Split-Cubic Curves from D(n)-Triples: A Bounded Finite-Field Census

Abstract

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.