$ and $δ=cn(b-a)$, and use them to audit the point counts. The archive records group orders, factorizations, cofactors, embedding degrees and transfer-field bit lengths. A worked example compares the main curve with its nonsquare twist. The census is deterministic and descriptive; it provides neither an asymptotic density estimate nor cryptographic parameter recommendations.
Comments14 pages, 7 numbered tables; data and verification code included as ancillary files
