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 Ea,b,c(n):y2=(ax+n)(bx+n)(cx+n) attached to positive integral D(n)-triples. For n∈{−3,3,5,8,12,20} there are $164$ such triples with 1≤ a<b<c≤100, allowing zero squares. Their reductions at the ten primes NextPrime(10k), 3≤ k≤12, 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.
Related
BibTeX
@misc{chuysurichay2026splitcubic,
title = {{Split-Cubic Curves from D(n)-Triples: A Bounded Finite-Field Census}},
author = {Sompong Chuysurichay and Sawian Jaidee and Chatchawan Panraksa and Teerapol Sukhonwimolmal},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03121},
eprint = {2610.03121},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03121},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Split-Cubic Curves from D(n)-Triples: A Bounded Finite-Field Census
AU - Sompong Chuysurichay
AU - Sawian Jaidee
AU - Chatchawan Panraksa
AU - Teerapol Sukhonwimolmal
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03121
UR - https://arxiv.org/abs/2610.03121
AB - 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∈{−3,3,5,8,12,20} there are $164$ such triples with 1≤ a<b<c≤100, allowing zero squares. Their reductions at the ten primes NextPrime(10^k), 3≤ k≤12, 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.
ER -